Mean, variance and standard deviation

Two numbers summarise a discrete distribution: where its “centre” lies and how much the values spread around it.

Property — Mean, variance, standard deviation

• Mean (expected value): $\mu = E[X] = \sum_i x_i\,p_i$.
• Variance: $\sigma^2 = E[(X-\mu)^2] = E[X^2]-\mu^2 = \sum_i (x_i-\mu)^2 p_i$.
• Standard deviation: $\sigma = \sqrt{\sigma^2}$.

$\mu$ measures the “centre” of the distribution; $\sigma$ measures the “width” (how much the values spread around the mean).

The alternative formula $\sigma^2 = E[X^2]-\mu^2$ is almost always the most convenient for calculations: you compute the mean of the squares and subtract the square of the mean.

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Topics: Probability distributions
Concepts: Standard deviation · Probability distribution · Expected value · Variance
Skills: Using formulae