Normal Approximation to the Binomial

For large $n$ , the binomial's bars trace out a bell curve. Replace $B(n,p)$ by $N(np,npq)$ , widen each whole number into an interval of width $1$ (the continuity correction), and standardise.

Cambridge 9709 Paper 5Edexcel YMA01 WST01Edexcel 9MA0 Paper 3AQA 7357 Paper 3≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

State the conditions $np>5$ and $nq>5$ for the approximation.
Use $N(np,\,npq)$ as an approximation to $B(n,p)$ .
Apply a continuity correction to any inequality.
Find approximate binomial probabilities by standardising.

Pause and think

$X\sim B(100,0.4)$ . Working out $P(X\le45)$ exactly means adding $46$ binomial terms. Why might a smooth curve do almost as well?

Section 2 of 14

Core idea

Start hereToo many terms to add

$X\sim B(100,0.5)$ : $P(X\le45)$ means adding 46 binomial terms. For large $n$ the binomial bars form a bell shape, so a normal curve with the same mean and variance gives an excellent, quick approximation.

The big idea

If $np>5$ and $n(1-p)>5$ : $B(n,p)\approx N(np,\,np(1-p))$ . Use a continuity correction: integer $k$ becomes the interval $k-0.5$ to $k+0.5$ .

1. Why it works

B(30, 0.4) bars with the normal curve N(12, 7.2)

Mean

np=12

, variance

np(1-p)=7.2

. The curve hugs the tops of the bars.

2. Continuity correction

Bars become areas

$P(X=k)\text{ is a bar of width }1\text{ centred at }k$
Discrete distribution as a histogram.
$\text{bar spans }k-0.5\text{ to }k+0.5$
Width 1.
$P(X\le10)\approx P(Y<10.5),\quad P(X<10)=P(X\le9)\approx P(Y<9.5)$
Include whole bars.

Worked example $X\sim B(30,0.4)$ , $P(X\le10)$

Approximate.

Goal: Probability.
Strategy: Check conditions; normal; correct; standardise.

$np=12,\ n(1-p)=18$
Both $>5$ ✓.
$Y\sim N(12,7.2)$
Same mean and variance.
$P(Y<10.5),\ z=\tfrac{10.5-12}{\sqrt{7.2}}=-0.559$
Correction; standardise.
$\Phi(-0.559)=0.288$
Table.

Answer

0.288

✓ CheckExact binomial

0.291

✓.

Common misconceptionForgetting the correction Without it:

\Phi\left(\frac{10-12}{2.683}\right)=0.228

, much worse.

Pause and think

$X\sim B(100,0.5)$ : approximate $P(45\le X\le55)$ .

Summary

$N(np,npq)$ when $np,nq>5$ .
±0.5 correction.

Continuity correction: Adjusting discrete values to intervals.

HistoryDe Moivre's discovery. In 1733 Abraham de Moivre wanted the probability of getting between, say,

480

and

520

heads in

1000

tosses. Adding hundreds of binomial terms by hand was hopeless, so he found a smooth curve that approximated them: the first appearance of the normal distribution.

Fun factWhy bell curves are everywhere. De Moivre's result is a special case of the central limit theorem: adding up many small, independent random effects tends to give a normal distribution. A binomial is a sum of

n

success-or-failure trials, which is why it becomes bell-shaped. Heights, measurement errors and exam totals are bell-shaped for the same reason.

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