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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Simple linear regression</span></span>

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</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></th></tr><tr><th class="sidebar-heading">

Models</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Linear_regression" title="Linear regression">Linear regression</a></li>

<li><a href="Polynomial_regression" title="Polynomial regression">Polynomial regression</a></li>

<li><a href="General_linear_model" title="General linear model">General linear model</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></li>

<li><a href="Vector_generalized_linear_model" title="Vector generalized linear model">Vector generalized linear model</a></li>

<li><a href="Discrete_choice" title="Discrete choice">Discrete choice</a></li>

<li><a href="Binomial_regression" title="Binomial regression">Binomial regression</a></li>

<li><a href="Binary_regression" title="Binary regression">Binary regression</a></li>

<li><a href="Logistic_regression" title="Logistic regression">Logistic regression</a></li>

<li><a href="Multinomial_logistic_regression" title="Multinomial logistic regression">Multinomial logistic regression</a></li>

<li><a href="Mixed_logit" title="Mixed logit">Mixed logit</a></li>

<li><a href="Probit_model" title="Probit model">Probit</a></li>

<li><a href="Multinomial_probit" title="Multinomial probit">Multinomial probit</a></li>

<li><a href="Ordered_logit" title="Ordered logit">Ordered logit</a></li>

<li><a href="Ordered_probit" class="mw-redirect" title="Ordered probit">Ordered probit</a></li>

<li><a href="Poisson_regression" title="Poisson regression">Poisson</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Multilevel_model" title="Multilevel model">Multilevel model</a></li>

<li><a href="Fixed_effects_model" title="Fixed effects model">Fixed effects</a></li>

<li><a href="Random_effects_model" title="Random effects model">Random effects</a></li>

<li><a href="Mixed_model" title="Mixed model">Linear mixed-effects model</a></li>

<li><a href="Nonlinear_mixed-effects_model" title="Nonlinear mixed-effects model">Nonlinear mixed-effects model</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li>

<li><a href="Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li>

<li><a href="Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li>

<li><a href="Robust_regression" title="Robust regression">Robust</a></li>

<li><a href="Quantile_regression" title="Quantile regression">Quantile</a></li>

<li><a href="Isotonic_regression" title="Isotonic regression">Isotonic</a></li>

<li><a href="Principal_component_regression" title="Principal component regression">Principal components</a></li>

<li><a href="Least-angle_regression" title="Least-angle regression">Least angle</a></li>

<li><a href="Local_regression" title="Local regression">Local</a></li>

<li><a href="Segmented_regression" title="Segmented regression">Segmented</a></li></ul></td>

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<ul><li><a href="Errors-in-variables_models" class="mw-redirect" title="Errors-in-variables models">Errors-in-variables</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Estimation</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Least_squares" title="Least squares">Least squares</a></li>

<li><a href="Linear_least_squares" title="Linear least squares">Linear</a></li>

<li><a href="Non-linear_least_squares" title="Non-linear least squares">Non-linear</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Ordinary_least_squares" title="Ordinary least squares">Ordinary</a></li>

<li><a href="Weighted_least_squares" title="Weighted least squares">Weighted</a></li>

<li><a href="Generalized_least_squares" title="Generalized least squares">Generalized</a></li>

<li><a href="Generalized_estimating_equation" title="Generalized estimating equation">Generalized estimating equation</a></li></ul></td>

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<ul><li><a href="Partial_least_squares_regression" title="Partial least squares regression">Partial</a></li>

<li><a href="Total_least_squares" title="Total least squares">Total</a></li>

<li><a href="Non-negative_least_squares" title="Non-negative least squares">Non-negative</a></li>

<li><a href="Tikhonov_regularization" class="mw-redirect" title="Tikhonov regularization">Ridge regression</a></li>

<li><a href="Regularized_least_squares" title="Regularized least squares">Regularized</a></li></ul></td>

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<ul><li><a href="Least_absolute_deviations" title="Least absolute deviations">Least absolute deviations</a></li>

<li><a href="Iteratively_reweighted_least_squares" title="Iteratively reweighted least squares">Iteratively reweighted</a></li>

<li><a href="Bayesian_linear_regression" title="Bayesian linear regression">Bayesian</a></li>

<li><a href="Bayesian_multivariate_linear_regression" title="Bayesian multivariate linear regression">Bayesian multivariate</a></li>

<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Background</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Regression_validation" title="Regression validation">Regression validation</a></li>

<li><a href="Mean_and_predicted_response" class="mw-redirect" title="Mean and predicted response">Mean and predicted response</a></li>

<li><a href="Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li>

<li><a href="Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></li>

<li><a href="Studentized_residual" title="Studentized residual">Studentized residual</a></li>

<li><a href="Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov theorem</a></li></ul></td>

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<ul><li><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul></td></tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

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<p>In <a href="Statistics" title="Statistics">statistics</a>, <b>simple linear regression</b> (<b>SLR</b>) is a <a href="Linear_regression" title="Linear regression">linear regression</a> model with a single <a href="Covariate" class="mw-redirect" title="Covariate">explanatory variable</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> That is, it concerns two-dimensional sample points with <a href="Dependent_and_independent_variables" title="Dependent and independent variables">one independent variable and one dependent variable</a> (conventionally, the <i>x</i> and <i>y</i> coordinates in a <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>) and finds a linear function (a non-vertical <a href="Straight_line" class="mw-redirect" title="Straight line">straight line</a>) that, as accurately as possible, predicts the dependent variable values as a function of the independent variable.

The adjective <i>simple</i> refers to the fact that the outcome variable is related to a single predictor.

</p><p>It is common to make the additional stipulation that the <a href="Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a> (OLS) method should be used: the accuracy of each predicted value is measured by its squared <i><a href="Errors_and_residuals" title="Errors and residuals">residual</a></i> (vertical distance between the point of the data set and the fitted line), and the goal is to make the sum of these squared deviations as small as possible.

In this case, the slope of the fitted line is equal to the <a href="Pearson_correlation_coefficient" title="Pearson correlation coefficient">correlation</a> between <span class="texhtml mvar" style="font-style:italic;">y</span> and <span class="texhtml mvar" style="font-style:italic;">x</span> corrected by the ratio of standard deviations of these variables. The intercept of the fitted line is such that the line passes through the center of mass <span class="texhtml">(<span style="text-decoration:overline;"><i>x</i></span>, <span style="text-decoration:overline;"><i>y</i></span>)</span> of the data points.

</p>

<div class="mw-heading mw-heading2"><h2 id="Formulation_and_computation">Formulation and computation</h2></div>

<p>Consider the <a href="Mathematical_model" title="Mathematical model">model</a> function

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<annotation encoding="application/x-tex">{\displaystyle y=\alpha +\beta x,}</annotation>

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which describes a line with slope <span class="texhtml mvar" style="font-style:italic;">β</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>-intercept <span class="texhtml mvar" style="font-style:italic;">α</span>. In general, such a relationship may not hold exactly for the largely unobserved population of values of the independent and dependent variables; we call the unobserved deviations from the above equation the <a href="Errors_and_residuals" title="Errors and residuals">errors</a>. Suppose we observe <span class="texhtml mvar" style="font-style:italic;">n</span> data pairs and call them <span class="texhtml">{(<i>x</i><sub><i>i</i></sub>, <i>y</i><sub><i>i</i></sub>), <i>i</i> = 1, ..., <i>n</i></span>}. We can describe the underlying relationship between <span class="texhtml"><i>y</i><sub><i>i</i></sub></span> and <span class="texhtml"><i>x</i><sub><i>i</i></sub></span> involving this error term <span class="texhtml"><i>ε</i><sub><i>i</i></sub></span> by

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}=\alpha +\beta x_{i}+\varepsilon _{i}.}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}=\alpha +\beta x_{i}+\varepsilon _{i}.}</annotation>

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</p><p>This relationship between the true (but unobserved) underlying parameters <span class="texhtml mvar" style="font-style:italic;">α</span> and <span class="texhtml mvar" style="font-style:italic;">β</span> and the data points is called a linear regression model.

</p><p>The goal is to find estimated values <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\alpha }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\alpha }}}</annotation>

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</math></span><img src="./77ca8a0303970de1e7553ee5eccd19e8ab5d2007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.343ex;" alt="{\displaystyle {\widehat {\alpha }}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\beta }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}}</annotation>

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</math></span><img src="./21fd425a5a1a245a101aae3ff48df531b4dc96ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.535ex; height:3.343ex;" alt="{\displaystyle {\widehat {\beta }}}" loading="lazy"></span> for the parameters <span class="texhtml mvar" style="font-style:italic;">α</span> and <span class="texhtml mvar" style="font-style:italic;">β</span> which would provide the "best" fit in some sense for the data points. As mentioned in the introduction, in this article the "best" fit will be understood as in the <a href="Ordinary_least_squares" title="Ordinary least squares">least-squares</a> approach: a line that minimizes the <a href="Residual_sum_of_squares" title="Residual sum of squares">sum of squared residuals</a> (see also <a href="Errors_and_residuals" title="Errors and residuals">Errors and residuals</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\varepsilon }}_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\varepsilon }}_{i}}</annotation>

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</math></span><img src="./11d456a180942b7e506cbd135b2281be14d34c83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.156ex; height:2.676ex;" alt="{\displaystyle {\widehat {\varepsilon }}_{i}}" loading="lazy"></span> (differences between actual and predicted values of the dependent variable <i>y</i>), each of which is given by, for any candidate parameter values <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span>,

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\varepsilon }}_{i}=y_{i}-\alpha -\beta x_{i}.}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\varepsilon }}_{i}=y_{i}-\alpha -\beta x_{i}.}</annotation>

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</p><p>In other words, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\alpha }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\alpha }}}</annotation>

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</math></span><img src="./77ca8a0303970de1e7553ee5eccd19e8ab5d2007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.343ex;" alt="{\displaystyle {\widehat {\alpha }}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\beta }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}}</annotation>

</semantics>

</math></span><img src="./21fd425a5a1a245a101aae3ff48df531b4dc96ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.535ex; height:3.343ex;" alt="{\displaystyle {\widehat {\beta }}}" loading="lazy"></span> solve the following <a href="Minimization_problem" class="mw-redirect" title="Minimization problem">minimization problem</a>:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\hat {\alpha }},\,{\hat {\beta }})=\operatorname {argmin} \left(Q(\alpha ,\beta )\right),}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mi>argmin</mi>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\hat {\alpha }},\,{\hat {\beta }})=\operatorname {argmin} \left(Q(\alpha ,\beta )\right),}</annotation>

</semantics>

</math></span></span>

where the <a href="Objective_function" class="mw-redirect" title="Objective function">objective function</a> <span class="texhtml mvar" style="font-style:italic;">Q</span> is:

</mrow>

</mrow>

</munderover>

<mover>

</mover>

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</msubsup>

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</munderover>

<msub>

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</msub>

<msub>

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</msub>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle Q(\alpha ,\beta )=\sum _{i=1}^{n}{\widehat {\varepsilon }}_{i}^{\,2}=\sum _{i=1}^{n}(y_{i}-\alpha -\beta x_{i})^{2}\ .}</annotation>

</semantics>

</math></span></span>

</p><p>By expanding to get a quadratic expression in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta ,}</annotation>

</semantics>

</math></span><img src="./59ab677d974cccb0132cac08bd67fc8ac765627e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.979ex; height:2.509ex;" alt="{\displaystyle \beta ,}" loading="lazy"></span> we can derive minimizing values of the function arguments, denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\alpha }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\alpha }}}</annotation>

