Normal Approximation to the Binomial Distribution — Mathematics

Introduction

In many real-world problems, the number of trials $n$ in a binomial distribution is so large that calculating exact binomial probabilities becomes impractical. The normal distribution provides a powerful and examinable approximation in these cases. The 9709 syllabus requires you to know precisely when the approximation is valid, which normal distribution to use, and how to apply the all-important continuity correction that adjusts for the fact that you are approximating a discrete distribution with a continuous one.

Core Concept

Suppose $X \sim \text{B}(n, p)$ , where $q = 1 - p$ . As $n$ grows large, the binomial distribution becomes increasingly bell-shaped and symmetric, and begins to resemble a normal distribution. This happens provided both tails of the distribution have enough probability mass — which the conditions $np > 5$ and $nq > 5$ ensure.

The approximation: When the conditions are met, we use

X \approx Y, \quad \text{where } Y \sim \text{N}(\mu, \sigma^2)

with $\mu = np$ and $\sigma^2 = npq$ .

The continuity correction is essential. Because $X$ is discrete (it takes integer values), the event $X = k$ corresponds to a unit-wide interval. When we switch to the continuous variable $Y$ , we replace each integer $k$ with the interval $(k - 0.5,\; k + 0.5)$ . This is not optional — failing to apply it in an exam will cost marks.

Binomial event	Approximating normal event
$P(X = k)$	$P(k - 0.5 < Y < k + 0.5)$
$P(X \leq k)$	$P(Y < k + 0.5)$
$P(X < k)$	$P(Y < k - 0.5)$
$P(X \geq k)$	$P(Y > k - 0.5)$
$P(X > k)$	$P(Y > k + 0.5)$
$P(a \leq X \leq b)$	$P(a - 0.5 < Y < b + 0.5)$

A useful mental check: inequalities that include $k$ move the boundary towards $k$ by 0.5; inequalities that exclude $k$ move the boundary away from $k$ by 0.5.

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Introduction

Core Concept

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