variance

(NL: variantie)

The variance is a measure of dispersion that is ideally suited for quantitative variables that are normally distributed.

The standard deviation (NL: standaardafwijking) is also calculated from the variance, which is used even more often.

The variance and standard deviation are sensitive to outliers. For data that is not normally distributed, mean and standard deviation are not a good summary of the data. In that case, the median and interquartile range can serve as alternatives.

variance of a sample

The variance \(s^2\) of a sample \(X = \{x_1, \ldots, x_n\}\) of size \(n\) with mean \(\overline{x}\) is calculated as follows:

\[s^2 = \frac{1}{n - 1} \sum_{i=1}^{n} (x_i - \overline{x})^2\]

standard deviation of a sample

The standard deviation is the square root of the variance:

\[s = \sqrt{s^2} = \sqrt{\frac{1}{n - 1} \sum_{i=1}^{n} (x_i - \overline{x})^2}\]

variance of a population

The variance \(\sigma\) of a population of size \(N\) with mean \(\mu\) is calculated as follows:

\[\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2\]

standard deviation of a population

The standard deviation is the square root of the variance:

\[\sigma = \sqrt{\sigma^2} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2}\]

variance and standard deviation of a discrete stochastic variable

The variance \(\sigma_X^2\) of a discrete stochastic variable \(X\) with expected value \(\mu_X\) is calculated as follows:

\[\sigma_X^2 = \sum_x (x - \mu_X)^2 \cdot P(X = x)\]

The standard deviation is then \(\sigma_X = \sqrt{\sigma_X^2}\).

variance and standard deviation of a continuous stochastic variable

The variance \(\sigma_X^2\) of a continuous stochastic variable \(X\) with expected value \(\mu_X\) is calculated as follows:

\[\sigma_X^2 = \int_{-\infty}^{+\infty} (x - \mu_X)^2 \cdot f_X(x) \, \mathrm{d}x\]

Again, the standard deviation is \(\sigma_X = \sqrt{\sigma_X^2}\).