</semantics>

</math></span><img src="./77ca8a0303970de1e7553ee5eccd19e8ab5d2007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.343ex;" alt="{\displaystyle {\widehat {\alpha }}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\beta }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}}</annotation>

</semantics>

</math></span><img src="./21fd425a5a1a245a101aae3ff48df531b4dc96ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.535ex; height:3.343ex;" alt="{\displaystyle {\widehat {\beta }}}" loading="lazy"></span>:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {\alpha }}&={\bar {y}}-{\widehat {\beta }}\,{\bar {x}},\\[5pt]{\widehat {\beta }}&={\frac {\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)\left(y_{i}-{\bar {y}}\right)}{\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}}}={\frac {\sum _{i=1}^{n}\Delta x_{i}\Delta y_{i}}{\sum _{i=1}^{n}\Delta x_{i}^{2}}}\end{aligned}}}">

<mtr>

<mtd>

<mover>

</mover>

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</mtd>

<mtd>

<mover>

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</mrow>

<mover>

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<mover>

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</mtd>

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<mtr>

<mtd>

<mover>

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</mtd>

<mtd>

<mfrac>

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</mrow>

</munderover>

<mrow>

<mrow>

<msub>

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</msub>

<mover>

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<mrow>

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<msub>

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</msub>

<mover>

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<mrow>

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</munderover>

<msup>

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<msub>

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</msub>

<mover>

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</msup>

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<mfrac>

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</munderover>

<msub>

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<msub>

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</msubsup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {\alpha }}&={\bar {y}}-{\widehat {\beta }}\,{\bar {x}},\\[5pt]{\widehat {\beta }}&={\frac {\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)\left(y_{i}-{\bar {y}}\right)}{\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}}}={\frac {\sum _{i=1}^{n}\Delta x_{i}\Delta y_{i}}{\sum _{i=1}^{n}\Delta x_{i}^{2}}}\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p><p>Here we have introduced

</p>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation>

</semantics>

</math></span><img src="./466e03e1c9533b4dab1b9949dad393883f385d80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {y}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {y}}}</annotation>

</semantics>

</math></span><img src="./6b298744237368f34e61ff7dc90b34016a7037af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.343ex;" alt="{\displaystyle {\bar {y}}}" loading="lazy"></span> as the average of the <span class="texhtml"><i>x</i><sub><i>i</i></sub></span> and <span class="texhtml"><i>y</i><sub><i>i</i></sub></span>, respectively</li><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta x_{i}}</annotation>

</semantics>

</math></span><img src="./8d03804c96ca38a2bff889eaf13470785aca25fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.065ex; height:2.509ex;" alt="{\displaystyle \Delta x_{i}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta y_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta y_{i}}</annotation>

</semantics>

</math></span><img src="./9feb0f4642f47ac452a0022ab753666d99761a97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.875ex; height:2.509ex;" alt="{\displaystyle \Delta y_{i}}" loading="lazy"></span> as the <a href="Deviation_(statistics)" title="Deviation (statistics)">deviations</a> in <span class="texhtml"><i>x</i><sub><i>i</i></sub></span> and <span class="texhtml"><i>y</i><sub><i>i</i></sub></span> with respect to their respective means.</li></ul></div>

<div class="mw-heading mw-heading3"><h3 id="Expanded_formulas">Expanded formulas</h3></div>

<p>The above equations are efficient to use if the mean of the x and y variables (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}{\text{ and }}{\bar {y}}}">

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}{\text{ and }}{\bar {y}}}</annotation>

</semantics>

</math></span><img src="./f3b1c25f1de38ecf78859f836ab3ff931522ebfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.541ex; height:2.509ex;" alt="{\displaystyle {\bar {x}}{\text{ and }}{\bar {y}}}" loading="lazy"></span>) are known. If the means are not known at the time of calculation, it may be more efficient to use the expanded version of the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\alpha }}{\text{ and }}{\widehat {\beta }}}">

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\alpha }}{\text{ and }}{\widehat {\beta }}}</annotation>

</semantics>

</math></span><img src="./3c916dd8515d7eced78ffd16c5b181e60fce83ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:7.931ex; height:3.343ex;" alt="{\displaystyle {\widehat {\alpha }}{\text{ and }}{\widehat {\beta }}}" loading="lazy"></span> equations. These expanded equations may be derived from the more general <a href="Polynomial_regression" title="Polynomial regression">polynomial regression</a> equations<sup id="cite_ref-:1_7-0" class="reference"><a href="#cite_note-:1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> by defining the regression polynomial to be of order 1, as follows.

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}n&\sum _{i=1}^{n}x_{i}\\[1ex]\sum _{i=1}^{n}x_{i}&\sum _{i=1}^{n}x_{i}^{2}\end{bmatrix}}{\begin{bmatrix}{\widehat {\alpha }}\\[1ex]{\widehat {\beta }}\end{bmatrix}}={\begin{bmatrix}\sum _{i=1}^{n}y_{i}\\[1ex]\sum _{i=1}^{n}y_{i}x_{i}\end{bmatrix}}}">

<mrow>

<mtr>

<mtd>

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</mrow>

</munderover>

<msub>

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<msub>

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<mtd>

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</munderover>

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</msubsup>

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<mrow>

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<mtd>

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<mtr>

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</mtable>

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<mrow>

<mtr>

<mtd>

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</munderover>

<msub>

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</msub>

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<mtr>

<mtd>

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</munderover>

<msub>

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</msub>

<msub>

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</msub>

</mtd>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}n&\sum _{i=1}^{n}x_{i}\\[1ex]\sum _{i=1}^{n}x_{i}&\sum _{i=1}^{n}x_{i}^{2}\end{bmatrix}}{\begin{bmatrix}{\widehat {\alpha }}\\[1ex]{\widehat {\beta }}\end{bmatrix}}={\begin{bmatrix}\sum _{i=1}^{n}y_{i}\\[1ex]\sum _{i=1}^{n}y_{i}x_{i}\end{bmatrix}}}</annotation>

</semantics>

</math></span></span>

</p><p>The above <a href="System_of_linear_equations" title="System of linear equations">system of linear equations</a> may be solved directly, or stand-alone equations for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\alpha }}{\text{ and }}{\widehat {\beta }}}">

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\alpha }}{\text{ and }}{\widehat {\beta }}}</annotation>

</semantics>

</math></span><img src="./3c916dd8515d7eced78ffd16c5b181e60fce83ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:7.931ex; height:3.343ex;" alt="{\displaystyle {\widehat {\alpha }}{\text{ and }}{\widehat {\beta }}}" loading="lazy"></span> may be derived by expanding the matrix equations above. The resultant equations are algebraically equivalent to the ones shown in the prior paragraph, and are shown below without proof.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_7-1" class="reference"><a href="#cite_note-:1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {\alpha }}&={\frac {\sum \limits _{i=1}^{n}y_{i}\sum \limits _{i=1}^{n}x_{i}^{2}-\sum \limits _{i=1}^{n}x_{i}\sum \limits _{i=1}^{n}x_{i}y_{i}}{n\sum \limits _{i=1}^{n}x_{i}^{2}-\left(\sum \limits _{i=1}^{n}x_{i}\right)^{2}}}\\[2ex]{\widehat {\beta }}&={\frac {n\sum \limits _{i=1}^{n}x_{i}y_{i}-\sum \limits _{i=1}^{n}x_{i}\sum \limits _{i=1}^{n}y_{i}}{n\sum \limits _{i=1}^{n}x_{i}^{2}-\left(\sum \limits _{i=1}^{n}x_{i}\right)^{2}}}\end{aligned}}}">

<mtr>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {\alpha }}&={\frac {\sum \limits _{i=1}^{n}y_{i}\sum \limits _{i=1}^{n}x_{i}^{2}-\sum \limits _{i=1}^{n}x_{i}\sum \limits _{i=1}^{n}x_{i}y_{i}}{n\sum \limits _{i=1}^{n}x_{i}^{2}-\left(\sum \limits _{i=1}^{n}x_{i}\right)^{2}}}\\[2ex]{\widehat {\beta }}&={\frac {n\sum \limits _{i=1}^{n}x_{i}y_{i}-\sum \limits _{i=1}^{n}x_{i}\sum \limits _{i=1}^{n}y_{i}}{n\sum \limits _{i=1}^{n}x_{i}^{2}-\left(\sum \limits _{i=1}^{n}x_{i}\right)^{2}}}\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="Interpretation">Interpretation</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Relationship_with_the_sample_covariance_matrix">Relationship with the sample covariance matrix</h3></div>

<p>The solution can be reformulated using elements of the <a href="Covariance_matrix" title="Covariance matrix">covariance matrix</a>:

<mover>

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</mrow>

<mfrac>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}={\frac {s_{x,y}}{s_{x}^{2}}}=r_{xy}{\frac {s_{y}}{s_{x}}}}</annotation>

</semantics>

</math></span></span>

</p><p>where

</p>

<div><ul><li><span class="texhtml"><i>r</i><sub><i>xy</i></sub></span> is the <a href="Correlation#Sample_correlation_coefficient" title="Correlation">sample correlation coefficient</a> between <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span></li><li><span class="texhtml"><i>s</i><sub><i>x</i></sub></span> and <span class="texhtml"><i>s<sub>y</sub></i></span> are the <a href="Standard_deviation#Uncorrected_sample_standard_deviation" title="Standard deviation">uncorrected sample standard deviations</a> of <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span></li><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{x}^{2}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{x}^{2}}</annotation>

</semantics>

</math></span><img src="./6537d007e370193c985936650f17b5a16b3700b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.843ex;" alt="{\displaystyle s_{x}^{2}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{x,y}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{x,y}}</annotation>

</semantics>

</math></span><img src="./103123da0287e0e601687983f4185b04298f6775.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.537ex; height:2.343ex;" alt="{\displaystyle s_{x,y}}" loading="lazy"></span> are the <a href="Variance#Sample_variance" title="Variance">sample variance</a> and <a href="Sample_mean_and_covariance#Sample_covariance" title="Sample mean and covariance">sample covariance</a>, respectively</li></ul></div>

<p>Substituting the above expressions for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\alpha }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\alpha }}}</annotation>

</semantics>

</math></span><img src="./77ca8a0303970de1e7553ee5eccd19e8ab5d2007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.343ex;" alt="{\displaystyle {\widehat {\alpha }}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\beta }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}}</annotation>

</semantics>

</math></span><img src="./21fd425a5a1a245a101aae3ff48df531b4dc96ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.535ex; height:3.343ex;" alt="{\displaystyle {\widehat {\beta }}}" loading="lazy"></span> into the original solution yields

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {y-{\bar {y}}}{s_{y}}}=r_{xy}{\frac {x-{\bar {x}}}{s_{x}}}.}">

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {y-{\bar {y}}}{s_{y}}}=r_{xy}{\frac {x-{\bar {x}}}{s_{x}}}.}</annotation>

</semantics>

</math></span></span>

</p><p>This shows that <span class="texhtml"><i>r</i><sub><i>xy</i></sub></span> is the slope of the regression line of the <a href="Standard_score" title="Standard score">standardized</a> data points (and that this line passes through the origin). Since <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq r_{xy}\leq 1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1\leq r_{xy}\leq 1}</annotation>

</semantics>

</math></span><img src="./1db80f05e5a0aaa3b8690ea6d6dc658d742569ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.368ex; height:2.843ex;" alt="{\displaystyle -1\leq r_{xy}\leq 1}" loading="lazy"></span> then we get that if x is some measurement and y is a followup measurement from the same item, then we expect that y (on average) will be closer to the mean measurement than it was to the original value of x. This phenomenon is known as <a href="Regression_toward_the_mean#Definition_for_simple_linear_regression_of_data_points" title="Regression toward the mean">regressions toward the mean</a>.

