3.6StatisticsStretch

Approximating a Binomial Distribution

When n is large and p is close to 0.5, a binomial distribution is well-approximated by a normal distribution. A continuity correction bridges the discrete and continuous models.

26 min The Normal Distribution

Edexcel A Level Maths: 3.6 Approximating the Binomial DistributionCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

State the conditions for a normal approximation
Find μ and σ from the binomial
Apply a continuity correction
Compute the approximate probability

The approximation

If $X \sim B(n, p)$ with $n$ large and $p$ near $0.5$ , then $X$ is approximately $N(np,\ np(1-p))$ — the normal with the same mean and variance.

X \sim B(n,p) \ \approx\ N\big(np,\ np(1-p)\big)

Mean np, variance np(1−p).

Continuity correction

Because binomial is discrete and normal is continuous, adjust by $\pm 0.5$ . For example, $P(X \ge 6)$ becomes $P(Y > 5.5)$ and $P(X \le 6)$ becomes $P(Y < 6.5)$ .

Example 1

X \sim B(100, 0.5)

. State the approximation and the correction for

P(X \le 60)

\mu = 50

\sigma^2 = 25

, so

Y \sim N(50, 25)

2Continuity:

P(Y < 60.5)

Answer

Y \sim N(50, 25)

, use

P(Y < 60.5)

Tip — Decide ±0.5 by widening the region to include the boundary integer.

Formula recap

\mu = np, \ \sigma^2 = np(1-p)

Matching parameters.

P(X \ge a) \to P(Y > a - 0.5)

Continuity correction.

Common mistakes to avoid

Omitting the continuity correction.

Always adjust by ±0.5 when approximating.

Using np(1−p) as σ instead of σ².

np(1−p) is the variance; σ is its square root.

Key takeaways

B(n,p) ≈ N(np, np(1−p)) for large n, p near 0.5.
Apply a ±0.5 continuity correction.
Widen the region to include the boundary integer.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

B(n, p) is approximated by:
- $N(np, np(1-p))$
- $N(p, n)$
- $N(np, np)$
- $N(0, 1)$
Show answer and explanation
Answer: $N(np, np(1-p))$
Same mean and variance.
The normal approximation is best when:
- n large, p near 0.5
- n small
- p near 0
- p near 1
Show answer and explanation
Answer: n large, p near 0.5
Roughly symmetric.
A continuity correction adjusts by:
- $\pm 0.5$
- $\pm 1$
- $\pm 0.1$
- $0$
Show answer and explanation
Answer: $\pm 0.5$
Discrete to continuous.
For B(100, 0.5), the mean and variance are:
- $50$ and $25$
- $50$ and $50$
- $100$ and $25$
- $25$ and $50$
Show answer and explanation
Answer: $50$ and $25$
np and np(1−p).
$P(X \le 60)$ with continuity correction becomes:
- $P(Y < 60.5)$
- $P(Y < 59.5)$
- $P(Y < 60)$
- $P(Y > 60.5)$
Show answer and explanation
Answer: $P(Y < 60.5)$
Include 60.

Next step

Check your understanding

Work through questions on Approximating a Binomial Distribution. Use the feedback to decide what needs another look.

1 past-paper question · 3 marks

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