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Data analysis

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</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on <a href="Statistics" title="Statistics">Statistics</a></td></tr><tr><th class="sidebar-title-with-pretitle" style="background:#ddddff;"><a href="Data_and_information_visualization" title="Data and information visualization">Data and information visualization</a></th></tr><tr><th class="sidebar-heading">

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

<ul><li><a href="Exploratory_data_analysis" title="Exploratory data analysis">Exploratory data analysis</a></li>

<li><a href="Information_design" title="Information design">Information design</a></li>

<li><a href="Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></li>

<li><a href="Statistical_inference" title="Statistical inference">Inferential statistics</a></li>

<li><a href="Statistical_graphics" title="Statistical graphics">Statistical graphics</a></li>

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

<li><a href="Data_science" title="Data science">Data science</a></li></ul></td>

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

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

<ul><li><a href="Tamara_Munzner" title="Tamara Munzner">Tamara Munzner</a></li>

<li><a href="Ben_Shneiderman" title="Ben Shneiderman">Ben Shneiderman</a></li>

<li><a href="John_Tukey" title="John Tukey">John Tukey</a></li>

<li><a href="Edward_Tufte" title="Edward Tufte">Edward Tufte</a></li>

<li><a href="Simon_Wardley" title="Simon Wardley">Simon Wardley</a></li>

<li><a href="Hans_Rosling" title="Hans Rosling">Hans Rosling</a></li>

<li><a href="David_McCandless" title="David McCandless">David McCandless</a></li>

<li><a href="Kim_Albrecht" title="Kim Albrecht">Kim Albrecht</a></li>

<li><a href="Alexander_Osterwalder" title="Alexander Osterwalder">Alexander Osterwalder</a></li>

<li><a href="Ed_Hawkins_(climatologist)" title="Ed Hawkins (climatologist)">Ed Hawkins</a></li>

<li><a href="Hadley_Wickham" title="Hadley Wickham">Hadley Wickham</a></li>

<li><a href="Leland_Wilkinson" title="Leland Wilkinson">Leland Wilkinson</a></li>

<li><a href="Mike_Bostock" title="Mike Bostock">Mike Bostock</a></li>

<li><a href="Jeffrey_Heer" title="Jeffrey Heer">Jeffrey Heer</a></li>

<li><a href="Ihab_Ilyas" title="Ihab Ilyas">Ihab Ilyas</a></li></ul></td>

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

Information graphic types</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Line_chart" title="Line chart">Line chart</a></li>

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

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

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

<li><a href="Pareto_chart" title="Pareto chart">Pareto chart</a></li>

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

<li><a href="Area_chart" title="Area chart">Area chart</a></li>

<li><a href="Bubble_chart" title="Bubble chart">Bubble chart</a></li>

<li><a href="Warming_stripes" title="Warming stripes">Stripe graphic</a></li>

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

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

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

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

<li><a href="Small_multiple" title="Small multiple">Small multiple</a></li>

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

<li><a href="Table_(information)" title="Table (information)">Table</a></li>

<li><a href="Mosaic_plot" title="Mosaic plot">Marimekko chart</a></li></ul></td>

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

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

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

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

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

<li><a href="Visual_perception" title="Visual perception">Visual perception</a></li>

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

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

<li><a href="Misleading_graph" title="Misleading graph">Misleading graph</a></li>

<li><a href="Topological_data_analysis" title="Topological data analysis">Topological data analysis</a></li></ul></td>

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<table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="Computational_physics" title="Computational physics">Computational physics</a></th></tr><tr><td class="sidebar-image"></td></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Computational_mechanics" title="Computational mechanics">Mechanics</a></li>

<li><a href="Computational_electromagnetics" title="Computational electromagnetics">Electromagnetics</a></li>

<li><a href="Multiphysics_simulation" title="Multiphysics simulation">Multiphysics</a></li>

<li><a href="Computational_particle_physics" title="Computational particle physics">Particle physics</a></li>

<li><a href="Computational_thermodynamics" title="Computational thermodynamics">Thermodynamics</a></li>

<li><a href="Computer_simulation" title="Computer simulation">Simulation</a> </li></ul></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Potentials</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Morse/Long-range_potential" title="Morse/Long-range potential">Morse/Long-range potential</a></li>

<li><a href="Lennard-Jones_potential" title="Lennard-Jones potential">Lennard-Jones potential</a></li>

<li><a href="Yukawa_potential" title="Yukawa potential">Yukawa potential</a></li>

<li><a href="Morse_potential" title="Morse potential">Morse potential</a></li></ul></div></div></td>

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

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="Computational_fluid_dynamics" title="Computational fluid dynamics">Fluid dynamics</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Finite_difference_method" title="Finite difference method">Finite difference</a></li>

<li><a href="Finite_volume_method" title="Finite volume method">Finite volume</a></li>

<li><a href="Finite_element_method" title="Finite element method">Finite element</a></li>

<li><a href="Boundary_element_method" title="Boundary element method">Boundary element</a></li>

<li><a href="Lattice_Boltzmann_methods" title="Lattice Boltzmann methods">Lattice Boltzmann</a></li>

<li><a href="Riemann_solver" title="Riemann solver">Riemann solver</a></li>

<li><a href="Dissipative_particle_dynamics" title="Dissipative particle dynamics">Dissipative particle dynamics</a></li>

<li><a href="Smoothed-particle_hydrodynamics" title="Smoothed-particle hydrodynamics">Smoothed particle hydrodynamics</a></li>

<li><a href="Turbulence_modeling" title="Turbulence modeling">Turbulence models</a></li></ul></div></div></td>

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

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="Monte_Carlo_method" title="Monte Carlo method">Monte Carlo methods</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Monte_Carlo_integration" title="Monte Carlo integration">Integration</a></li>

<li><a href="Gibbs_sampling" title="Gibbs sampling">Gibbs sampling</a></li>

<li><a href="Metropolis%E2%80%93Hastings_algorithm" title="Metropolis–Hastings algorithm">Metropolis algorithm</a></li></ul></div></div></td>

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

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Particle</div><div class="sidebar-list-content mw-collapsible-content hlist">

<li><a href="Particle-in-cell" title="Particle-in-cell">Particle-in-cell</a></li>

<li><a href="Molecular_dynamics" title="Molecular dynamics">Molecular dynamics</a></li></ul></div></div></td>

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

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Sergei_K._Godunov" class="mw-redirect" title="Sergei K. Godunov">Godunov</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">von Neumann</a></li>

<li><a href="Boris_Galerkin" title="Boris Galerkin">Galerkin</a></li>

<li><a href="Edward_Norton_Lorenz" title="Edward Norton Lorenz">Lorenz</a></li>

<li><a href="Kenneth_G._Wilson" title="Kenneth G. Wilson">Wilson</a></li>

<li><a href="Berni_Alder" title="Berni Alder">Alder</a></li>

<li><a href="Robert_D._Richtmyer" title="Robert D. Richtmyer">Richtmyer</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-navbar"></td></tr></tbody></table>

Data analysis is the process of inspecting, <a href="Data_cleansing" title="Data cleansing">cleansing</a>, <a href="Data_transformation" class="mw-redirect" title="Data transformation">transforming</a>, and <a href="Data_modeling" title="Data modeling">modeling</a> <a href="Data" title="Data">data</a> with the goal of discovering useful information, informing conclusions, and supporting <a href="Decision-making" title="Decision-making">decision-making</a>.<a href="#cite_note-Auerbach_Publications-1">[1]</a> Data analysis has multiple facets and approaches, encompassing diverse techniques under a variety of names, and is used in different business, science, and social science domains.<a href="#cite_note-2">[2]</a> In today's business world, data analysis plays a role in making decisions more scientific and helping businesses operate more effectively.<a href="#cite_note-3">[3]</a>

