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A real number x {\displaystyle x} can be mutated using normal distribution N ( 0 , σ σ ) {\displaystyle {\mathcal {N}}(0,\sigma )} by adding the generated random value to the old value of the gene, resulting in the mutated value x ′ {\displaystyle x'} :

x ′ = x + N ( 0 , σ σ ) {\displaystyle x'=x+{\mathcal {N}}(0,\sigma )}

In the case of genes with a restricted range of values, it is a good idea to choose the step size of the mutation σ σ {\displaystyle \sigma } so that it reasonably fits the range [ x min , x max ] {\displaystyle [x_{\min },x_{\max }]} of the gene to be changed, e.g.:

σ σ = x max − − x min 6 {\displaystyle \sigma ={\frac {x_{\text{max}}-x_{\text{min}}}{6}}}

The step size can also be adjusted to the smaller permissible change range depending on the current value. In any case, however, it is likely that the new value x ′ {\displaystyle x'} of the gene will be outside the permissible range of values. Such a case must be considered a lethal mutation, since the obvious repair by using the respective violated limit as the new value of the gene would lead to a drift. This is because the limit value would then be selected with the entire probability of the values beyond the limit of the value range.

The evolution strategy works with real numbers and mutation based on normal distribution. The step sizes are part of the chromosome and are subject to evolution together with the actual decision variables.cite-ref-15[15]cite-ref-16[16]

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