</p><p>Generalizing the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation>

</semantics>

</math></span><img src="./466e03e1c9533b4dab1b9949dad393883f385d80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}}" loading="lazy"></span> notation, we can write a horizontal bar over an expression to indicate the average value of that expression over the set of samples. For example:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {xy}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}y_{i}.}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {xy}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}y_{i}.}</annotation>

</semantics>

</math></span></span>

</p><p>This notation allows us a concise formula for <span class="texhtml"><i>r</i><sub><i>xy</i></sub></span>:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}={\frac {{\overline {xy}}-{\bar {x}}{\bar {y}}}{\sqrt {\left({\overline {x^{2}}}-{\bar {x}}^{2}\right)\left({\overline {y^{2}}}-{\bar {y}}^{2}\right)}}}.}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msqrt>

<mrow>

<mrow>

<mover>

<msup>

</mrow>

</msup>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

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</mrow>

<mrow>

<mrow>

<mover>

<msup>

</mrow>

</msup>

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</mrow>

<msup>

<mover>

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</msup>

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</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{xy}={\frac {{\overline {xy}}-{\bar {x}}{\bar {y}}}{\sqrt {\left({\overline {x^{2}}}-{\bar {x}}^{2}\right)\left({\overline {y^{2}}}-{\bar {y}}^{2}\right)}}}.}</annotation>

</semantics>

</math></span></span>

</p><p>The <a href="Coefficient_of_determination" title="Coefficient of determination">coefficient of determination</a> ("R squared") is equal to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}^{2}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{xy}^{2}}</annotation>

</semantics>

</math></span><img src="./94187277cf924c115ef5bc61e27c8844f8a0e811.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.038ex; height:3.176ex;" alt="{\displaystyle r_{xy}^{2}}" loading="lazy"></span> when the model is linear with a single independent variable. See <a href="Correlation#Pearson's_product-moment_coefficient" title="Correlation">sample correlation coefficient</a> for additional details.

</p>

<div class="mw-heading mw-heading3"><h3 id="Interpretation_about_the_slope">Interpretation about the slope</h3></div>

<p>By multiplying all members of the summation in the numerator by : <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x_{i}-{\bar {x}}}{x_{i}-{\bar {x}}}}=1}">

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {x_{i}-{\bar {x}}}{x_{i}-{\bar {x}}}}=1}</annotation>

</semantics>

</math></span><img src="./e4a51cab42672d5b54ce51441c441e9c593affb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.396ex; height:5.343ex;" alt="{\displaystyle {\frac {x_{i}-{\bar {x}}}{x_{i}-{\bar {x}}}}=1}" loading="lazy"></span> (thereby not changing it):

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {\beta }}&={\frac {\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)\left(y_{i}-{\bar {y}}\right)}{\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}}}\\[1ex]&={\frac {\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}{\frac {y_{i}-{\bar {y}}}{x_{i}-{\bar {x}}}}}{\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}}}\\[1ex]&=\sum _{i=1}^{n}{\frac {\left(x_{i}-{\bar {x}}\right)^{2}}{\sum _{j=1}^{n}\left(x_{j}-{\bar {x}}\right)^{2}}}{\frac {y_{i}-{\bar {y}}}{x_{i}-{\bar {x}}}}\\[6pt]\end{aligned}}}">

<mtr>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

<mfrac>

<mrow>

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</mrow>

</munderover>

<mrow>

<mrow>

<msub>

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</msub>

<mover>

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</mrow>

</mrow>

<mrow>

<mrow>

<msub>

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</msub>

<mover>

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<mrow>

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</munderover>

<msup>

<mrow>

<mrow>

<msub>

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</msub>

<mover>

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</mrow>

</msup>

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</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

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</munderover>

<msup>

<mrow>

<mrow>

<msub>

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</msub>

<mover>

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</mrow>

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</mrow>

</msup>

<mfrac>

<mrow>

<msub>

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</msub>

<mover>

</mover>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

<mrow>

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</mrow>

</munderover>

<msup>

<mrow>

<mrow>

<msub>

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</msub>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

<mrow>

<mrow>

<msub>

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</msub>

<mover>

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</munderover>

<msup>

<mrow>

<mrow>

<msub>

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</msub>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {\beta }}&={\frac {\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)\left(y_{i}-{\bar {y}}\right)}{\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}}}\\[1ex]&={\frac {\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}{\frac {y_{i}-{\bar {y}}}{x_{i}-{\bar {x}}}}}{\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}}}\\[1ex]&=\sum _{i=1}^{n}{\frac {\left(x_{i}-{\bar {x}}\right)^{2}}{\sum _{j=1}^{n}\left(x_{j}-{\bar {x}}\right)^{2}}}{\frac {y_{i}-{\bar {y}}}{x_{i}-{\bar {x}}}}\\[6pt]\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p><p>We can see that the slope (tangent of angle) of the regression line is the weighted average of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {y_{i}-{\bar {y}}}{x_{i}-{\bar {x}}}}}">

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {y_{i}-{\bar {y}}}{x_{i}-{\bar {x}}}}}</annotation>

</semantics>

</math></span><img src="./9843ffbd3d91e7e8c2c19d72afd6a2fce46547b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.136ex; height:5.509ex;" alt="{\displaystyle {\frac {y_{i}-{\bar {y}}}{x_{i}-{\bar {x}}}}}" loading="lazy"></span> that is the slope (tangent of angle) of the line that connects the i-th point to the average of all points, weighted by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i}-{\bar {x}})^{2}}">

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{i}-{\bar {x}})^{2}}</annotation>

</semantics>

</math></span><img src="./f0774da93861931721c359addb1462c2fbe99117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.163ex; height:3.176ex;" alt="{\displaystyle (x_{i}-{\bar {x}})^{2}}" loading="lazy"></span> because the further the point is the more "important" it is, since small errors in its position will affect the slope connecting it to the center point more.

</p>

<div class="mw-heading mw-heading3"><h3 id="Interpretation_about_the_intercept">Interpretation about the intercept</h3></div>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

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</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {\alpha }}&={\bar {y}}-{\widehat {\beta }}\,{\bar {x}},\\[5pt]\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p><p>Given <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\beta }}=\tan(\theta )=dy/dx\rightarrow dy={\widehat {\beta }}\,dx}">

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}=\tan(\theta )=dy/dx\rightarrow dy={\widehat {\beta }}\,dx}</annotation>

</semantics>

</math></span><img src="./9fbd215fedb7abfd49b1e613d878a7f5bed894c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.585ex; height:3.509ex;" alt="{\displaystyle {\widehat {\beta }}=\tan(\theta )=dy/dx\rightarrow dy={\widehat {\beta }}\,dx}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math></span><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span> the angle the line makes with the positive x axis,

we have <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{\rm {intersection}}={\bar {y}}-{\widehat {\beta }}\,dx={\bar {y}}-dy}">

<msub>

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</msub>

<mover>

</mover>

</mrow>

<mover>

</mover>

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<mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{\rm {intersection}}={\bar {y}}-{\widehat {\beta }}\,dx={\bar {y}}-dy}</annotation>

</semantics>

</math></span><img src="./97f8732544c3ec02bdd048e0af37787176c41a6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.001ex; height:3.343ex;" alt="{\displaystyle y_{\rm {intersection}}={\bar {y}}-{\widehat {\beta }}\,dx={\bar {y}}-dy}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Interpretation_about_the_correlation">Interpretation about the correlation</h3></div>

<p>In the above formulation, notice that each <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math></span><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"></span> is a constant ("known upfront") value, while the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation>

</semantics>

</math></span><img src="./67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"></span> are random variables that depend on the linear function of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math></span><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"></span> and the random term <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i}}</annotation>

</semantics>

</math></span><img src="./00e1b6ad3cbad4af49bf21a3ad2dc379ff045079.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{i}}" loading="lazy"></span>. This assumption is used when deriving the standard error of the slope and showing that it is <a href="Proofs_involving_ordinary_least_squares" title="Proofs involving ordinary least squares">unbiased</a>.

</p><p>In this framing, when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math></span><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"></span> is not actually a <a href="Random_variable" title="Random variable">random variable</a>, what type of parameter does the empirical correlation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{xy}}</annotation>

</semantics>

</math></span><img src="./eca3bfa846f27e4f6c993bd259805d3ce59f4882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.038ex; height:2.343ex;" alt="{\displaystyle r_{xy}}" loading="lazy"></span> estimate? The issue is that for each value i we'll have: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(x_{i})=x_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(x_{i})=x_{i}}</annotation>

</semantics>

</math></span><img src="./11920e8a46de1d8c76171f1e307b27cd1f0284c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.942ex; height:2.843ex;" alt="{\displaystyle E(x_{i})=x_{i}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Var(x_{i})=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Var(x_{i})=0}</annotation>

</semantics>

</math></span><img src="./6136706f8273c483b20e2ba1bfa1ecf572ef59a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.265ex; height:2.843ex;" alt="{\displaystyle Var(x_{i})=0}" loading="lazy"></span>. A possible interpretation of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{xy}}</annotation>

</semantics>

</math></span><img src="./eca3bfa846f27e4f6c993bd259805d3ce59f4882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.038ex; height:2.343ex;" alt="{\displaystyle r_{xy}}" loading="lazy"></span> is to imagine that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math></span><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"></span> defines a random variable drawn from the <a href="Empirical_distribution_function" title="Empirical distribution function">empirical distribution</a> of the x values in our sample. For example, if x had 10 values from the <a href="Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>: [1,2,3...,10], then we can imagine x to be a <a href="Discrete_uniform_distribution" title="Discrete uniform distribution">Discrete uniform distribution</a>. Under this interpretation all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math></span><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"></span> have the same expectation and some positive variance. With this interpretation we can think of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{xy}}</annotation>

</semantics>

</math></span><img src="./eca3bfa846f27e4f6c993bd259805d3ce59f4882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.038ex; height:2.343ex;" alt="{\displaystyle r_{xy}}" loading="lazy"></span> as the estimator of the <a href="Pearson_correlation_coefficient#Definition" title="Pearson correlation coefficient">Pearson's correlation</a> between the random variable y and the random variable x (as we just defined it).

