How Do You Find the Mean and Variance of a Binomial Distribution?

For a binomial distribution $X \sim B(n,p)$, the mean is $E(X)=np$ and the variance is $\operatorname{Var}(X)=np(1-p)$. Here, $n$ is the number of trials and $p$ is the constant probability of success on each trial.

The Reasoning

A binomial random variable counts successes across $n$ trials. The model is appropriate only when there is a fixed number of independent trials, each trial has two possible outcomes, and the probability of success remains constant.

It is often useful to define $q=1-p$, the probability of failure. The formulas can then be written as $E(X)=np$ and $\operatorname{Var}(X)=npq$. This notation emphasizes that the spread depends on both success and failure probabilities, not only on the expected number of successes.

The expected number of successes in one trial is $p$. Across $n$ trials, expected values add, producing the mean:

$$\mu=E(X)=np.$$

For one trial, the variance is $p(1-p)$. Since the trials are independent, their variances add, producing the variance:

$$\sigma^2=\operatorname{Var}(X)=np(1-p).$$

Worked Example

Suppose $X \sim B(20,0.3)$. Then

$$E(X)=20(0.3)=6,$$

so the expected number of successes is $6$. Also,

$$\operatorname{Var}(X)=20(0.3)(1-0.3)=4.2.$$

In the example, a variance of 4.2 is not a count. If the standard deviation is required,

$$\sigma=\sqrt{4.2}=2.05 \quad \text{(3 s.f.)}.$$

A common misconception is that $np(1-p)$ is the standard deviation. It is the variance; take its square root to obtain the standard deviation.

Exam Technique

This is SL 4.8: Binomial distribution. Identify $n$ and $p$, show the formula before evaluating it, and interpret the mean in context. Check whether the command asks for variance $\sigma^2$ or standard deviation $\sigma$, and retain full GDC precision until the final answer.