<a href="Data_mining" title="Data mining">Data mining</a> is a particular data analysis technique that focuses on statistical modeling and knowledge discovery for predictive rather than purely descriptive purposes, while <a href="Business_intelligence" title="Business intelligence">business intelligence</a> covers data analysis that relies heavily on aggregation, focusing mainly on business information. In statistical applications, data analysis can be divided into <a href="Descriptive_statistics" title="Descriptive statistics">descriptive statistics</a>, <a href="Exploratory_data_analysis" title="Exploratory data analysis">exploratory data analysis</a> (EDA), and <a href="Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">confirmatory data analysis</a> (CDA).<a href="#cite_note-4">[4]</a> EDA focuses on discovering new features in the data while CDA focuses on confirming or falsifying existing <a href="Hypotheses" class="mw-redirect" title="Hypotheses">hypotheses</a>.<a href="#cite_note-5">[5]</a> <a href="Predictive_analytics" title="Predictive analytics">Predictive analytics</a> focuses on the application of statistical models for predictive forecasting or classification, while <a href="Text_analytics" class="mw-redirect" title="Text analytics">text analytics</a> applies statistical, linguistic, and structural techniques to extract and classify information from textual sources, a variety of <a href="Unstructured_data" title="Unstructured data">unstructured data</a>. All of the above are varieties of data analysis.<a href="#cite_note-6">[6]</a>

<div class="mw-heading mw-heading2"><h2 id="Data_analysis_process">Data analysis process</h2></div>

Data analysis is a <a href="Process_theory" title="Process theory">process</a> for obtaining <a href="Raw_data" title="Raw data">raw data</a>, and subsequently converting it into information useful for decision-making by users.<a href="#cite_note-Auerbach_Publications-1">[1]</a> Statistician <a href="John_Tukey" title="John Tukey">John Tukey</a>, defined data analysis in 1961, as:<blockquote>"Procedures for analyzing data, techniques for interpreting the results of such procedures, ways of planning the gathering of data to make its analysis easier, more precise or more accurate, and all the machinery and results of (mathematical) statistics which apply to analyzing data."<a href="#cite_note-7">[7]</a></blockquote>

There are several phases, and they are <a href="Iteration" title="Iteration">iterative</a>, in that feedback from later phases may result in additional work in earlier phases.<a href="#cite_note-Schutt_&_O'Neil-8">[8]</a>

<div class="mw-heading mw-heading3"><h3 id="Data_requirements">Data requirements</h3></div>

The data is necessary as inputs to the analysis, which is specified based upon the requirements of those directing the analytics (or customers, who will use the finished product of the analysis).<a href="#cite_note-9">[9]</a> The general type of entity upon which the data will be collected is referred to as an <a href="Statistical_unit" title="Statistical unit">experimental unit</a> (e.g., a person or population of people). Specific variables regarding a population (e.g., age and income) may be specified and obtained. Data may be numerical or categorical (i.e., a text label for numbers).<a href="#cite_note-Schutt_&_O'Neil-8">[8]</a>

<div class="mw-heading mw-heading3"><h3 id="Data_collection">Data collection</h3></div>

Data may be collected from a variety of sources.<a href="#cite_note-10">[10]</a> A <a href="List_of_datasets_for_machine-learning_research" title="List of datasets for machine-learning research">list of data sources</a> are available for study & research. The requirements may be communicated by analysts to <a href="Data_custodian" title="Data custodian">custodians</a> of the data; such as, <a href="Information_systems_technician" title="Information systems technician">Information Technology personnel</a> within an organization.<a href="#cite_note-11">[11]</a> Data collection or data gathering is the process of gathering and <a href="Measuring" class="mw-redirect" title="Measuring">measuring</a> <a href="Information" title="Information">information</a> on targeted variables in an established system, which then enables one to answer relevant questions and evaluate outcomes. The data may also be collected from sensors in the environment, including traffic cameras, satellites, recording devices, etc. It may also be obtained through interviews, downloads from online sources, or reading documentation.<a href="#cite_note-Schutt_&_O'Neil-8">[8]</a>

<div class="mw-heading mw-heading3"><h3 id="Data_processing">Data processing</h3></div>

<a href="Data_integration" title="Data integration">Data integration</a> is a precursor to data analysis: Data, when initially obtained, must be processed or organized for analysis. For instance, this may involve placing data into rows and columns in a table format (known as <a href="Data_model" title="Data model">structured data</a>) for further analysis, often through the use of spreadsheet(excel) or statistical software.<a href="#cite_note-Schutt_&_O'Neil-8">[8]</a>

<div class="mw-heading mw-heading3"><h3 id="Data_cleaning">Data cleaning</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Data_cleansing" title="Data cleansing">Data cleansing</a></div>

Once processed and organized, the data may be incomplete, contain duplicates, or contain errors.<a href="#cite_note-Bohannon-12">[12]</a> The need for data cleaning will arise from problems in the way that the data is entered and stored.<a href="#cite_note-Bohannon-12">[12]</a><a href="#cite_note-13">[13]</a> Data cleaning is the process of preventing and correcting these errors. Common tasks include record matching, identifying inaccuracy of data, overall quality of existing data, deduplication, and column segmentation.<a href="#cite_note-14">[14]</a><a href="#cite_note-15">[15]</a>

Such data problems can also be identified through a variety of analytical techniques. For example; with financial information, the totals for particular variables may be compared against separately published numbers that are believed to be reliable.<a href="#cite_note-Koomey1-16">[16]</a> Unusual amounts, above or below predetermined thresholds, may also be reviewed. There are several types of data cleaning that are dependent upon the type of data in the set; this could be phone numbers, email addresses, employers, or other values.<a href="#cite_note-17">[17]</a> Quantitative data methods for outlier detection can be used to get rid of data that appears to have a higher likelihood of being input incorrectly. Text data spell checkers can be used to lessen the amount of mistyped words. However, it is harder to tell if the words are contextually (i.e., semantically and idiomatically) correct.

<div class="mw-heading mw-heading3"><h3 id="Exploratory_data_analysis">Exploratory data analysis</h3></div>

Once the datasets are cleaned, they can then begin to be analyzed using <a href="Exploratory_data_analysis" title="Exploratory data analysis">exploratory data analysis</a>. The process of data exploration may result in additional data cleaning or additional requests for data; thus, the initialization of the iterative phases mentioned above.<a href="#cite_note-18">[18]</a> <a href="Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a>, such as the average, median, and standard deviation, are often used to broadly characterize the data.<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a> <a href="Data_visualization" class="mw-redirect" title="Data visualization">Data visualization</a> is also used, in which the analyst is able to examine the data in a graphical format in order to obtain additional insights about messages within the data.<a href="#cite_note-Schutt_&_O'Neil-8">[8]</a>

<div class="mw-heading mw-heading3"><h3 id="Modeling_and_algorithms">Modeling and algorithms</h3></div>

Mathematical formulas or models (also known as <a href="Algorithms" class="mw-redirect" title="Algorithms">algorithms</a>), may be applied to the data in order to identify relationships among the variables; for example, checking for <a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">correlation</a> and by determining whether or not there is the presence of <a href="Causality" title="Causality">causality</a>. In general terms, models may be developed to evaluate a specific variable based on other variable(s) contained within the dataset, with some <a href="Residual_bit_error_rate" title="Residual bit error rate">residual error</a> depending on the implemented model's accuracy (e.g., Data = Model + Error).<a href="#cite_note-21">[21]</a>

