Year 12 Maths Extension 1 (2027) Statistical analysis

Mean & variance of a binomial distribution

Q: What are the mean and variance of a binomial?

The mean is n times p and the variance is n times p times one minus p.

Q: What is the standard deviation?

The square root of n times p times one minus p.

Q: Why do the formulas work?

X is a sum of n independent Bernoulli variables, so means and variances add.

Q: How do you recover p from mean and variance?

Use the variance divided by the mean equals one minus p.

20 practice questions 2 video lessons Theory + worked examples

Learn the mean and variance of a binomial distribution for NSW Year 12 Mathematics Extension 1. From the number of trials and the probability of success, you can predict a binomial variable's average and spread directly.

You will learn to apply the results for the expected value and variance of a binomial distribution, and use them to solve practical probability problems — an efficient statistical tool in the Extension 1 course.

Practice 20 questions

Practice questions

Every question with a fully worked solution.

Start practising

Watch 2 video(s)

Binomial distribution expected value variance and standard deviation Watch
The Binomial Distribution: Mathematically Deriving the Mean and Variance Watch

Theory

For $X \sim Bin (n, p)$ , the mean is $n p$ , the variance $n p (1 - p)$ , and the standard deviation $\sqrt{n p (1 - p)}$ — because $X$ is a sum of $n$ Bernoulli trials. This NSW Year 12 Mathematics Extension 1 topic is NESA outcome ME1-12-06.

For a binomial variable $X \sim Bin (n, p)$ , the mean and variance follow simple formulas, because $X$ is the sum of $n$ identical Bernoulli variables.

E (X) = μ = n p, Var (X) = σ^{2} = n p (1 - p), σ = \sqrt{n p (1 - p)} .

Since $X = X_{1} + \dots + X_{n}$ (each Bernoulli, mean $p$ , variance $p q$ ), the means add and — because the trials are independent — the variances add, giving $n p$ and $n p q$ .

NESA link. Part of the Year 12 The binomial distribution and sampling distribution of the mean focus area, outcome ME1-12-06 ("solves problems involving binomial distributions, sampling distribution of the mean and the central limit theorem") with MAO-WM-01.

Centre

μ = n p

; spread

σ = \sqrt{n p (1 - p)}

X

is a sum of

n

Bernoulli trials, so means and variances add.

E (X) = n p, Var (X) = n p (1 - p), σ = \sqrt{n p (1 - p)} .

E(X) = np, Var(X) = np(1-p), sigma = sqrt(np(1-p))

For reverse problems, the ratio recovers $p$ in one step:

\frac{Var (X)}{μ} = \frac{n p (1 - p)}{n p} = 1 - p .

Var(X)/mean = 1 - p

These are for $Bin (n, p)$ only. The standard deviation is $\sqrt{n p (1 - p)}$ ; don't forget the square root.

How to find or use mean and variance

Forward: $μ = n p$ , $σ^{2} = n p (1 - p)$ , $σ = \sqrt{n p (1 - p)}$ .
Reverse: use $\frac{Var}{mean} = 1 - p$ to find $p$ .
Then find $n$ from $n = \frac{μ}{p}$ .
Check the values are consistent (\(0

Example 1 — All three

X \sim Bin (48, 0.25)

. Find the mean, variance and standard deviation.

Solution

$μ$	$=$	$48 \times 0.25 = 12$
$σ^{2}$	$=$	$48 \times 0.25 \times 0.75 = 9$
$σ$	$=$	$3$

mean 12, variance 9, sd 3

$μ = 12$ , $σ^{2} = 9$ , $σ = 3$ .

Example 2 — Fractional p

X \sim Bin (45, \frac{1}{3})

. Find the mean and standard deviation.

Solution

$μ$	$=$	$45 \times \frac{1}{3} = 15$
$σ^{2}$	$=$	$45 \times \frac{1}{3} \times \frac{2}{3} = 10$
$σ$	$=$	$\sqrt{10}$

mean 15, sd sqrt(10)

$μ = 15$ , $σ = \sqrt{10}$ .

Example 3 — Reverse problem

X \sim Bin (n, p)

has mean

15

and variance

12

. Find

n

and

p

Solution

$1 - p$	$=$	$\frac{12}{15} = 0.8 \Rightarrow p = 0.2$
$n$	$=$	$\frac{15}{0.2} = 75$

n = 75, p = 0.2

$n = 75$ , $p = 0.2$ .

Example 4 — Find n and variance

X \sim Bin (n, 0.4)

has mean

10

. Find

n

and the variance.

Solution

$n$	$=$	$\frac{10}{0.4} = 25$
$σ^{2}$	$=$	$25 \times 0.4 \times 0.6 = 6$

n = 25, variance 6

$n = 25$ , $σ^{2} = 6$ .

Common pitfalls

Standard deviation. It is

\sqrt{n p (1 - p)}

— take the square root of the variance.

Reverse shortcut.

\frac{Var}{mean} = 1 - p

gives

p

in one step.

Only for binomials.

n p

and

n p (1 - p)

apply to

Bin (n, p)

, not a general distribution.

Consistency. Check \(0

Frequently asked questions

What are the mean and variance of a binomial?

$E (X) = n p$ and $Var (X) = n p (1 - p)$ .

What is the standard deviation?

$σ = \sqrt{n p (1 - p)}$ .

Why do the formulas work?

$X$ is a sum of $n$ independent Bernoulli variables, so means and variances add.

How do you recover p from mean and variance?

Use $\frac{Var}{mean} = 1 - p$ .

How do you find n?

Divide the mean by $p$ : $n = \frac{μ}{p}$ .

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Sampling distribution of the mean & Central Limit Theorem

Mean & variance of a binomial distribution

📖 Theory

How to find or use mean and variance

Common pitfalls

Frequently asked questions

Theory