</p>

<div class="mw-heading mw-heading2"><h2 id="Numerical_properties">Numerical properties</h2></div>

<div><ol><li>The regression line goes through the <i>center of mass</i> point, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\bar {x}},\,{\bar {y}})}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\bar {x}},\,{\bar {y}})}</annotation>

</semantics>

</math></span><img src="./98e8d534e7501e84cb44f4442fdd58abe2616651.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.862ex; height:2.843ex;" alt="{\displaystyle ({\bar {x}},\,{\bar {y}})}" loading="lazy"></span>, if the model includes an intercept term (i.e., not forced through the origin).</li><li>The sum of the residuals is zero if the model includes an intercept term:

</mrow>

</mrow>

</munderover>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}{\widehat {\varepsilon }}_{i}=0.}</annotation>

</semantics>

</math></span></span></li><li>The residuals and <span class="texhtml mvar" style="font-style:italic;">x</span> values are uncorrelated (whether or not there is an intercept term in the model), meaning:

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}x_{i}{\widehat {\varepsilon }}_{i}\;=\;0}</annotation>

</semantics>

</math></span></span></li><li>The relationship between <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{xy}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{xy}}</annotation>

</semantics>

</math></span><img src="./e18e4804b339fd6e032bf29ef36b8fcf7a4bf6a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.191ex; height:2.343ex;" alt="{\displaystyle \rho _{xy}}" loading="lazy"></span> (the <a href="Pearson_correlation_coefficient#For_a_population" title="Pearson correlation coefficient">correlation coefficient for the population</a>) and the population variances of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}^{2}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{y}^{2}}</annotation>

</semantics>

</math></span><img src="./690d699292b74ee1b030d5720a17595baaea5227.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.385ex; height:3.176ex;" alt="{\displaystyle \sigma _{y}^{2}}" loading="lazy"></span>) and the error term of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math></span><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"></span> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\varepsilon }^{2}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{\varepsilon }^{2}}</annotation>

</semantics>

</math></span><img src="./eec379b86e73255492d3266c76f6e17acfdfabd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.385ex; height:2.843ex;" alt="{\displaystyle \sigma _{\varepsilon }^{2}}" loading="lazy"></span>) is:<sup id="cite_ref-Valliant2013_10-0" class="reference"><a href="#cite_note-Valliant2013-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 401">: 401 </span></sup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{\varepsilon }^{2}=(1-\rho _{xy}^{2})\sigma _{y}^{2}}</annotation>

</semantics>

</math></span></span>

For extreme values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{xy}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{xy}}</annotation>

</semantics>

</math></span><img src="./e18e4804b339fd6e032bf29ef36b8fcf7a4bf6a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.191ex; height:2.343ex;" alt="{\displaystyle \rho _{xy}}" loading="lazy"></span> this is self evident. Since when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{xy}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{xy}=0}</annotation>

</semantics>

</math></span><img src="./98edc22541e142f86dd7e3959b7f046d2768ad8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.452ex; height:2.843ex;" alt="{\displaystyle \rho _{xy}=0}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\varepsilon }^{2}=\sigma _{y}^{2}}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{\varepsilon }^{2}=\sigma _{y}^{2}}</annotation>

</semantics>

</math></span><img src="./4c5e087c9dec5e739811e2a5eeb1ec214aa72ea9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.868ex; height:3.176ex;" alt="{\displaystyle \sigma _{\varepsilon }^{2}=\sigma _{y}^{2}}" loading="lazy"></span>. And when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{xy}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{xy}=1}</annotation>

</semantics>

</math></span><img src="./b347491b81e5431fa458c86b6ac55bb1f4bec543.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.452ex; height:2.843ex;" alt="{\displaystyle \rho _{xy}=1}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\varepsilon }^{2}=0}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{\varepsilon }^{2}=0}</annotation>

</semantics>

</math></span><img src="./fa8f48c8ed11665062bb207d8ce39962dc84a212.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.646ex; height:2.843ex;" alt="{\displaystyle \sigma _{\varepsilon }^{2}=0}" loading="lazy"></span>.</li></ol></div>

<div class="mw-heading mw-heading2"><h2 id="Statistical_properties">Statistical properties</h2></div>

<p>Description of the statistical properties of estimators from the simple linear regression estimates requires the use of a <a href="Statistical_model" title="Statistical model">statistical model</a>. The following is based on assuming the validity of a model under which the estimates are optimal. It is also possible to evaluate the properties under other assumptions, such as <a href="Homoscedasticity" class="mw-redirect" title="Homoscedasticity">inhomogeneity</a>, but this is discussed elsewhere.

</p>

<div class="mw-heading mw-heading3"><h3 id="Unbiasedness">Unbiasedness</h3></div>

<p>The estimators <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\alpha }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\alpha }}}</annotation>

</semantics>

</math></span><img src="./77ca8a0303970de1e7553ee5eccd19e8ab5d2007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.343ex;" alt="{\displaystyle {\widehat {\alpha }}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\beta }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}}</annotation>

</semantics>

</math></span><img src="./21fd425a5a1a245a101aae3ff48df531b4dc96ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.535ex; height:3.343ex;" alt="{\displaystyle {\widehat {\beta }}}" loading="lazy"></span> are <a href="Estimator_bias" class="mw-redirect" title="Estimator bias">unbiased</a>.

</p><p>To formalize this assertion we must define a framework in which these estimators are random variables. We consider the residuals <span class="texhtml"><i>ε</i><sub>i</sub></span> as random variables drawn independently from some distribution with mean zero. In other words, for each value of <span class="texhtml mvar" style="font-style:italic;">x</span>, the corresponding value of <span class="texhtml mvar" style="font-style:italic;">y</span> is generated as a mean response <span class="texhtml"><i>α</i> + <i>βx</i></span> plus an additional random variable <span class="texhtml mvar" style="font-style:italic;">ε</span> called the <i>error term</i>, equal to zero on average. Under such interpretation, the least-squares estimators <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\alpha }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\alpha }}}</annotation>

</semantics>

</math></span><img src="./77ca8a0303970de1e7553ee5eccd19e8ab5d2007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.343ex;" alt="{\displaystyle {\widehat {\alpha }}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\beta }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}}</annotation>

</semantics>

</math></span><img src="./21fd425a5a1a245a101aae3ff48df531b4dc96ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.535ex; height:3.343ex;" alt="{\displaystyle {\widehat {\beta }}}" loading="lazy"></span> will themselves be random variables whose means will equal the "true values" <span class="texhtml mvar" style="font-style:italic;">α</span> and <span class="texhtml mvar" style="font-style:italic;">β</span>. This is the definition of an unbiased estimator.

</p>

<div class="mw-heading mw-heading3"><h3 id="Variance_of_the_mean_response">Variance of the mean response</h3></div>

<p>Since the data in this context is defined to be (<i>x</i>, <i>y</i>) pairs for every observation, the <i>mean response</i> at a given value of <i>x</i>, say <i>x<sub>d</sub></i>, is an estimate of the mean of the <i>y</i> values in the population at the <i>x</i> value of <i>x<sub>d</sub></i>, that is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {E}}(y\mid x_{d})\equiv {\hat {y}}_{d}\!}">

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {E}}(y\mid x_{d})\equiv {\hat {y}}_{d}\!}</annotation>

</semantics>

</math></span><img src="./28799f248bacdccf4a736ea30322bfd3ceba2e23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:14.592ex; height:3.343ex;" alt="{\displaystyle {\hat {E}}(y\mid x_{d})\equiv {\hat {y}}_{d}\!}" loading="lazy"></span>. The variance of the mean response is given by:<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} \left({\hat {\alpha }}+{\hat {\beta }}x_{d}\right)=\operatorname {Var} \left({\hat {\alpha }}\right)+\left(\operatorname {Var} {\hat {\beta }}\right)x_{d}^{2}+2x_{d}\operatorname {Cov} \left({\hat {\alpha }},{\hat {\beta }}\right).}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} \left({\hat {\alpha }}+{\hat {\beta }}x_{d}\right)=\operatorname {Var} \left({\hat {\alpha }}\right)+\left(\operatorname {Var} {\hat {\beta }}\right)x_{d}^{2}+2x_{d}\operatorname {Cov} \left({\hat {\alpha }},{\hat {\beta }}\right).}</annotation>

</semantics>

</math></span></span>

</p><p>This expression can be simplified to

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} \left({\hat {\alpha }}+{\hat {\beta }}x_{d}\right)=\sigma ^{2}\left({\frac {1}{m}}+{\frac {\left(x_{d}-{\bar {x}}\right)^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}\right),}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} \left({\hat {\alpha }}+{\hat {\beta }}x_{d}\right)=\sigma ^{2}\left({\frac {1}{m}}+{\frac {\left(x_{d}-{\bar {x}}\right)^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}\right),}</annotation>

</semantics>

</math></span></span>

</p><p>where <i>m</i> is the number of data points.

</p><p>To demonstrate this simplification, one can make use of the identity

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}(x_{i}-{\bar {x}})^{2}=\sum _{i}x_{i}^{2}-{\frac {1}{m}}\left(\sum _{i}x_{i}\right)^{2}.}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i}(x_{i}-{\bar {x}})^{2}=\sum _{i}x_{i}^{2}-{\frac {1}{m}}\left(\sum _{i}x_{i}\right)^{2}.}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Variance_of_the_predicted_response">Variance of the predicted response</h3></div>

/* start https://en.wikipedia.org/ */

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</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Prediction_interval" title="Prediction interval">Prediction interval</a></div>

<p>The <i>predicted response</i> distribution is the predicted distribution of the residuals at the given point <i>x<sub>d</sub></i>. So the variance is given by

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {Var} \left(y_{d}-\left[{\hat {\alpha }}+{\hat {\beta }}x_{d}\right]\right)&=\operatorname {Var} (y_{d})+\operatorname {Var} \left({\hat {\alpha }}+{\hat {\beta }}x_{d}\right)-2\operatorname {Cov} \left(y_{d},\left[{\hat {\alpha }}+{\hat {\beta }}x_{d}\right]\right)\\&=\operatorname {Var} (y_{d})+\operatorname {Var} \left({\hat {\alpha }}+{\hat {\beta }}x_{d}\right).\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Var} \left(y_{d}-\left[{\hat {\alpha }}+{\hat {\beta }}x_{d}\right]\right)&=\operatorname {Var} (y_{d})+\operatorname {Var} \left({\hat {\alpha }}+{\hat {\beta }}x_{d}\right)-2\operatorname {Cov} \left(y_{d},\left[{\hat {\alpha }}+{\hat {\beta }}x_{d}\right]\right)\\&=\operatorname {Var} (y_{d})+\operatorname {Var} \left({\hat {\alpha }}+{\hat {\beta }}x_{d}\right).\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p><p>The second line follows from the fact that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cov} \left(y_{d},\left[{\hat {\alpha }}+{\hat {\beta }}x_{d}\right]\right)}">

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} \left(y_{d},\left[{\hat {\alpha }}+{\hat {\beta }}x_{d}\right]\right)}</annotation>

</semantics>

</math></span><img src="./fecd895228c749dd21da71a3cd5c661be5293f8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.505ex; height:4.843ex;" alt="{\displaystyle \operatorname {Cov} \left(y_{d},\left[{\hat {\alpha }}+{\hat {\beta }}x_{d}\right]\right)}" loading="lazy"></span> is zero because the new prediction point is independent of the data used to fit the model. Additionally, the term <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} \left({\hat {\alpha }}+{\hat {\beta }}x_{d}\right)}">

<mrow>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} \left({\hat {\alpha }}+{\hat {\beta }}x_{d}\right)}</annotation>

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</math></span><img src="./35608f7529f3483a8338d6323c21e3c34d3c0637.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.794ex; height:4.843ex;" alt="{\displaystyle \operatorname {Var} \left({\hat {\alpha }}+{\hat {\beta }}x_{d}\right)}" loading="lazy"></span> was calculated earlier for the mean response.

