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 Video by Zeeshan Zamurred The Normal Distribution

Edexcel A Level Maths: 3.6 Approximating the Binomial DistributionWatch the full walkthrough 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 $N (n p, n p (1 - p))$ — the normal with the same mean and variance.

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

Mean np, variance np(1−p).

Continuity correction

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

μ = 50

σ^{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

μ = n p, σ^{2} = n p (1 - p)

Matching parameters.

P (X \geq 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

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