<a href="Inferential_statistics" class="mw-redirect" title="Inferential statistics">Inferential statistics</a> utilizes techniques that measure the relationships between particular variables.<a href="#cite_note-22">[22]</a> For example, <a href="Regression_analysis" title="Regression analysis">regression analysis</a> may be used to model whether a change in advertising (independent variable X), provides an explanation for the variation in sales (dependent variable Y), i.e. is Y a function of X? This can be described as (Y = aX + b + error), where the model is designed such that (a) and (b) minimize the error when the model predicts Y for a given range of values of X.<a href="#cite_note-23">[23]</a>

<div class="mw-heading mw-heading3"><h3 id="Data_product">Data product</h3></div>

A data product is a computer application that takes data inputs and generates outputs, feeding them back into the environment.<a href="#cite_note-24">[24]</a> It may be based on a model or algorithm. For instance, an application that analyzes data about customer purchase history, and uses the results to recommend other purchases the customer might enjoy.<a href="#cite_note-25">[25]</a><a href="#cite_note-Schutt_&_O'Neil-8">[8]</a>

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

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Data_and_information_visualization" title="Data and information visualization">Data and information visualization</a></div>

Once data is analyzed, it may be reported in many formats to the users of the analysis to support their requirements.<a href="#cite_note-27">[27]</a> The users may have feedback, which results in additional analysis.

When determining how to communicate the results, the analyst may consider implementing a variety of data visualization techniques to help communicate the message more clearly and efficiently to the audience. Data visualization uses <a href="Information_displays" class="mw-redirect" title="Information displays">information displays</a> (graphics such as, tables and charts) to help communicate key messages contained in the data. <a href="Table_(information)" title="Table (information)">Tables</a> are a valuable tool by enabling the ability of a user to query and focus on specific numbers; while charts (e.g., bar charts or line charts), may help explain the quantitative messages contained in the data.<a href="#cite_note-28">[28]</a>

<div class="mw-heading mw-heading2"><h2 id="Quantitative_messages">Quantitative messages</h2></div>

Stephen Few described eight types of quantitative messages that users may attempt to communicate from a set of data, including the associated graphs.<a href="#cite_note-Few_GraphType-29">[29]</a><a href="#cite_note-30">[30]</a>

<ol><li>Time-series: A single variable is captured over a period of time, such as the unemployment rate over a 10-year period. A <a href="Line_chart" title="Line chart">line chart</a> may be used to demonstrate the trend.</li>

<li>Ranking: Categorical subdivisions are ranked in ascending or descending order, such as a ranking of sales performance (the measure) by salespersons (the category, with each salesperson a categorical subdivision) during a single period. A <a href="Bar_chart" title="Bar chart">bar chart</a> may be used to show the comparison across the salespersons.<a href="#cite_note-31">[31]</a></li>

<li>Part-to-whole: Categorical subdivisions are measured as a ratio to the whole (i.e., a percentage out of 100%). A <a href="Pie_chart" title="Pie chart">pie chart</a> or bar chart can show the comparison of ratios, such as the market share represented by competitors in a market.<a href="#cite_note-32">[32]</a></li>

<li>Deviation: Categorical subdivisions are compared against a reference, such as a comparison of actual vs. budget expenses for several departments of a business for a given time period. A bar chart can show the comparison of the actual versus the reference amount.<a href="#cite_note-33">[33]</a></li>

<li>Frequency distribution: Shows the number of observations of a particular variable for a given interval, such as the number of years in which the stock market return is between intervals such as 0–10%, 11–20%, etc. A <a href="Histogram" title="Histogram">histogram</a>, a type of bar chart, may be used for this analysis.</li>

<li>Correlation: Comparison between observations represented by two variables (X,Y) to determine if they tend to move in the same or opposite directions. For example, plotting unemployment (X) and inflation (Y) for a sample of months. A <a href="Scatter_plot" title="Scatter plot">scatter plot</a> is typically used for this message.<a href="#cite_note-34">[34]</a></li>

<li>Nominal comparison: Comparing categorical subdivisions in no particular order, such as the sales volume by product code. A bar chart may be used for this comparison.<a href="#cite_note-35">[35]</a></li>

<li>Geographic or geo-spatial: Comparison of a variable across a map or layout, such as the unemployment rate by state or the number of persons on the various floors of a building. A <a href="Cartogram" title="Cartogram">cartogram</a> is typically used.<a href="#cite_note-Few_GraphType-29">[29]</a></li></ol>

<div class="mw-heading mw-heading2"><h2 id="Analyzing_quantitative_data_in_finance">Analyzing quantitative data in finance</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Problem_solving" title="Problem solving">Problem solving</a></div>

Author <a href="Jonathan_Koomey" title="Jonathan Koomey">Jonathan Koomey</a> has recommended a series of best practices for understanding quantitative data. These include:<a href="#cite_note-Koomey1-16">[16]</a>

<ul><li>Check raw data for anomalies prior to performing an analysis;</li>

<li>Re-perform important calculations, such as verifying columns of data that are formula-driven;</li>

<li>Confirm main totals are the sum of subtotals;</li>

<li>Check relationships between numbers that should be related in a predictable way, such as ratios over time;</li>

<li>Normalize numbers to make comparisons easier, such as analyzing amounts per person or relative to GDP or as an index value relative to a base year;</li>

<li>Break problems into component parts by analyzing factors that led to the results, such as <a href="DuPont_analysis" title="DuPont analysis">DuPont analysis</a> of return on equity.</li></ul>

For the variables under examination, analysts typically obtain <a href="Descriptive_statistics" title="Descriptive statistics">descriptive statistics</a>, such as the mean (average), <a href="Median" title="Median">median</a>, and <a href="Standard_deviation" title="Standard deviation">standard deviation</a>. They may also analyze the <a href="Probability_distribution" title="Probability distribution">distribution</a> of the key variables to see how the individual values cluster around the mean.<a href="#cite_note-Koomey1-16">[16]</a>

<a href="McKinsey_and_Company" class="mw-redirect" title="McKinsey and Company">McKinsey and Company</a> named a technique for breaking down a quantitative problem into its component parts called the <a href="MECE_principle" title="MECE principle">MECE principle</a>. MECE means "Mutually Exclusive and Collectively Exhaustive".<a href="#cite_note-36">[36]</a> Each layer can be broken down into its components; each of the sub-components must be <a href="Mutually_exclusive_events" class="mw-redirect" title="Mutually exclusive events">mutually exclusive</a> of each other and <a href="Collectively_exhaustive_events" title="Collectively exhaustive events">collectively</a> add up to the layer above them. For example, profit by definition can be broken down into total revenue and total cost.<a href="#cite_note-37">[37]</a>

Analysts may use robust statistical measurements to solve certain analytical problems. <a href="Hypothesis_testing" class="mw-redirect" title="Hypothesis testing">Hypothesis testing</a> is used when a particular hypothesis about the true state of affairs is made by the analyst and data is gathered to determine whether that hypothesis is true or false.<a href="#cite_note-38">[38]</a> For example, the hypothesis might be that "Unemployment has no effect on inflation", which relates to an economics concept called the <a href="Phillips_Curve" class="mw-redirect" title="Phillips Curve">Phillips Curve</a>.<a href="#cite_note-39">[39]</a> Hypothesis testing involves considering the likelihood of <a href="Type_I_and_type_II_errors" title="Type I and type II errors">Type I and type II errors</a>, which relate to whether the data supports accepting or rejecting the hypothesis.<a href="#cite_note-40">[40]</a>

<a href="Regression_analysis" title="Regression analysis">Regression analysis</a> may be used when the analyst is trying to determine the extent to which independent variable X affects dependent variable Y (e.g., "To what extent do changes in the unemployment rate (X) affect the inflation rate (Y)?").<a href="#cite_note-Yanamandra_57–68-41">[41]</a>

