Mean and variance of the binomial distribution

7 minAS Level5.4.25.4.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can state and use $E(X) = np$ and $Var(X) = np(1-p)$ for $X \sim B(n, p)$ , and work backwards from a given mean or variance to find $n$ or $p$ .

Two formulae worth memorising

For $X \sim B(n, p)$ there is no need to build a table and apply $\sum x P(X = x)$ . Two standard results give the mean and variance directly:

$E(X) = np \qquad Var(X) = np(1 - p)$

The syllabus does not require proofs, but the mean is easy to believe: if each of $n$ trials succeeds with probability $p$ , then in the long run you expect $np$ successes out of $n$ . Ten trials with $p = 0.3$ should give about $3$ successes.

Why the variance has the (1 - p) factor

Read $p(1-p)$ as the uncertainty in a single trial. It is largest when $p = 0.5$ , where the outcome is as unpredictable as possible, and it falls to $0$ when $p$ is $0$ or $1$ , where the result is certain.

The $n$ factor says that the uncertainty builds up across the trials: twice as many trials gives twice the variance.

So a binomial variable is most spread out when $p = 0.5$ and least spread out when the trial is nearly certain either way. That is exactly what the formula predicts.

Worked example: Example 1: mean and variance from n and p

$X \sim B(20, 0.4)$ . Find $E(X)$ , $Var(X)$ and the standard deviation of $X$ .

Have a go first?

Check 1 — applying the formulae

CalculateQuestion 1 · Recall

CalculateQuestion 2 · Apply

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