</p><p>Since <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} (y_{d})=\sigma ^{2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (y_{d})=\sigma ^{2}}</annotation>

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</math></span><img src="./ec1b19f0cacec3d3a92ef8260e444b6d776e7232.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.341ex; height:3.176ex;" alt="{\displaystyle \operatorname {Var} (y_{d})=\sigma ^{2}}" loading="lazy"></span> (a fixed but unknown parameter that can be estimated), the variance of the predicted response is given by

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {Var} \left(y_{d}-\left[{\hat {\alpha }}+{\hat {\beta }}x_{d}\right]\right)&=\sigma ^{2}+\sigma ^{2}\left({\frac {1}{m}}+{\frac {\left(x_{d}-{\bar {x}}\right)^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}\right)\\[4pt]&=\sigma ^{2}\left(1+{\frac {1}{m}}+{\frac {(x_{d}-{\bar {x}})^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}\right).\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Var} \left(y_{d}-\left[{\hat {\alpha }}+{\hat {\beta }}x_{d}\right]\right)&=\sigma ^{2}+\sigma ^{2}\left({\frac {1}{m}}+{\frac {\left(x_{d}-{\bar {x}}\right)^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}\right)\\[4pt]&=\sigma ^{2}\left(1+{\frac {1}{m}}+{\frac {(x_{d}-{\bar {x}})^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}\right).\end{aligned}}}</annotation>

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</p>

<div class="mw-heading mw-heading3"><h3 id="Confidence_intervals">Confidence intervals</h3></div>

<p>The formulas given in the previous section allow one to calculate the <i>point estimates</i> of <span class="texhtml mvar" style="font-style:italic;">α</span> and <span class="texhtml mvar" style="font-style:italic;">β</span> — that is, the coefficients of the regression line for the given set of data. However, those formulas do not tell us how precise the estimates are, i.e., how much the estimators <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\alpha }}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\alpha }}}</annotation>

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</math></span><img src="./77ca8a0303970de1e7553ee5eccd19e8ab5d2007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.343ex;" alt="{\displaystyle {\widehat {\alpha }}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\beta }}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}}</annotation>

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</math></span><img src="./21fd425a5a1a245a101aae3ff48df531b4dc96ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.535ex; height:3.343ex;" alt="{\displaystyle {\widehat {\beta }}}" loading="lazy"></span> vary from sample to sample for the specified sample size. <a href="Confidence_interval" title="Confidence interval">Confidence intervals</a> were devised to give a plausible set of values to the estimates one might have if one repeated the experiment a very large number of times.

</p><p>The standard method of constructing confidence intervals for linear regression coefficients relies on the normality assumption, which is justified if either:

</p>

<ol><li>the errors in the regression are <a href="Normal_distribution" title="Normal distribution">normally distributed</a> (the so-called <i>classic regression</i> assumption), or</li>

<li>the number of observations <span class="texhtml mvar" style="font-style:italic;">n</span> is sufficiently large, in which case the estimator is approximately normally distributed.</li></ol>

<p>The latter case is justified by the <a href="Central_limit_theorem" title="Central limit theorem">central limit theorem</a>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Normality_assumption">Normality assumption</h4></div>

<p>Under the first assumption above, that of the normality of the error terms, the estimator of the slope coefficient will itself be normally distributed with mean <span class="texhtml mvar" style="font-style:italic;">β</span> and variance <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma ^{2}\left/\sum _{i}(x_{i}-{\bar {x}})^{2}\right.,}">

<msup>

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<msub>

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<annotation encoding="application/x-tex">{\textstyle \sigma ^{2}\left/\sum _{i}(x_{i}-{\bar {x}})^{2}\right.,}</annotation>

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</math></span><img src="./7dadd587e0b4928301ce4c981291665b201e047c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -1.005ex; width:17.566ex; height:3.176ex;" aria-hidden="true" alt="{\textstyle \sigma ^{2}\left/\sum _{i}(x_{i}-{\bar {x}})^{2}\right.,}" loading="lazy"></span> where <span class="texhtml"><i>σ</i><sup>2</sup></span> is the variance of the error terms (see <a href="Proofs_involving_ordinary_least_squares" title="Proofs involving ordinary least squares">Proofs involving ordinary least squares</a>). At the same time the sum of squared residuals <span class="texhtml mvar" style="font-style:italic;">Q</span> is distributed proportionally to <span class="texhtml"><a href="Chi-squared_distribution" title="Chi-squared distribution"><i>χ</i><sup>2</sup></a></span> with <span class="texhtml"><i>n</i> − 2</span> degrees of freedom, and independently from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\beta }}}">

<mover>

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</math></span><img src="./21fd425a5a1a245a101aae3ff48df531b4dc96ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.535ex; height:3.343ex;" alt="{\displaystyle {\widehat {\beta }}}" loading="lazy"></span>. This allows us to construct a <span class="texhtml mvar" style="font-style:italic;">t</span>-value

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={\frac {{\widehat {\beta }}-\beta }{s_{\widehat {\beta }}}}\ \sim \ t_{n-2},}">

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<annotation encoding="application/x-tex">{\displaystyle t={\frac {{\widehat {\beta }}-\beta }{s_{\widehat {\beta }}}}\ \sim \ t_{n-2},}</annotation>

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</p><p>where

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{\widehat {\beta }}={\sqrt {\frac {{\frac {1}{n-2}}\sum _{i=1}^{n}{\widehat {\varepsilon }}_{i}^{\,2}}{\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}}}">

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<annotation encoding="application/x-tex">{\displaystyle s_{\widehat {\beta }}={\sqrt {\frac {{\frac {1}{n-2}}\sum _{i=1}^{n}{\widehat {\varepsilon }}_{i}^{\,2}}{\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}}}</annotation>

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</p><p>is the unbiased <i>standard error</i> estimator of the estimator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\beta }}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}}</annotation>

</semantics>

</math></span><img src="./21fd425a5a1a245a101aae3ff48df531b4dc96ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.535ex; height:3.343ex;" alt="{\displaystyle {\widehat {\beta }}}" loading="lazy"></span>.

</p><p>This <span class="texhtml mvar" style="font-style:italic;">t</span>-value has a <a href="Student's_t-distribution" title="Student's t-distribution">Student's <span class="texhtml mvar" style="font-style:italic;">t</span></a>-distribution with <span class="texhtml"><i>n</i> − 2</span> degrees of freedom. Using it we can construct a confidence interval for <span class="texhtml mvar" style="font-style:italic;">β</span>:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \in \left[{\widehat {\beta }}-s_{\widehat {\beta }}t_{n-2}^{*},\ {\widehat {\beta }}+s_{\widehat {\beta }}t_{n-2}^{*}\right],}">

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta \in \left[{\widehat {\beta }}-s_{\widehat {\beta }}t_{n-2}^{*},\ {\widehat {\beta }}+s_{\widehat {\beta }}t_{n-2}^{*}\right],}</annotation>

</semantics>

</math></span></span>

</p><p>at confidence level <span class="texhtml">(1 − <i>γ</i>)</span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n-2}^{*}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{n-2}^{*}}</annotation>

</semantics>

</math></span><img src="./08406c7b78aa0ab33bc2a76cb932c246c1ae54ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.159ex; height:3.009ex;" alt="{\displaystyle t_{n-2}^{*}}" loading="lazy"></span> is the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \left(1\;-\;{\frac {\gamma }{2}}\right){\text{-th}}}">

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left(1\;-\;{\frac {\gamma }{2}}\right){\text{-th}}}</annotation>

</semantics>

</math></span><img src="./f9015da49756ed5b15dec6846974f0c0a1ba76d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.016ex; height:3.509ex;" alt="{\displaystyle \scriptstyle \left(1\;-\;{\frac {\gamma }{2}}\right){\text{-th}}}" loading="lazy"></span> quantile of the <span class="texhtml"><i>t</i><sub><i>n</i>−2</sub></span> distribution. For example, if <span class="texhtml"><i>γ</i> = 0.05</span> then the confidence level is 95%.

</p><p>Similarly, the confidence interval for the intercept coefficient <span class="texhtml mvar" style="font-style:italic;">α</span> is given by

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \left[{\widehat {\alpha }}-s_{\widehat {\alpha }}t_{n-2}^{*},\ {\widehat {\alpha }}+s_{\widehat {\alpha }}t_{n-2}^{*}\right],}">

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in \left[{\widehat {\alpha }}-s_{\widehat {\alpha }}t_{n-2}^{*},\ {\widehat {\alpha }}+s_{\widehat {\alpha }}t_{n-2}^{*}\right],}</annotation>

</semantics>

</math></span></span>

</p><p>at confidence level (1 − <i>γ</i>), where

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{\widehat {\alpha }}=s_{\widehat {\beta }}{\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{2}}}={\sqrt {{\frac {1}{n(n-2)}}\left(\sum _{i=1}^{n}{\widehat {\varepsilon }}_{i}^{\,2}\right){\frac {\sum _{i=1}^{n}x_{i}^{2}}{\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}}}}">

<msub>

<mover>

</mover>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</msub>

<msqrt>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mrow>

<msqrt>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{\widehat {\alpha }}=s_{\widehat {\beta }}{\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{2}}}={\sqrt {{\frac {1}{n(n-2)}}\left(\sum _{i=1}^{n}{\widehat {\varepsilon }}_{i}^{\,2}\right){\frac {\sum _{i=1}^{n}x_{i}^{2}}{\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}}}}</annotation>

</semantics>

</math></span></span>

</p>

<p>The confidence intervals for <span class="texhtml mvar" style="font-style:italic;">α</span> and <span class="texhtml mvar" style="font-style:italic;">β</span> give us the general idea where these regression coefficients are most likely to be. For example, in the <a href="Okun's_law" title="Okun's law">Okun's law</a> regression shown here the point estimates are

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\alpha }}=0.859,\qquad {\widehat {\beta }}=-1.817.}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\alpha }}=0.859,\qquad {\widehat {\beta }}=-1.817.}</annotation>

</semantics>

</math></span></span>

</p><p>The 95% confidence intervals for these estimates are

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \left[\,0.76,0.96\right],\qquad \beta \in \left[-2.06,-1.58\,\right].}">

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in \left[\,0.76,0.96\right],\qquad \beta \in \left[-2.06,-1.58\,\right].}</annotation>

</semantics>

</math></span></span>

</p><p>In order to represent this information graphically, in the form of the confidence bands around the regression line, one has to proceed carefully and account for the joint distribution of the estimators. It can be shown<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> that at confidence level (1 − <i>γ</i>) the confidence band has hyperbolic form given by the equation

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\alpha +\beta \xi )\in \left[\,{\widehat {\alpha }}+{\widehat {\beta }}\xi \pm t_{n-2}^{*}{\sqrt {\left({\frac {1}{n-2}}\sum {\widehat {\varepsilon }}_{i}^{\,2}\right)\cdot \left({\frac {1}{n}}+{\frac {(\xi -{\bar {x}})^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}\right)}}\,\right].}">

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

<msqrt>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</msqrt>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\alpha +\beta \xi )\in \left[\,{\widehat {\alpha }}+{\widehat {\beta }}\xi \pm t_{n-2}^{*}{\sqrt {\left({\frac {1}{n-2}}\sum {\widehat {\varepsilon }}_{i}^{\,2}\right)\cdot \left({\frac {1}{n}}+{\frac {(\xi -{\bar {x}})^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}\right)}}\,\right].}</annotation>

</semantics>

</math></span></span>

</p><p>When the model assumed the intercept is fixed and equal to 0 (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =0}</annotation>