<a href="Necessary_condition_analysis" title="Necessary condition analysis">Necessary condition analysis</a> (NCA) may be used when the analyst is trying to determine the extent to which independent variable X allows variable Y (e.g., "To what extent is a certain unemployment rate (X) necessary for a certain inflation rate (Y)?").<a href="#cite_note-Yanamandra_57–68-41">[41]</a> Whereas (multiple) regression analysis uses additive logic where each X-variable can produce the outcome and the X's can compensate for each other (they are sufficient but not necessary),<a href="#cite_note-42">[42]</a> necessary condition analysis (NCA) uses necessity logic, where one or more X-variables allow the outcome to exist, but may not produce it (they are necessary but not sufficient). Each single necessary condition must be present and compensation is not possible.<a href="#cite_note-43">[43]</a>

<div class="mw-heading mw-heading2"><h2 id="Analytical_activities_of_data_users">Analytical activities of data users</h2></div>

Users may have particular data points of interest within a data set, as opposed to the general messaging outlined above. Such low-level user analytic activities are presented in the following table. The taxonomy can also be organized by three poles of activities: retrieving values, finding data points, and arranging data points.<a href="#cite_note-44">[44]</a><a href="#cite_note-45">[45]</a><a href="#cite_note-46">[46]</a>

<th>General description</th>

<th>Pro forma abstract</th>

<th width="35%">Examples

</th></tr>

<tr>

<td align="center">1

</td>

<td>Retrieve Value

</td>

<td>Given a set of specific cases, find attributes of those cases.

</td>

<td>What are the values of attributes {X, Y, Z, ...} in the data cases {A, B, C, ...}?

</td>

<td>- What is the mileage per gallon of the Ford Mondeo?

- How long is the movie Gone with the Wind?

</td></tr>

<tr>

<td align="center">2

</td>

<td> Filter

</td>

<td>Given some concrete conditions on attribute values, find data cases satisfying those conditions.

</td>

<td>Which data cases satisfy conditions {A, B, C...}?

</td>

<td>- What Kellogg's cereals have high fiber?

- What comedies have won awards?

- Which funds underperformed the SP-500?

</td></tr>

<tr>

<td align="center">3

</td>

<td>Compute Derived Value

</td>

<td>Given a set of data cases, compute an aggregate numeric representation of those data cases.

</td>

<td>What is the value of aggregation function F over a given set S of data cases?

</td>

<td>- What is the average calorie content of Post cereals?

- What is the gross income of all stores combined?

- How many manufacturers of cars are there?

</td></tr>

<tr>

<td align="center">4

</td>

<td>Find Extremum

</td>

<td>Find data cases possessing an extreme value of an attribute over its range within the data set.

</td>

<td>What are the top/bottom N data cases with respect to attribute A?

</td>

<td>- What is the car with the highest MPG?

- What director/film has won the most awards?

- What Marvel Studios film has the most recent release date?

</td></tr>

<tr>

<td align="center">5

</td>

<td>Sort

</td>

<td>Given a set of data cases, rank them according to some ordinal metric.

</td>

<td>What is the sorted order of a set S of data cases according to their value of attribute A?

</td>

<td>- Order the cars by weight.

- Rank the cereals by calories.

</td></tr>

<tr>

<td align="center">6

</td>

<td>Determine Range

</td>

<td>Given a set of data cases and an attribute of interest, find the span of values within the set.

</td>

<td>What is the range of values of attribute A in a set S of data cases?

</td>

<td>- What is the range of film lengths?

- What is the range of car horsepowers?

- What actresses are in the data set?

</td></tr>

<tr>

<td align="center">7

</td>

<td>Characterize Distribution

</td>

<td>Given a set of data cases and a quantitative attribute of interest, characterize the distribution of that attribute's values over the set.

</td>

<td>What is the distribution of values of attribute A in a set S of data cases?

</td>

<td>- What is the distribution of carbohydrates in cereals?

- What is the age distribution of shoppers?

</td></tr>

<tr>

<td align="center">8

</td>

<td>Find Anomalies

</td>

<td>Identify any anomalies within a given set of data cases with respect to a given relationship or expectation, e.g. statistical outliers.

</td>

<td>Which data cases in a set S of data cases have unexpected/exceptional values?

</td>

<td>- Are there exceptions to the relationship between horsepower and acceleration?

- Are there any outliers in protein?

</td></tr>

<tr>

<td align="center">9

</td>

<td>Cluster

</td>

<td>Given a set of data cases, find clusters of similar attribute values.

</td>

<td>Which data cases in a set S of data cases are similar in value for attributes {X, Y, Z, ...}?

</td>

<td>- Are there groups of cereals w/ similar fat/calories/sugar?

- Is there a cluster of typical film lengths?

</td></tr>

<tr>

<td align="center">10

</td>

<td>Correlate

</td>

<td>Given a set of data cases and two attributes, determine useful relationships between the values of those attributes.

</td>

<td>What is the correlation between attributes X and Y over a given set S of data cases?

</td>

<td>- Is there a correlation between carbohydrates and fat?

- Is there a correlation between country of origin and MPG?

- Do different genders have a preferred payment method?

- Is there a trend of increasing film length over the years?

</td></tr>

<tr>

<td align="center">11

</td>

<td><a href="Contextualization_(computer_science)" title="Contextualization (computer science)">Contextualization</a>

</td>

<td>Given a set of data cases, find contextual relevancy of the data to the users.

</td>

<td>Which data cases in a set S of data cases are relevant to the current users' context?

</td>

<td>- Are there groups of restaurants that have foods based on my current caloric intake?

</td></tr>

</tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Barriers_to_effective_analysis">Barriers to effective analysis</h2></div>

Barriers to effective analysis may exist among the analysts performing the data analysis or among the audience. Distinguishing fact from opinion, cognitive biases, and innumeracy are all challenges to sound data analysis.<a href="#cite_note-47">[47]</a>

<div class="mw-heading mw-heading3"><h3 id="Confusing_fact_and_opinion">Confusing fact and opinion</h3></div>

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

.mw-parser-output .quotebox{background-color:#F9F9F9;border:1px solid #aaa;box-sizing:border-box;padding:10px;font-size:88%;max-width:100%}.mw-parser-output .quotebox.floatleft{margin:.5em 1.4em .8em 0}.mw-parser-output .quotebox.floatright{margin:.5em 0 .8em 1.4em}.mw-parser-output .quotebox.centered{overflow:hidden;position:relative;margin:.5em auto .8em auto}.mw-parser-output .quotebox.floatleft span,.mw-parser-output .quotebox.floatright span{font-style:inherit}.mw-parser-output .quotebox>blockquote{margin:0;padding:0;border-left:0;font-family:inherit;font-size:inherit}.mw-parser-output .quotebox-title{text-align:center;font-size:110%;font-weight:bold}.mw-parser-output .quotebox-quote>:first-child{margin-top:0}.mw-parser-output .quotebox-quote:last-child>:last-child{margin-bottom:0}.mw-parser-output .quotebox-quote.quoted:before{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" “ ";vertical-align:-45%;line-height:0}.mw-parser-output .quotebox-quote.quoted:after{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" ” ";line-height:0}.mw-parser-output .quotebox .left-aligned{text-align:left}.mw-parser-output .quotebox .right-aligned{text-align:right}.mw-parser-output .quotebox .center-aligned{text-align:center}.mw-parser-output .quotebox .quote-title,.mw-parser-output .quotebox .quotebox-quote{display:block}.mw-parser-output .quotebox cite{display:block;font-style:normal}@media screen and (max-width:640px){.mw-parser-output .quotebox{width:100%!important;margin:0 0 .8em!important;float:none!important}}

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</style><div class="quotebox pullquote floatright" style="width:250px; ;">

You are entitled to your own opinion, but you are not entitled to your own facts.

