LearnYMA01WST02 · Statistics 2Continuous distributionsNormal approximations to the binomial and Poisson

Continuous distributions · Subtopic 2 of 2

Normal Approximations to the Binomial and Poisson

For large $n$ with $p$ near $0.5$ , $B(n,p)$ is close to $N(np,\,np(1-p))$ . For large $\lambda$ , $\text{Po}(\lambda)$ is close to $N(\lambda,\lambda)$ . Because you're replacing whole numbers with a continuous curve, widen each value to an interval of half a unit either side: the continuity correction.

Edexcel YMA01 WST02≈ 40 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Know when a normal approximation is suitable.
Write down the approximating normal distribution.
Apply the continuity correction correctly.
Choose between a Poisson and a normal approximation for a binomial.

Pause and think

On a bar chart, the bar for $X=20$ stretches from $19.5$ to $20.5$ . If you replace the bars with a smooth curve, which interval of the curve should stand for " $X\le20$ "?

Section 2 of 14

Core idea

Start hereBars into a curve

The bars of $B(50,0.4)$ form a bell shape. A normal curve fits them closely, but bars are discrete: the bar for 15 covers $14.5$ to $15.5$ . Remembering that half-unit is the continuity correction.

The big idea

$B(n,p)\approx N(np,np(1-p))$ ( $n$ large, $p$ near 0.5); $\operatorname{Po}(\lambda)\approx N(\lambda,\lambda)$ ( $\lambda$ large); $B(n,p)\approx\operatorname{Po}(np)$ ( $n$ large, $p$ small). Normal approximations need ±0.5.

1. Continuity correction

Continuity correction: P(X ≤ 15) uses the area up to 15.5

X\sim\text{Po}(16)

. The bar for

15

runs from

14.5

15.5

, so "

X\le15

" includes everything up to

15.5

P(X\le k)\approx P(Y<k+0.5)\qquad P(X\ge k)\approx P(Y>k-0.5)

$X$: discrete variable.
$Y$: approximating normal.

Worked example $X\sim\operatorname{Po}(16)$ , $P(X\le15)$

Approximate.

Goal: Probability.
Strategy: $N(16,16)$ with correction.

$P(Y<15.5)$
Include the whole bar for 15.
$z=\frac{15.5-16}4=-0.125$
SD $4$ .
$\Phi(-0.125)=0.450$
Table.

Answer

0.450

✓ CheckJust below

0.5

since 15 is just below the mean ✓.

Worked example $X\sim B(50,0.4)$ , $P(X>25)$

Approximate.

Goal: Probability.
Strategy: $N(20,12)$ .

$X>25\iff X\ge26\to P(Y>25.5)$
Strict inequality: start at 26.
$z=\frac{5.5}{3.464}=1.588$
$\sqrt{12}=3.464$ .
$1-0.9438=0.056$
Upper tail.

Answer

0.056

✓ Check

np=20

nq=30

, both

>5

: approximation valid ✓.

Common misconception $X<10$ Means

X\le9

: use

P(Y<9.5)

, not

10.5

Pause and think

Continuity-correct $P(X=12)$ .

Summary

Match mean and variance.
±0.5 correction.

Continuity correction: Adjusting discrete values to intervals.

HistoryDe Moivre's discovery. In 1733 Abraham de Moivre found that the binomial distribution for many coin tosses is closely approximated by a bell-shaped curve. It was the first appearance of the normal distribution, decades before Gauss used it for errors.

Fun factThe Galton board. Drop balls through rows of pegs, each bouncing left or right at random, and they pile up in a bell shape. Each ball's final position is binomial; with many rows the pile looks normal.

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