</semantics>

</math></span><img src="./30cc00f65bbc630448311dd2dc82e7ce5e90985a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =0}" loading="lazy"></span>), the standard error of the slope turns into:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{\widehat {\beta }}={\sqrt {{\frac {1}{n-1}}{\frac {\sum _{i=1}^{n}{\widehat {\varepsilon }}_{i}^{\,2}}{\sum _{i=1}^{n}x_{i}^{2}}}}}}">

<msub>

<mover>

</mover>

</mrow>

</msub>

<msqrt>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</munderover>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{\widehat {\beta }}={\sqrt {{\frac {1}{n-1}}{\frac {\sum _{i=1}^{n}{\widehat {\varepsilon }}_{i}^{\,2}}{\sum _{i=1}^{n}x_{i}^{2}}}}}}</annotation>

</semantics>

</math></span></span>

</p><p>With: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\varepsilon }}_{i}=y_{i}-{\hat {y}}_{i}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\varepsilon }}_{i}=y_{i}-{\hat {y}}_{i}}</annotation>

</semantics>

</math></span><img src="./502cba1055d51c500bef4d8b2b363c9f6198e98f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.135ex; height:2.843ex;" alt="{\displaystyle {\hat {\varepsilon }}_{i}=y_{i}-{\hat {y}}_{i}}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading4"><h4 id="Asymptotic_assumption">Asymptotic assumption</h4></div>

<p>The alternative second assumption states that when the number of points in the dataset is "large enough", the <a href="Law_of_large_numbers" title="Law of large numbers">law of large numbers</a> and the <a href="Central_limit_theorem" title="Central limit theorem">central limit theorem</a> become applicable, and then the distribution of the estimators is approximately normal. Under this assumption all formulas derived in the previous section remain valid, with the only exception that the quantile <i>t*</i><sub style="position:relative;left:-0.4em"><i>n</i>−2</sub> of <a href="Student's_t-distribution" title="Student's t-distribution">Student's <i>t</i></a> distribution is replaced with the quantile <i>q*</i> of the <a href="Standard_normal_distribution" class="mw-redirect" title="Standard normal distribution">standard normal distribution</a>. Occasionally the fraction <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>n</i>−2</span></span>⁠</span></span> is replaced with <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>n</i></span></span>⁠</span></span>. When <span class="texhtml mvar" style="font-style:italic;">n</span> is large such a change does not alter the results appreciably.

</p>

<div class="mw-heading mw-heading2"><h2 id="Numerical_example">Numerical example</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Ordinary_least_squares#Example" title="Ordinary least squares">Ordinary least squares § Example</a>, and <a href="Linear_least_squares#Example" title="Linear least squares">Linear least squares § Example</a></div>

<p>This data set gives average masses for women as a function of their height in a sample of American women of age 30–39. Although the <a href="Ordinary_least_squares" title="Ordinary least squares">OLS</a> article argues that it would be more appropriate to run a quadratic regression for this data, the simple linear regression model is applied here instead.

</p>

<th style="text-align:left;">Height (m), <i>x<sub>i</sub></i>

</th>

<td>1.83

</td></tr>

<tr>

</th>

<td>74.46

</td></tr></tbody></table>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span></th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math></span><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"></span></th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation>

</semantics>

</math></span><img src="./67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"></span></th>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}^{2}}</annotation>

</semantics>

</math></span><img src="./0ea4ea44d82d72c37a4f22ba288ffef326f925d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.384ex; height:3.176ex;" alt="{\displaystyle x_{i}^{2}}" loading="lazy"></span></th>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}y_{i}}</annotation>

</semantics>

</math></span><img src="./c5ce6e068edada97dc9c321cb6d3c7b45cffbd48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.068ex; height:2.009ex;" alt="{\displaystyle x_{i}y_{i}}" loading="lazy"></span></th>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{i}^{2}}</annotation>

</semantics>

</math></span><img src="./10dbc846f3e3db5f84e6495efbf8626ceac33191.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.215ex; height:3.176ex;" alt="{\displaystyle y_{i}^{2}}" loading="lazy"></span>

</th></tr>

<tr>

<td>2725.8841

</td></tr>

<tr>

<td>2821.7344

</td></tr>

<tr>

<td>2968.0704

</td></tr>

<tr>

<td>3118.1056

</td></tr>

<tr>

<td>3271.8400

</td></tr>

<tr>

<td>3430.4449

</td></tr>

<tr>

<td>3591.6049

</td></tr>

<tr>

<td>3756.4641

</td></tr>

<tr>

<td>3982.8721

</td></tr>

<tr>

<td>4156.3809

</td></tr>

<tr>

<td>4393.0384

</td></tr>

<tr>

<td>4637.6100

</td></tr>

<tr>

<td>4888.8064

</td></tr>

<tr>

<td>5211.3961

</td></tr>

<tr>

<td>5544.2916

</td></tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math></span><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"></span></td>

<td>58498.5439

</td></tr></tbody></table>

<p>There are <i>n</i> = 15 points in this data set. Hand calculations would be started by finding the following five sums:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S_{x}&=\sum _{i}x_{i}\,=24.76,&\qquad S_{y}&=\sum _{i}y_{i}\,=931.17,\\[5pt]S_{xx}&=\sum _{i}x_{i}^{2}=41.0532,&\;\;\,S_{yy}&=\sum _{i}y_{i}^{2}=58498.5439,\\[5pt]S_{xy}&=\sum _{i}x_{i}y_{i}=1548.2453&\end{aligned}}}">

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

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</msub>

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<mtd>

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</msubsup>

</mtd>

<mtd>

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</munder>

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<mtr>

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<msub>

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</munder>

<msub>

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<msub>

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</msub>

</mtd>

</mtr>

</mtable>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{x}&=\sum _{i}x_{i}\,=24.76,&\qquad S_{y}&=\sum _{i}y_{i}\,=931.17,\\[5pt]S_{xx}&=\sum _{i}x_{i}^{2}=41.0532,&\;\;\,S_{yy}&=\sum _{i}y_{i}^{2}=58498.5439,\\[5pt]S_{xy}&=\sum _{i}x_{i}y_{i}=1548.2453&\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p><p>These quantities would be used to calculate the estimates of the regression coefficients, and their standard errors.

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {\beta }}&={\frac {nS_{xy}-S_{x}S_{y}}{nS_{xx}-S_{x}^{2}}}=61.272\\[8pt]{\widehat {\alpha }}&={\frac {1}{n}}S_{y}-{\widehat {\beta }}{\frac {1}{n}}S_{x}=-39.062\\[8pt]s_{\varepsilon }^{2}&={\frac {1}{n(n-2)}}\left[nS_{yy}-S_{y}^{2}-{\widehat {\beta }}^{2}(nS_{xx}-S_{x}^{2})\right]=0.5762\\[8pt]s_{\widehat {\beta }}^{2}&={\frac {ns_{\varepsilon }^{2}}{nS_{xx}-S_{x}^{2}}}=3.1539\\[8pt]s_{\widehat {\alpha }}^{2}&=s_{\widehat {\beta }}^{2}{\frac {1}{n}}S_{xx}=8.63185\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {\beta }}&={\frac {nS_{xy}-S_{x}S_{y}}{nS_{xx}-S_{x}^{2}}}=61.272\\[8pt]{\widehat {\alpha }}&={\frac {1}{n}}S_{y}-{\widehat {\beta }}{\frac {1}{n}}S_{x}=-39.062\\[8pt]s_{\varepsilon }^{2}&={\frac {1}{n(n-2)}}\left[nS_{yy}-S_{y}^{2}-{\widehat {\beta }}^{2}(nS_{xx}-S_{x}^{2})\right]=0.5762\\[8pt]s_{\widehat {\beta }}^{2}&={\frac {ns_{\varepsilon }^{2}}{nS_{xx}-S_{x}^{2}}}=3.1539\\[8pt]s_{\widehat {\alpha }}^{2}&=s_{\widehat {\beta }}^{2}{\frac {1}{n}}S_{xx}=8.63185\end{aligned}}}</annotation>

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</p>

<p>The 0.975 quantile of Student's <i>t</i>-distribution with 13 degrees of freedom is <span class="texhtml"><i>t</i><sup>*</sup><sub style="position:relative; left:-0.3em;">13</sub> = 2.1604</span>, and thus the 95% confidence intervals for <span class="texhtml mvar" style="font-style:italic;">α</span> and <span class="texhtml mvar" style="font-style:italic;">β</span> are

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\alpha \in [\,{\widehat {\alpha }}\mp t_{13}^{*}s_{\widehat {\alpha }}\,]=[\,{-45.4},\ {-32.7}\,]\\[5pt]&\beta \in [\,{\widehat {\beta }}\mp t_{13}^{*}s_{\widehat {\beta }}\,]=[\,57.4,\ 65.1\,]\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\alpha \in [\,{\widehat {\alpha }}\mp t_{13}^{*}s_{\widehat {\alpha }}\,]=[\,{-45.4},\ {-32.7}\,]\\[5pt]&\beta \in [\,{\widehat {\beta }}\mp t_{13}^{*}s_{\widehat {\beta }}\,]=[\,57.4,\ 65.1\,]\end{aligned}}}</annotation>

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</p><p>The <a href="Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">product-moment correlation coefficient</a> might also be calculated:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {r}}={\frac {nS_{xy}-S_{x}S_{y}}{\sqrt {(nS_{xx}-S_{x}^{2})(nS_{yy}-S_{y}^{2})}}}=0.9946}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {r}}={\frac {nS_{xy}-S_{x}S_{y}}{\sqrt {(nS_{xx}-S_{x}^{2})(nS_{yy}-S_{y}^{2})}}}=0.9946}</annotation>

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</p>

<div class="mw-heading mw-heading2"><h2 id="Alternatives">Alternatives</h2></div>

<p>In SLR, there is an underlying assumption that only the dependent variable contains measurement error; if the explanatory variable is also measured with error, then simple regression is not appropriate for estimating the underlying relationship because it will be biased due to <a href="Regression_dilution" title="Regression dilution">regression dilution</a>.

</p><p>Other estimation methods that can be used in place of ordinary least squares include <a href="Least_absolute_deviations" title="Least absolute deviations">least absolute deviations</a> (minimizing the sum of absolute values of residuals) and the <a href="Theil%E2%80%93Sen_estimator" title="Theil–Sen estimator">Theil–Sen estimator</a> (which chooses a line whose <a href="Slope" title="Slope">slope</a> is the <a href="Median" title="Median">median</a> of the slopes determined by pairs of sample points).

</p><p><a href="Deming_regression" title="Deming regression">Deming regression</a> (total least squares) also finds a line that fits a set of two-dimensional sample points, but (unlike ordinary least squares, least absolute deviations, and median slope regression) it is not really an instance of simple linear regression, because it does not separate the coordinates into one dependent and one independent variable and could potentially return a vertical line as its fit. can lead to a model that attempts to fit the outliers more than the data.

</p>

<div class="mw-heading mw-heading3"><h3 id="Line_fitting">Line fitting</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="Line_fitting" title="Line fitting">Line fitting</a>.<span class="mw-editsection-like "><span class="mw-editsection-bracket">[</span><a class="external text external" href="https://en.wikipedia.org/w/index.php?title=Line_fitting&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt">

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<p><a href="Line_fitting" title="Line fitting">Line fitting</a> is the process of constructing a <a href="Straight_line" class="mw-redirect" title="Straight line">straight line</a> that has the best fit to a series of data points.