</blockquote>

<div style="padding-bottom: 0; padding-top: 0.5em"><cite class="left-aligned" style=""><a href="Daniel_Patrick_Moynihan" title="Daniel Patrick Moynihan">Daniel Patrick Moynihan</a></cite></div>

</div>

Effective analysis requires obtaining relevant <a href="Fact" title="Fact">facts</a> to answer questions, support a conclusion or formal <a href="Opinion" title="Opinion">opinion</a>, or test <a href="Hypotheses" class="mw-redirect" title="Hypotheses">hypotheses</a>.<a href="#cite_note-48">[48]</a> Facts by definition are irrefutable, meaning that any person involved in the analysis should be able to agree upon them. The auditor of a public company must arrive at a formal opinion on whether financial statements of publicly traded corporations are "fairly stated, in all material respects".<a href="#cite_note-49">[49]</a> This requires extensive analysis of factual data and evidence to support their opinion.

<div class="mw-heading mw-heading3"><h3 id="Cognitive_biases">Cognitive biases</h3></div>

There are a variety of <a href="Cognitive_bias" title="Cognitive bias">cognitive biases</a> that can adversely affect analysis. For example, <a href="Confirmation_bias" title="Confirmation bias">confirmation bias</a> is the tendency to search for or interpret information in a way that confirms one's preconceptions.<a href="#cite_note-50">[50]</a> In addition, individuals may discredit information that does not support their views.<a href="#cite_note-51">[51]</a>

Analysts may be trained specifically to be aware of these biases and how to overcome them.<a href="#cite_note-52">[52]</a> In his book Psychology of Intelligence Analysis, retired CIA analyst <a href="Richards_Heuer" title="Richards Heuer">Richards Heuer</a> wrote that analysts should clearly delineate their assumptions and chains of inference and specify the degree and source of the uncertainty involved in the conclusions.<a href="#cite_note-53">[53]</a> He emphasized procedures to help surface and debate alternative points of view.<a href="#cite_note-Heuer1-54">[54]</a>

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

Effective analysts are generally adept with a variety of numerical techniques. However, audiences may not have such literacy with numbers or <a href="Numeracy" title="Numeracy">numeracy</a>; they are said to be innumerate.<a href="#cite_note-55">[55]</a> Persons communicating the data may also be attempting to mislead or misinform, deliberately using bad numerical techniques.<a href="#cite_note-56">[56]</a>

For example, whether a number is rising or falling may not be the key factor. More important may be the number relative to another number, such as the size of government revenue or spending relative to the size of the economy (GDP) or the amount of cost relative to revenue in corporate financial statements.<a href="#cite_note-57">[57]</a> This numerical technique is referred to as normalization<a href="#cite_note-Koomey1-16">[16]</a> or common-sizing. There are many such techniques employed by analysts, whether adjusting for inflation (i.e., comparing real vs. nominal data) or considering population increases, demographics, etc.<a href="#cite_note-58">[58]</a>

Analysts may also analyze data under different assumptions or scenarios. For example, when analysts perform <a href="Financial_statement_analysis" title="Financial statement analysis">financial statement analysis</a>, they will often recast the financial statements under different assumptions to help arrive at an estimate of future cash flow, which they then discount to present value based on some interest rate, to determine the valuation of the company or its stock.<a href="#cite_note-59">[59]</a> Similarly, the CBO analyzes the effects of various policy options on the government's revenue, outlays and deficits, creating alternative future scenarios for key measures.<a href="#cite_note-60">[60]</a>

<div class="mw-heading mw-heading2"><h2 id="Other_applications">Other applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Analytics_and_business_intelligence">Analytics and business intelligence</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Analytics" title="Analytics">Analytics</a></div>

Analytics is the "extensive use of data, statistical and quantitative analysis, explanatory and predictive models, and fact-based management to drive decisions and actions." It is a subset of <a href="Business_intelligence" title="Business intelligence">business intelligence</a>, which is a set of technologies and processes that uses data to understand and analyze business performance to drive decision-making.<a href="#cite_note-Competing_on_Analytics_2007-61">[61]</a>

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

In <a href="Education" title="Education">education</a>, most educators have access to a <a href="Data_system" title="Data system">data system</a> for the purpose of analyzing student data.<a href="#cite_note-62">[62]</a> These data systems present data to educators in an <a href="Over-the-counter_data" title="Over-the-counter data">over-the-counter data</a> format (embedding labels, supplemental documentation, and a help system and making key package/display and content decisions) to improve the accuracy of educators' data analyses.<a href="#cite_note-63">[63]</a>

<div class="mw-heading mw-heading2"><h2 id="Practitioner_notes">Practitioner notes</h2></div>

This section contains rather technical explanations that may assist practitioners but are beyond the typical scope of a Wikipedia article.<a href="#cite_note-64">[64]</a>

<div class="mw-heading mw-heading3"><h3 id="Initial_data_analysis">Initial data analysis</h3></div>

The most important distinction between the initial data analysis phase and the main analysis phase is that during initial data analysis one refrains from any analysis that is aimed at answering the original research question. The initial data analysis phase is guided by the following four questions:<a href="#cite_note-FOOTNOTEAdèr2008a337-65">[65]</a>

<div class="mw-heading mw-heading4"><h4 id="Quality_of_data">Quality of data</h4></div>

The quality of the data should be checked as early as possible. Data quality can be assessed in several ways, using different types of analysis: frequency counts, descriptive statistics (mean, standard deviation, median), normality (skewness, kurtosis, frequency histograms), normal <a href="Imputation_(statistics)" title="Imputation (statistics)">imputation</a> is needed.<a href="#cite_note-66">[66]</a>

<ul><li>Analysis of <a href="Outlier" title="Outlier">extreme observations</a>: outlying observations in the data are analyzed to see if they seem to disturb the distribution.<a href="#cite_note-67">[67]</a></li>

<li>Comparison and correction of differences in coding schemes: variables are compared with coding schemes of variables external to the data set, and possibly corrected if coding schemes are not comparable.<a href="#cite_note-68">[68]</a></li>

<li>Test for <a href="Common-method_variance" title="Common-method variance">common-method variance</a>. The choice of analyses to assess the data quality during the initial data analysis phase depends on the analyses that will be conducted in the main analysis phase.<a href="#cite_note-FOOTNOTEAdèr2008a338–341-69">[69]</a></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Quality_of_measurements">Quality of measurements</h4></div>

The quality of the <a href="Measuring_instrument" class="mw-redirect" title="Measuring instrument">measurement instruments</a> should only be checked during the initial data analysis phase when this is not the focus or research question of the study.<a href="#cite_note-70">[70]</a> One should check whether structure of measurement instruments corresponds to structure reported in the literature.

