# Variance

the variance, denoted by σ2, is calculated as

σ2 = E{[X – E(X)]2}

= P(Xi)∙[Xi – E(X)]2

= P(X1)∙[X1 – E(X)]2 + P(X2)∙[X2 – E(X)]2 + …

+ P(Xn)∙[Xn – E(X)]2

We calculate the standard deviation, denoted by σ, by taking the square root of the variance:

σ = σ2.

### Methode

Hier klicken zum Ausklappen

The standard deviation is still a better indicator of dispersion than the variance, because its dimension is the same as the dimension of the dataset.

### Beispiel

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Consider the following probability distribution from the former Example. Find the variance and the standard deviation.

We calculate

σ2 = E{[X – E(X)]2}

= P(Xi)∙[Xi – E(X)]2

= P(X1)∙[X1 – E(X)]2 + P(X2)∙[X2 – E(X)]2 + …

+ P(Xn)∙[Xn – E(X)]2

= P(X1)∙[X1 – E(X)]2 + P(X2)∙[X2 – E(X)]2 +

= P(X1)∙[X1 – E(X)]2 + P(X2)∙[X2 – E(X)]2 +

+ P(X3)∙[X3 – E(X)]2+ P(X4)∙[X4 – E(X)]2

= 0.25∙[-3 - 0.05]2 + 0.4∙[0 - 0.05]2 + 0.3∙[2 - 0.05]2

+ 0.05∙[4 - 0.05]2

= 4.2475.

Therefore, the standard deviation is equal to

σ = σ2 = √4.2475= 2.0609.