ProbabilityConcept reference

Variance

Variance is the expected squared deviation from the mean.

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01Variance measures spread around the mean μ=E[X]#

Variance measures spread around the mean $\mu=E[X]$ . Take each deviation $x-\mu$ , square it, and average those squares using the probabilities.

\begin{gathered}\operatorname{Var}(X)\\=E[(X-\mu)^2]\end{gathered}

If $1$ and $5$ are equally likely, the mean is $3$ . Both squared deviations are $4$ , so the variance is $4$ .

Check your reasoning

PMF: $p(0)=0.5$ , $p(2)=0.5$ . Mean $\mu=1$ . Find $\operatorname{Var}(X)$ .

Show answer and explanation

Squared deviations: $1, 1$ . Weighted sum: $0.5+0.5=1$ .

Unequal probabilities still matter after squaring. A rare large deviation contributes its squared size multiplied by its probability.

These exercises describe a probability model. Use its supplied masses; no sample correction is involved.

Check your reasoning

A two-state model gives $X=0$ with probability 0.75 and $X=4$ otherwise. Mean $\mu=1$ . Find $\operatorname{Var}(X)$ .

Show answer and explanation

Squared deviations: $1, 9$ . Weighted sum: $0.75+2.25=3$ .

Signed deviations average to zero because positive and negative terms cancel. Squaring before averaging prevents that cancellation.

Variance is nonnegative. It is zero when the model puts all probability at its mean. Its units are the original units squared.

Check your reasoning

PMF: $p(-3)=0.5$ , $p(3)=0.5$ . Mean $\mu=0$ . A learner averages signed deviations and gets zero. Find the variance.

Show answer and explanation

Squared deviations: $9, 9$ . Weighted sum: $4.5+4.5=9$ .

Key takeaway

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Pishro-Nik, Introduction to Probability: Variance ↗Pishro-Nik: Introduction to Probability · Article