There are two ways to assess measurement quality:

<ul><li>Confirmatory factor analysis</li>

<li>Analysis of homogeneity (<a href="Internal_consistency" title="Internal consistency">internal consistency</a>), which gives an indication of the <a href="Reliability_(statistics)" title="Reliability (statistics)">reliability</a> of a measurement instrument.<a href="#cite_note-71">[71]</a> During this analysis, one inspects the variances of the items and the scales, the <a href="Cronbach's_alpha" title="Cronbach's alpha">Cronbach's α</a> of the scales, and the change in the Cronbach's alpha when an item would be deleted from a scale<a href="#cite_note-FOOTNOTEAdèr2008a341–342-72">[72]</a></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Initial_transformations">Initial transformations</h4></div>

After assessing the quality of the data and of the measurements, one might decide to impute missing data, or to perform initial transformations of one or more variables, although this can also be done during the main analysis phase.<a href="#cite_note-FOOTNOTEAdèr2008a344-73">[73]</a>

Possible transformations of variables are:<a href="#cite_note-74">[74]</a>

<ul><li>Square root transformation (if the distribution differs moderately from normal)</li>

<li>Log-transformation (if the distribution differs substantially from normal)</li>

<li>Inverse transformation (if the distribution differs severely from normal)</li>

<li>Make categorical (ordinal / dichotomous) (if the distribution differs severely from normal, and no transformations help)</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Did_the_implementation_of_the_study_fulfill_the_intentions_of_the_research_design?">Did the implementation of the study fulfill the intentions of the research design?</h4></div>

One should check the success of the <a href="Randomization" title="Randomization">randomization</a> procedure, for instance by checking whether background and substantive variables are equally distributed within and across groups. If the study did not need or use a randomization procedure, one should check the success of the non-random sampling, for instance by checking whether all subgroups of the population of interest are represented in the sample.<a href="#cite_note-75">[75]</a> Other possible data distortions that should be checked are:

<ul><li><a href="Dropout_(electronics)" class="mw-redirect" title="Dropout (electronics)">dropout</a> (this should be identified during the initial data analysis phase)</li>

<li>Item <a href="Response_rate_(survey)" title="Response rate (survey)">non-response</a> (whether this is random or not should be assessed during the initial data analysis phase)</li>

<li>Treatment quality (using <a href="Manipulation_check" title="Manipulation check">manipulation checks</a>).<a href="#cite_note-FOOTNOTEAdèr2008a344–345-76">[76]</a></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Characteristics_of_data_sample">Characteristics of data sample</h4></div>

In any report or article, the structure of the sample must be accurately described. It is especially important to exactly determine the size of the subgroup when subgroup analyses will be performed during the main analysis phase.<a href="#cite_note-77">[77]</a> The characteristics of the data sample can be assessed by looking at:

<ul><li>Basic statistics of important variables</li>

<li>Scatter plots</li>

<li>Correlations and associations</li>

<li>Cross-tabulations<a href="#cite_note-FOOTNOTEAdèr2008a345-78">[78]</a></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Final_stage_of_the_initial_data_analysis">Final stage of the initial data analysis</h4></div>

During the final stage, the findings of the initial data analysis are documented, and necessary, preferable, and possible corrective actions are taken. Also, the original plan for the main data analyses can and should be specified in more detail or rewritten. In order to do this, several decisions about the main data analyses can and should be made:

<ul><li>In the case of non-<a href="Normal_distribution" title="Normal distribution">normals</a>: should one <a href="Data_transformation_(statistics)" title="Data transformation (statistics)">transform</a> variables; make variables categorical (ordinal/dichotomous); adapt the analysis method?</li>

<li>In the case of <a href="Missing_data" title="Missing data">missing data</a>: should one neglect or impute the missing data; which imputation technique should be used?</li>

<li>In the case of <a href="Outlier" title="Outlier">outliers</a>: should one use robust analysis techniques?</li>

<li>In case items do not fit the scale: should one adapt the measurement instrument by omitting items, or rather ensure comparability with other (uses of the) measurement instrument(s)?</li>

<li>In the case of (too) small subgroups: should one drop the hypothesis about inter-group differences, or use small sample techniques, like exact tests or <a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">bootstrapping</a>?</li>

<li>In case the <a href="Randomization" title="Randomization">randomization</a> procedure seems to be defective: can and should one calculate <a href="Propensity_score_matching" title="Propensity score matching">propensity scores</a> and include them as covariates in the main analyses?<a href="#cite_note-FOOTNOTEAdèr2008a345–346-79">[79]</a></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Analysis">Analysis</h4></div>

Several analyses can be used during the initial data analysis phase:<a href="#cite_note-FOOTNOTEAdèr2008a346–347-80">[80]</a>

<ul><li>Univariate statistics (single variable)</li>

<li>Bivariate associations (correlations)</li>

<li>Graphical techniques (scatter plots)</li></ul>

It is important to take the measurement levels of the variables into account for the analyses, as special statistical techniques are available for each level:<a href="#cite_note-FOOTNOTEAdèr2008a349–353-81">[81]</a>

<ul><li>Nominal and ordinal variables

<ul><li>Frequency counts (numbers and percentages)</li>

<li>Associations

<ul><li>circumambulations (crosstabulations)</li>

<li>hierarchical loglinear analysis (restricted to a maximum of 8 variables)</li>

<li>loglinear analysis (to identify relevant/important variables and possible confounders)</li></ul></li>

<li>Exact tests or bootstrapping (in case subgroups are small)</li>

<li>Computation of new variables</li></ul></li>

<li>Continuous variables

<ul><li>Distribution

<ul><li>Statistics (M, SD, variance, skewness, kurtosis)</li>

<li>Stem-and-leaf displays</li>

<li>Box plots</li></ul></li></ul></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Nonlinear_analysis">Nonlinear analysis</h4></div>

Nonlinear analysis is often necessary when the data is recorded from a <a href="Nonlinear_system" title="Nonlinear system">nonlinear system</a>. Nonlinear systems can exhibit complex dynamic effects including <a href="Bifurcation_theory" title="Bifurcation theory">bifurcations</a>, <a href="Chaos_theory" title="Chaos theory">chaos</a>, <a href="Harmonics" class="mw-redirect" title="Harmonics">harmonics</a> and <a href="Subharmonics" class="mw-redirect" title="Subharmonics">subharmonics</a> that cannot be analyzed using simple linear methods. Nonlinear data analysis is closely related to <a href="Nonlinear_system_identification" title="Nonlinear system identification">nonlinear system identification</a>.<a href="#cite_note-SAB1-82">[82]</a>

<div class="mw-heading mw-heading3"><h3 id="Main_data_analysis">Main data analysis</h3></div>

In the main analysis phase, analyses aimed at answering the research question are performed as well as any other relevant analysis needed to write the first draft of the research report.<a href="#cite_note-FOOTNOTEAdèr2008b363-83">[83]</a>

<div class="mw-heading mw-heading4"><h4 id="Exploratory_and_confirmatory_approaches">Exploratory and confirmatory approaches</h4></div>

In the main analysis phase, either an exploratory or confirmatory approach can be adopted. Usually the approach is decided before data is collected.<a href="#cite_note-84">[84]</a> In an exploratory analysis no clear hypothesis is stated before analysing the data, and the data is searched for models that describe the data well.<a href="#cite_note-85">[85]</a> In a confirmatory analysis, clear hypotheses about the data are tested.<a href="#cite_note-86">[86]</a>

<a href="Exploratory_data_analysis" title="Exploratory data analysis">Exploratory data analysis</a> should be interpreted carefully. When testing multiple models at once there is a high chance on finding at least one of them to be significant, but this can be due to a <a href="Type_1_error" class="mw-redirect" title="Type 1 error">type 1 error</a>. It is important to always adjust the significance level when testing multiple models with, for example, a <a href="Bonferroni_correction" title="Bonferroni correction">Bonferroni correction</a>.<a href="#cite_note-87">[87]</a> Also, one should not follow up an exploratory analysis with a confirmatory analysis in the same dataset.<a href="#cite_note-Mcardle_2008-88">[88]</a> An exploratory analysis is used to find ideas for a theory, but not to test that theory as well.<a href="#cite_note-Mcardle_2008-88">[88]</a> When a model is found exploratory in a dataset, then following up that analysis with a confirmatory analysis in the same dataset could simply mean that the results of the confirmatory analysis are due to the same <a href="Type_1_error" class="mw-redirect" title="Type 1 error">type 1 error</a> that resulted in the exploratory model in the first place.<a href="#cite_note-Mcardle_2008-88">[88]</a> The confirmatory analysis therefore will not be more informative than the original exploratory analysis.<a href="#cite_note-FOOTNOTEAdèr2008b361–362-89">[89]</a>

<div class="mw-heading mw-heading4"><h4 id="Stability_of_results">Stability of results</h4></div>

It is important to obtain some indication about how generalizable the results are.<a href="#cite_note-FOOTNOTEAdèr2008b361–371-90">[90]</a> While this is often difficult to check, one can look at the stability of the results. Are the results reliable and reproducible? There are two main ways of doing that.