</p><p>Several methods exist, considering:

</p>

<ul><li>Vertical distance: Simple linear regression</li>

<li>Resistance to <a href="Outlier_(statistics)" class="mw-redirect" title="Outlier (statistics)">outliers</a>: <a href="Robust_simple_linear_regression" class="mw-redirect" title="Robust simple linear regression">Robust simple linear regression</a></li>

<li><a href="Perpendicular_distance" title="Perpendicular distance">Perpendicular distance</a>: <a href="Orthogonal_regression" class="mw-redirect" title="Orthogonal regression">Orthogonal regression</a> (this is not scale-invariant i.e. changing the measurement units leads to a different line.)</li>

<li>Weighted geometric distance: <a href="Deming_regression" title="Deming regression">Deming regression</a></li>

<li>Scale invariant approach: <a href="Major_axis_regression" class="mw-redirect" title="Major axis regression">Major axis regression</a> This allows for measurement error in both variables, and gives an equivalent equation if the measurement units are altered.</li></ul></div></div>

<div class="mw-heading mw-heading3"><h3 id="Simple_linear_regression_without_the_intercept_term_(single_regressor)">Simple linear regression without the intercept term (single regressor)</h3></div>

<p>Sometimes it is appropriate to force the regression line to pass through the origin, because <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> are assumed to be proportional. For the model without the intercept term, <span class="texhtml"><i>y</i> = <i>βx</i></span>, the OLS estimator for <span class="texhtml mvar" style="font-style:italic;">β</span> simplifies to

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\beta }}={\frac {\sum _{i=1}^{n}x_{i}y_{i}}{\sum _{i=1}^{n}x_{i}^{2}}}={\frac {\overline {xy}}{\overline {x^{2}}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}={\frac {\sum _{i=1}^{n}x_{i}y_{i}}{\sum _{i=1}^{n}x_{i}^{2}}}={\frac {\overline {xy}}{\overline {x^{2}}}}}</annotation>

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</p><p>Substituting <span class="texhtml">(<i>x</i> − <i>h</i>, <i>y</i> − <i>k</i>)</span> in place of <span class="texhtml">(<i>x</i>, <i>y</i>)</span> gives the regression through <span class="texhtml">(<i>h</i>, <i>k</i>)</span>:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {\beta }}&={\frac {\sum _{i=1}^{n}(x_{i}-h)(y_{i}-k)}{\sum _{i=1}^{n}(x_{i}-h)^{2}}}={\frac {\overline {(x-h)(y-k)}}{\overline {(x-h)^{2}}}}\\[6pt]&={\frac {{\overline {xy}}-k{\bar {x}}-h{\bar {y}}+hk}{{\overline {x^{2}}}-2h{\bar {x}}+h^{2}}}\\[6pt]&={\frac {{\overline {xy}}-{\bar {x}}{\bar {y}}+({\bar {x}}-h)({\bar {y}}-k)}{{\overline {x^{2}}}-{\bar {x}}^{2}+({\bar {x}}-h)^{2}}}\\[6pt]&={\frac {\operatorname {Cov} (x,y)+({\bar {x}}-h)({\bar {y}}-k)}{\operatorname {Var} (x)+({\bar {x}}-h)^{2}}},\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {\beta }}&={\frac {\sum _{i=1}^{n}(x_{i}-h)(y_{i}-k)}{\sum _{i=1}^{n}(x_{i}-h)^{2}}}={\frac {\overline {(x-h)(y-k)}}{\overline {(x-h)^{2}}}}\\[6pt]&={\frac {{\overline {xy}}-k{\bar {x}}-h{\bar {y}}+hk}{{\overline {x^{2}}}-2h{\bar {x}}+h^{2}}}\\[6pt]&={\frac {{\overline {xy}}-{\bar {x}}{\bar {y}}+({\bar {x}}-h)({\bar {y}}-k)}{{\overline {x^{2}}}-{\bar {x}}^{2}+({\bar {x}}-h)^{2}}}\\[6pt]&={\frac {\operatorname {Cov} (x,y)+({\bar {x}}-h)({\bar {y}}-k)}{\operatorname {Var} (x)+({\bar {x}}-h)^{2}}},\end{aligned}}}</annotation>

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</p><p>where Cov and Var refer to the covariance and variance of the sample data (uncorrected for bias).

The last form above demonstrates how moving the line away from the center of mass of the data points affects the slope.

</p>

<ul><li><a href="Design_matrix#Simple_linear_regression" title="Design matrix">Design matrix § Simple linear regression</a></li>

<li><a href="Linear_trend_estimation" title="Linear trend estimation">Linear trend estimation</a></li>

<li><a href="Segmented_regression" title="Segmented regression">Linear segmented regression</a></li>

<li><a href="Proofs_involving_ordinary_least_squares" title="Proofs involving ordinary least squares">Proofs involving ordinary least squares</a>—derivation of all formulas used in this article in general multidimensional case</li>

<li><a href="Newey%E2%80%93West_estimator" title="Newey–West estimator">Newey–West estimator</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFSeltman2008" class="citation book cs1">Seltman, Howard J. (2008-09-08). <a rel="nofollow" class="external text" href="http://www.stat.cmu.edu/~hseltman/309/Book/Book.pdf"><i>Experimental Design and Analysis</i></a> <span class="cs1-format">(PDF)</span>. p. 227.</cite></span>

</li>

<li id="cite_note-:0-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_2-0">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://ci.columbia.edu/ci/premba_test/c0331/s7/s7_6.html">"Statistical Sampling and Regression: Simple Linear Regression"</a>. Columbia University<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-10-17</span></span>. <q>When one independent variable is used in a regression, it is called a simple regression;(...)</q></cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFLane" class="citation book cs1">Lane, David M. <a rel="nofollow" class="external text" href="http://onlinestatbook.com/Online_Statistics_Education.pdf"><i>Introduction to Statistics</i></a> <span class="cs1-format">(PDF)</span>. p. 462.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFZou_KHTuncali_KSilverman_SG2003" class="citation journal cs1">Zou KH; Tuncali K; Silverman SG (2003). <a rel="nofollow" class="external text" href="https://repositorio.unal.edu.co/handle/unal/81200">"Correlation and simple linear regression"</a>. <i>Radiology</i>. <b>227</b> (3): <span class="nowrap">617–</span>22. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1148%2Fradiol.2273011499">10.1148/radiol.2273011499</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0033-8419">0033-8419</a>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/110941167">110941167</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/12773666">12773666</a>.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFAltmanKrzywinski2015" class="citation journal cs1">Altman, Naomi; Krzywinski, Martin (2015). <a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnmeth.3627">"Simple linear regression"</a>. <i>Nature Methods</i>. <b>12</b> (11): <span class="nowrap">999–</span>1000. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnmeth.3627">10.1038/nmeth.3627</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1548-7091">1548-7091</a>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/5912005539">5912005539</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26824102">26824102</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:261269711">261269711</a>.</cite></span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Kenney, J. F. and Keeping, E. S. (1962) "Linear Regression and Correlation." Ch. 15 in <i>Mathematics of Statistics</i>, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 252–285</span>

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<li id="cite_note-:1-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFMuthukrishnan2018" class="citation web cs1">Muthukrishnan, Gowri (17 Jun 2018). <a rel="nofollow" class="external text" href="https://muthu.co/maths-behind-polynomial-regression/">"Maths behind Polynomial regression, Muthukrishnan"</a>. <i>Maths behind Polynomial regression</i><span class="reference-accessdate">. Retrieved <span class="nowrap">30 Jan</span> 2024</span>.</cite></span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://polynomialregression.drque.net/math.html">"Mathematics of Polynomial Regression"</a>. <i>Polynomial Regression, A PHP regression class</i>.</cite></span>

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<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.ncl.ac.uk/webtemplate/ask-assets/external/maths-resources/statistics/regression-and-correlation/simple-linear-regression.html">"Numeracy, Maths and Statistics - Academic Skills Kit, Newcastle University"</a>. <i>Simple Linear Regression</i><span class="reference-accessdate">. Retrieved <span class="nowrap">30 Jan</span> 2024</span>.</cite></span>

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<li id="cite_note-Valliant2013-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Valliant2013_10-0">^</a></b></span> <span class="reference-text">Valliant, Richard, Jill A. Dever, and Frauke Kreuter. Practical tools for designing and weighting survey samples. New York: Springer, 2013.</span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFDraperSmith1998" class="citation book cs1">Draper, N. R.; Smith, H. (1998). <i>Applied Regression Analysis</i> (3rd ed.). John Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-17082-8</bdi>.</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Casella, G. and Berger, R. L. (2002), "Statistical Inference" (2nd Edition), Cengage, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-534-24312-8</bdi>, pp. 558–559.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/LeastSquaresFitting.html">Wolfram MathWorld's explanation of Least Squares Fitting, and how to calculate it</a></li>

<li><a rel="nofollow" class="external text" href="http://people.duke.edu/~rnau/mathreg.htm">Mathematics of simple regression (Robert Nau, Duke University)</a></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Statistics654" style="padding:3px"><table class="nowraplinks hlist mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistics654" style="font-size:114%;margin:0 4em"><a href="Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="Outline_of_statistics" title="Outline of statistics">Outline</a></li>

<li><a href="List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics654" style="font-size:114%;margin:0 4em"><a href="Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li>

<li><a href="Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li>

<li><a href="Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li>

<li><a href="Cubic_mean" title="Cubic mean">Cubic</a></li>

<li><a href="Generalized_mean" title="Generalized mean">Generalized/power</a></li>

<li><a href="Geometric_mean" title="Geometric mean">Geometric</a></li>

<li><a href="Harmonic_mean" title="Harmonic mean">Harmonic</a></li>

<li><a href="Heronian_mean" title="Heronian mean">Heronian</a></li>

<li><a href="Heinz_mean" title="Heinz mean">Heinz</a></li>

<li><a href="Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li>

<li><a href="Median" title="Median">Median</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li>

<li><a href="Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li>

<li><a href="Interquartile_range" title="Interquartile range">Interquartile range</a></li>

<li><a href="Percentile" title="Percentile">Percentile</a></li>

<li><a href="Range_(statistics)" title="Range (statistics)">Range</a></li>

<li><a href="Standard_deviation" title="Standard deviation">Standard deviation</a></li>

<li><a href="Variance#Sample_variance" title="Variance">Variance</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Moment_(mathematics)" title="Moment (mathematics)">Moments</a>

<ul><li><a href="Kurtosis" title="Kurtosis">Kurtosis</a></li>

<li><a href="L-moment" title="L-moment">L-moments</a></li>

<li><a href="Skewness" title="Skewness">Skewness</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li>

<li><a href="Grouped_data" title="Grouped data">Grouped data</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li>

<li><a href="Rank_correlation" title="Rank correlation">Rank correlation</a>

<ul><li><a href="Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li>

<li><a href="Spearman's_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bar_chart" title="Bar chart">Bar chart</a></li>

<li><a href="Biplot" title="Biplot">Biplot</a></li>

<li><a href="Control_chart" title="Control chart">Control chart</a></li>

<li><a href="Correlogram" title="Correlogram">Correlogram</a></li>

<li><a href="Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li>

<li><a href="Forest_plot" title="Forest plot">Forest plot</a></li>

<li><a href="Histogram" title="Histogram">Histogram</a></li>

<li><a href="Pie_chart" title="Pie chart">Pie chart</a></li>

<li><a href="Radar_chart" title="Radar chart">Radar chart</a></li>

<li><a href="Run_chart" title="Run chart">Run chart</a></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li>