<ul><li><a href="Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross-validation</a>. By splitting the data into multiple parts, we can check if an analysis (like a fitted model) based on one part of the data generalizes to another part of the data as well.<a href="#cite_note-91">[91]</a> Cross-validation is generally inappropriate, though, if there are correlations within the data, e.g. with <a href="Panel_data" title="Panel data">panel data</a>.<a href="#cite_note-92">[92]</a> Hence other methods of validation sometimes need to be used. For more on this topic, see <a href="Statistical_model_validation" title="Statistical model validation">statistical model validation</a>.<a href="#cite_note-93">[93]</a></li>

<li><a href="Sensitivity_analysis" title="Sensitivity analysis">Sensitivity analysis</a>. A procedure to study the behavior of a system or model when global parameters are (systematically) varied. One way to do that is via <a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">bootstrapping</a>.<a href="#cite_note-94">[94]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Free_software_for_data_analysis">Free software for data analysis</h2></div>

Free software for data analysis include:

<ul><li><a href="DevInfo" title="DevInfo">DevInfo</a> – A database system endorsed by the <a href="United_Nations_Development_Group" class="mw-redirect" title="United Nations Development Group">United Nations Development Group</a> for monitoring and analyzing human development.<a href="#cite_note-95">[95]</a></li>

<li><a href="ELKI" title="ELKI">ELKI</a> – Data mining framework in Java with data mining oriented visualization functions.</li>

<li><a href="KNIME" title="KNIME">KNIME</a> – The Konstanz Information Miner, a user friendly and comprehensive data analytics framework.</li>

<li><a href="Orange_(software)" title="Orange (software)">Orange</a> – A visual programming tool featuring <a href="Interactive_data_visualization" class="mw-redirect" title="Interactive data visualization">interactive data visualization</a> and methods for statistical data analysis, <a href="Data_mining" title="Data mining">data mining</a>, and <a href="Machine_learning" title="Machine learning">machine learning</a>.</li>

<li><a href="Pandas_(software)" title="Pandas (software)">Pandas</a> – Python library for data analysis.</li>

<li><a href="Physics_Analysis_Workstation" title="Physics Analysis Workstation">PAW</a> – FORTRAN/C data analysis framework developed at <a href="CERN" title="CERN">CERN</a>.</li>

<li><a href="R_(programming_language)" title="R (programming language)">R</a> – A programming language and software environment for statistical computing and graphics.<a href="#cite_note-96">[96]</a></li>

<li><a href="ROOT" title="ROOT">ROOT</a> – C++ data analysis framework developed at <a href="CERN" title="CERN">CERN</a>.</li>

<li><a href="SciPy" title="SciPy">SciPy</a> – Python library for scientific computing.</li>

<li><a href="Julia_(programming_language)" title="Julia (programming language)">Julia</a> – A programming language well-suited for numerical analysis and computational science.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Reproducible_analysis">Reproducible analysis</h2></div>

The typical data analysis workflow involves collecting data, running analyses, creating visualizations, and writing reports. However, this workflow presents challenges, including a separation between analysis scripts and data, as well as a gap between analysis and documentation. Often, the correct order of running scripts is only described informally or resides in the data scientist's memory. The potential for losing this information creates issues for reproducibility.

To address these challenges, it is essential to document analysis script content and workflow. Additionally, overall documentation is crucial, as well as providing reports that are understandable by both machines and humans, and ensuring accurate representation of the analysis workflow even as scripts evolve.<a href="#cite_note-97">[97]</a>

<div class="mw-heading mw-heading2"><h2 id="Data_analysis_contests">Data analysis contests</h2></div>

Different companies and organizations hold data analysis contests to encourage researchers to utilize their data or to solve a particular question using data analysis. A few examples of well-known international data analysis contests are:

<ul><li><a href="Kaggle" title="Kaggle">Kaggle</a> competitions; the <a href="Kaggle" title="Kaggle">Kaggle</a> platform is owned and run by <a href="Google" title="Google">Google</a>.<a href="#cite_note-98">[98]</a></li>

<li><a href="LTPP_International_Data_Analysis_Contest" class="mw-redirect" title="LTPP International Data Analysis Contest">LTPP data analysis contest</a><a href="#cite_note-99">[99]</a> held by <a href="FHWA" class="mw-redirect" title="FHWA">FHWA</a> and <a href="ASCE" class="mw-redirect" title="ASCE">ASCE</a>.<a href="#cite_note-Nehme_2016-09-29-100">[100]</a></li></ul>

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

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

<li><a href="Augmented_Analytics" title="Augmented Analytics">Augmented Analytics</a></li>

<li><a href="Business_intelligence" title="Business intelligence">Business intelligence</a></li>

<li><a href="Data_presentation_architecture" class="mw-redirect" title="Data presentation architecture">Data presentation architecture</a></li>

<li><a href="Exploratory_data_analysis" title="Exploratory data analysis">Exploratory data analysis</a></li>

<li><a href="Machine_learning" title="Machine learning">Machine learning</a></li>

<li><a href="Multiway_data_analysis" title="Multiway data analysis">Multiway data analysis</a></li>

<li><a href="Qualitative_research" title="Qualitative research">Qualitative research</a></li>

<li><a href="Structured_data_analysis_(statistics)" title="Structured data analysis (statistics)">Structured data analysis (statistics)</a></li>

<li><a href="Text_mining" title="Text mining">Text mining</a></li>

<li><a href="Unstructured_data" title="Unstructured data">Unstructured data</a></li>

<li><a href="List_of_datasets_for_machine-learning_research" title="List of datasets for machine-learning research">List of datasets for machine-learning research</a></li></ul>

</div>

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

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

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

<ul><li><cite id="CITEREFAdèr2008a" class="citation book cs1"><a href="Herman_J._Ad%C3%A8r" title="Herman J. Adèr">Adèr, Herman J.</a> (2008a). "Chapter 14: Phases and initial steps in data analysis". In Adèr, Herman J.; <a href="Gideon_J._Mellenbergh" title="Gideon J. Mellenbergh">Mellenbergh, Gideon J.</a>; <a href="David_Hand_(statistician)" title="David Hand (statistician)">Hand, David J</a> (eds.). Advising on research methods : a consultant's companion. Huizen, Netherlands: Johannes van Kessel Pub. pp. 333–356. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9789079418015</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/905799857">905799857</a>.</cite></li>

<li><cite id="CITEREFAdèr2008b" class="citation book cs1"><a href="Herman_J._Ad%C3%A8r" title="Herman J. Adèr">Adèr, Herman J.</a> (2008b). "Chapter 15: The main analysis phase". In Adèr, Herman J.; <a href="Gideon_J._Mellenbergh" title="Gideon J. Mellenbergh">Mellenbergh, Gideon J.</a>; <a href="David_Hand_(statistician)" title="David Hand (statistician)">Hand, David J</a> (eds.). Advising on research methods : a consultant's companion. Huizen, Netherlands: Johannes van Kessel Pub. pp. 357–386. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9789079418015</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/905799857">905799857</a>.</cite></li>

<li>Tabachnick, B.G. & Fidell, L.S. (2007). Chapter 4: Cleaning up your act. Screening data prior to analysis. In B.G. Tabachnick & L.S. Fidell (Eds.), Using Multivariate Statistics, Fifth Edition (pp. 60–116). Boston: Pearson Education, Inc. / Allyn and Bacon.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<div class="side-box-text plainlist">Wikiversity has learning resources about <a href="https://en.wikiversity.org/wiki/Special:Search/Data_analysis" class="extiw external" title="v:Special:Search/Data analysis">Data analysis</a></div></div>

</div>

<ul><li><a href="Ad%C3%A8r%2C_H.J." class="mw-redirect" title="Adèr, H.J.">Adèr, H.J.</a> & <a href="Gideon_J._Mellenbergh" title="Gideon J. Mellenbergh">Mellenbergh, G.J.</a> (with contributions by D.J. Hand) (2008). Advising on Research Methods: A Consultant's Companion. Huizen, the Netherlands: Johannes van Kessel Publishing. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-90-79418-01-5</bdi></li>

<li>Chambers, John M.; Cleveland, William S.; Kleiner, Beat; Tukey, Paul A. (1983). Graphical Methods for Data Analysis, Wadsworth/Duxbury Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-534-98052-X</bdi></li>

<li>Fandango, Armando (2017). Python Data Analysis, 2nd Edition. Packt Publishers. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1787127487</bdi></li>

<li>Juran, Joseph M.; Godfrey, A. Blanton (1999). Juran's Quality Handbook, 5th Edition. New York: McGraw Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-034003-X</bdi></li>

<li>Lewis-Beck, Michael S. (1995). Data Analysis: an Introduction, Sage Publications Inc, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8039-5772-6</bdi></li>

<li>NIST/SEMATECH (2008) <a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/">Handbook of Statistical Methods</a></li>

<li>Pyzdek, T, (2003). Quality Engineering Handbook, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8247-4614-7</bdi></li>

<li><a href="Richard_Veryard" title="Richard Veryard">Richard Veryard</a> (1984). Pragmatic Data Analysis. Oxford : Blackwell Scientific Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-632-01311-7</bdi></li>

<li>Tabachnick, B.G.; Fidell, L.S. (2007). Using Multivariate Statistics, 5th Edition. Boston: Pearson Education, Inc. / Allyn and Bacon, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-205-45938-4</bdi></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Data8" style="padding:3px"><table class="nowraplinks 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="Data8" style="font-size:114%;margin:0 4em"><a href="Data" title="Data">Data</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Data_acquisition" title="Data acquisition">Acquisition</a></li>

<li><a href="Data_augmentation" title="Data augmentation">Augmentation</a></li>

<li><a href="Data_anonymization" title="Data anonymization">Anonymization</a></li>

<li><a href="Data_archaeology" title="Data archaeology">Archaeology</a></li>

<li><a href="Data_cleansing" title="Data cleansing">Cleansing</a></li>

<li><a href="Data_collection" title="Data collection">Collection</a></li>

<li><a href="Data_compression" title="Data compression">Compression</a></li>

<li><a href="Data_corruption" title="Data corruption">Corruption</a></li>

<li><a href="Data_curation" title="Data curation">Curation</a></li>

<li><a href="Data_deduplication" title="Data deduplication">Deduplication</a></li>

<li><a href="Data_degradation" title="Data degradation">Degradation</a></li>

<li><a href="Data_de-identification" class="mw-redirect" title="Data de-identification">De-identification</a></li>

<li><a href="Data_ecosystem" title="Data ecosystem">Ecosystem</a></li>

<li><a href="Data_editing" title="Data editing">Editing</a></li>

<li><a href="Data_engineering" title="Data engineering">Engineering</a></li>

<li><a href="Data_erasure" title="Data erasure">Erasure</a></li>

<ul><li><a href="Data_extraction" title="Data extraction">Extract</a></li>

<li><a href="Data_transformation" class="mw-redirect" title="Data transformation">Transform</a></li>

<li><a href="Data_ethics" class="mw-redirect" title="Data ethics">Ethics</a></li>

<li><a href="Data_exhaust" title="Data exhaust">Exhaust</a></li>

<li><a href="Data_exploration" title="Data exploration">Exploration</a></li>

<li><a href="Data_farming" title="Data farming">Farming</a></li>

<li><a href="Data_format_management" title="Data format management">Format management</a></li>

<li><a href="Data_fusion" title="Data fusion">Fusion</a></li>

<li><a href="Data_governance" title="Data governance">Governance</a>

<ul><li><a href="Data_cooperative" title="Data cooperative">Cooperatives</a></li></ul></li>

<li><a href="Data_infrastructure" title="Data infrastructure">Infrastructure</a></li>

<li><a href="Data_integration" title="Data integration">Integration</a></li>

<li><a href="Data_integrity" title="Data integrity">Integrity</a></li>

<li><a href="Data_library" class="mw-redirect" title="Data library">Library</a></li>

<li><a href="Data_lineage" title="Data lineage">Lineage</a></li>

<li><a href="Data_management" title="Data management">Management</a></li>

<li><a href="Data_migration" title="Data migration">Migration</a></li>

<li><a href="Data_mining" title="Data mining">Mining</a></li>

<li><a href="Data_philanthropy" title="Data philanthropy">Philanthropy</a></li>

<li><a href="Data_pre-processing" class="mw-redirect" title="Data pre-processing">Pre-processing</a></li>

<li><a href="Data_preservation" title="Data preservation">Preservation</a></li>

<li><a href="Data_processing" title="Data processing">Processing</a></li>

<li><a href="Information_privacy" title="Information privacy">Protection (privacy)</a></li>

<li><a href="Data_publishing" title="Data publishing">Publishing</a>

<li><a href="Data_recovery" title="Data recovery">Recovery</a></li>

<li><a href="Data_reduction" title="Data reduction">Reduction</a></li>

<li><a href="Data_redundancy" title="Data redundancy">Redundancy</a></li>

<li><a href="Data_re-identification" title="Data re-identification">Re-identification</a></li>

<li><a href="Data_remanence" title="Data remanence">Remanence</a></li>

<li><a href="Data_rescue" title="Data rescue">Rescue</a></li>

<li><a href="Data_retention" title="Data retention">Retention</a></li>

<li><a href="Data_quality" title="Data quality">Quality</a></li>

<li><a href="Data_science" title="Data science">Science</a></li>

<li><a href="Data_scraping" title="Data scraping">Scraping</a></li>

<li><a href="Data_scrubbing" title="Data scrubbing">Scrubbing</a></li>

<li><a href="Data_security" title="Data security">Security</a></li>

<li><a href="Data_sharing" title="Data sharing">Sharing</a></li>

<li><a href="Data_steward" title="Data steward">Stewardship</a></li>

<li><a href="Data_storage" title="Data storage">Storage</a></li>

<li><a href="Data_structure" title="Data structure">Structure</a></li>

<li><a href="Data_synchronization" title="Data synchronization">Synchronization</a></li>

<li><a href="Topological_data_analysis" title="Topological data analysis">Topological data analysis</a></li>

<li><a href="Data_validation" title="Data validation">Validation</a></li>

<li><a href="Data_warehouse" title="Data warehouse">Warehouse</a></li>

<li><a href="Data_wrangling" title="Data wrangling">Wrangling/munging</a></li></ul>

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

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