<li><a href="Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li>

<li><a href="Violin_plot" title="Violin plot">Violin plot</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection654" style="font-size:114%;margin:0 4em"><a href="Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Effect_size" title="Effect size">Effect size</a></li>

<li><a href="Missing_data" title="Missing data">Missing data</a></li>

<li><a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li>

<li><a href="Statistical_population" title="Statistical population">Population</a></li>

<li><a href="Replication_(statistics)" title="Replication (statistics)">Replication</a></li>

<li><a href="Sample_size_determination" title="Sample size determination">Sample size determination</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sampling_(statistics)" title="Sampling (statistics)">Sampling</a>

<ul><li><a href="Cluster_sampling" title="Cluster sampling">Cluster</a></li>

<li><a href="Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li>

<li><a href="Opinion_poll" title="Opinion poll">Opinion poll</a></li>

<li><a href="Questionnaire" title="Questionnaire">Questionnaire</a></li>

<li><a href="Standard_error" title="Standard error">Standard error</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li>

<li><a href="Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li>

<li><a href="Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li>

<li><a href="Random_assignment" title="Random assignment">Random assignment</a></li>

<li><a href="Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li>

<li><a href="Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li>

<li><a href="Scientific_control" title="Scientific control">Scientific control</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li>

<li><a href="Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li>

<li><a href="Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cohort_study" title="Cohort study">Cohort study</a></li>

<li><a href="Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li>

<li><a href="Natural_experiment" title="Natural experiment">Natural experiment</a></li>

<li><a href="Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference654" style="font-size:114%;margin:0 4em"><a href="Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a></li>

<li><a href="Sampling_distribution" title="Sampling distribution">Sampling distribution</a>

<ul><li><a href="Order_statistic" title="Order statistic">Order statistic</a></li></ul></li>

<li><a href="Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a>

<ul><li><a href="Density_estimation" title="Density estimation">Density estimation</a></li></ul></li>

<li><a href="Statistical_model" title="Statistical model">Statistical model</a>

<ul><li><a href="Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li>

<li><a href="Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li>

<li><a href="Statistical_parameter" title="Statistical parameter">Parameter</a>

<ul><li><a href="Location_parameter" title="Location parameter">location</a></li>

<li><a href="Scale_parameter" title="Scale parameter">scale</a></li>

<li><a href="Shape_parameter" title="Shape parameter">shape</a></li></ul></li>

<li><a href="Parametric_statistics" title="Parametric statistics">Parametric family</a>

<ul><li><a href="Likelihood_function" title="Likelihood function">Likelihood</a> <a href="Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size: 85%;">(monotone)</span></a></li>

<li><a href="Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li>

<li><a href="Exponential_family" title="Exponential family">Exponential family</a></li></ul></li>

<li><a href="Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li>

<li><a href="Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li>

<li><a href="Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="Optimal_decision" title="Optimal decision">Optimal decision</a>

<ul><li><a href="Loss_function" title="Loss function">loss function</a></li></ul></li>

<li><a href="Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li>

<li><a href="Statistical_distance" title="Statistical distance">Statistical distance</a>

<ul><li><a href="Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li>

<li><a href="Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li>

<li><a href="Robust_statistics" title="Robust statistics">Robustness</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Estimating_equations" title="Estimating equations">Estimating equations</a>

<ul><li><a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li>

<li><a href="Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li>

<li><a href="M-estimator" title="M-estimator">M-estimator</a></li>

<li><a href="Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li>

<li><a href="Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a>

<ul><li><a href="Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a>

<ul><li><a href="Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li>

<li><a href="Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li>

<li><a href="Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Confidence_interval" title="Confidence interval">Confidence interval</a></li>

<li><a href="Pivotal_quantity" title="Pivotal quantity">Pivot</a></li>

<li><a href="Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li>

<li><a href="Prediction_interval" title="Prediction interval">Prediction interval</a></li>

<li><a href="Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li>

<li><a href="Resampling_(statistics)" title="Resampling (statistics)">Resampling</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li>

<li><a href="Power_(statistics)" title="Power (statistics)">Power</a>

<ul><li><a href="Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li>

<li><a href="Permutation_test" title="Permutation test">Permutation test</a>

<ul><li><a href="Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li>

<li><a href="Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li>

<li><a href="Score_test" title="Score test">Score/Lagrange multiplier</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Z-test" title="Z-test"><i>Z</i>-test <span style="font-size: 85%;">(normal)</span></a></li>

<li><a href="Student's_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chi-squared_test" title="Chi-squared test">Chi-squared</a></li>

<li><a href="Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li>

<li><a href="Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li>

<li><a href="Lilliefors_test" title="Lilliefors test">Lilliefors</a></li>

<li><a href="Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li>

<li><a href="Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size: 85%;">(Shapiro–Wilk)</span></a></li>

<li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li>

<li><a href="Model_selection" title="Model selection">Model selection</a>

<ul><li><a href="Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li>

<li><a href="Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size: 85%;">(Wilcoxon)</span></a>

<ul><li><a href="Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li>

<li><a href="Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size: 85%;">(Mann–Whitney)</span></a></li>

<li><a href="Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="Analysis_of_variance" title="Analysis of variance">anova</a>

<ul><li><a href="Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size: 85%;">(Kruskal–Wallis)</span></a></li>

<li><a href="Friedman_test" title="Friedman test">2-way <span style="font-size: 85%;">(Friedman)</span></a></li>

<li><a href="Jonckheere's_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size: 85%;">(Jonckheere–Terpstra)</span></a></li></ul></li>

<li><a href="Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bayesian_probability" title="Bayesian probability">Bayesian probability</a>

<ul><li><a href="Prior_probability" title="Prior probability">prior</a></li>

<li><a href="Posterior_probability" title="Posterior probability">posterior</a></li></ul></li>

<li><a href="Credible_interval" title="Credible interval">Credible interval</a></li>

<li><a href="Bayes_factor" title="Bayes factor">Bayes factor</a></li>

<li><a href="Bayes_estimator" title="Bayes estimator">Bayesian estimator</a>

<ul><li><a href="Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis654" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li>

<li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="Confounding" title="Confounding">Confounding variable</a></li>

<li><a href="Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li>

<li><a href="Regression_validation" title="Regression validation">Regression validation</a></li>

<li><a href="Mixed_model" title="Mixed model">Mixed effects models</a></li>

<li><a href="Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li>

<li><a href="Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul>

<li><a href="Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li>

<li><a href="General_linear_model" title="General linear model">General linear model</a></li>

<li><a href="Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li>

<li><a href="Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li>

<li><a href="Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li>

<li><a href="Isotonic_regression" title="Isotonic regression">Isotonic</a></li>

<li><a href="Robust_regression" title="Robust regression">Robust</a></li>

<li><a href="Homoscedasticity_and_heteroscedasticity" title="Homoscedasticity and heteroscedasticity">Homoscedasticity and Heteroscedasticity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Exponential_family" title="Exponential family">Exponential families</a></li>

<li><a href="Logistic_regression" title="Logistic regression">Logistic <span style="font-size: 85%;">(Bernoulli)</span></a> / <a href="Binomial_regression" title="Binomial regression">Binomial</a> / <a href="Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li>

<li><a href="Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li>

<li><a href="Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_multivariate_/_time-series_/_survival_analysis654" style="font-size:114%;margin:0 4em"><a href="Categorical_variable" title="Categorical variable">Categorical</a> / <a href="Multivariate_statistics" title="Multivariate statistics">multivariate</a> / <a href="Time_series" title="Time series">time-series</a> / <a href="Survival_analysis" title="Survival analysis">survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cohen's_kappa" title="Cohen's kappa">Cohen's kappa</a></li>

<li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Graphical_model" title="Graphical model">Graphical model</a></li>

<li><a href="Poisson_regression" title="Poisson regression">Log-linear model</a></li>

<li><a href="McNemar's_test" title="McNemar's test">McNemar's test</a></li>

<li><a href="Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="General_linear_model" title="General linear model">Regression</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li>

<li><a href="Principal_component_analysis" title="Principal component analysis">Principal components</a></li>

<li><a href="Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li>

<li><a href="Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li>

<li><a href="Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li>

<li><a href="Statistical_classification" title="Statistical classification">Classification</a></li>

<li><a href="Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a>

<ul><li><a href="Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li>

<li><a href="Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a>

<ul><li><a href="Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a>

<ul><li><a href="Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li>

<li><a href="Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationarity</a></li>

<li><a href="Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li>

<li><a href="Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li>

<li><a href="Cointegration" title="Cointegration">Cointegration</a></li>

<li><a href="Structural_break" title="Structural break">Structural break</a></li>

<li><a href="Granger_causality" title="Granger causality">Granger causality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li>

<li><a href="Johansen_test" title="Johansen test">Johansen</a></li>

<li><a href="Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size: 85%;">(Ljung–Box)</span></a></li>

<li><a href="Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li>

<li><a href="Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a>

<ul><li><a href="Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li>

<li><a href="Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li>

<li><a href="Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size: 85%;">(Box–Jenkins)</span></a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li>

<li><a href="Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li>

<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>

<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li>

<li><a href="Wavelet" title="Wavelet">Wavelet</a></li>

<li><a href="Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li>

<li><a href="Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li>

<li><a href="Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li>

<li><a href="First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications654" style="font-size:114%;margin:0 4em"><a href="List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bioinformatics" title="Bioinformatics">Bioinformatics</a></li>

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<ul><li><a href="Chemometrics" title="Chemometrics">Chemometrics</a></li>

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<li><a href="System_identification" title="System identification">System identification</a></li></ul>

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<ul><li><a href="Actuarial_science" title="Actuarial science">Actuarial science</a></li>

<li><a href="Census" title="Census">Census</a></li>

<li><a href="Crime_statistics" title="Crime statistics">Crime statistics</a></li>

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</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Spatial_analysis" title="Spatial analysis">Spatial statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cartography" title="Cartography">Cartography</a></li>

<li><a href="Environmental_statistics" title="Environmental statistics">Environmental statistics</a></li>

<li><a href="Geographic_information_system" title="Geographic information system">Geographic information system</a></li>

<li><a href="Geostatistics" title="Geostatistics">Geostatistics</a></li>

<li><a href="Kriging" title="Kriging">Kriging</a></li></ul>

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<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span><b><a href="https://commons.wikimedia.org/wiki/Category:Statistics" class="extiw external" title="commons:Category:Statistics">Commons</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> <b>WikiProject</b></li></ul>

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<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Quantitative_forecasting_methods36" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Quantitative_forecasting_methods36" style="font-size:114%;margin:0 4em">Quantitative <a href="Forecasting" title="Forecasting">forecasting</a> methods</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">Historical data forecasts

<ul><li><a href="Moving_average" title="Moving average">Moving average</a></li>

<li><a href="Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li>

<li><a href="Trend_analysis" title="Trend analysis">Trend analysis</a></li>

<li><a href="Decomposition_of_time_series" title="Decomposition of time series">Decomposition of time series</a></li>

<li><a href="Forecasting#Na.C3.AFve_approach" title="Forecasting">Naïve approach</a></li></ul></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">Associative (causal) forecasts

<li><a href="Moving_average" title="Moving average">Moving average</a></li>

<li><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></li>

<li><a href="Econometric_model" title="Econometric model">Econometric model</a></li></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-04" href="https://en.wikipedia.org/wiki/?title=Simple_linear_regression&oldid=1304234792">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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