The idea

Approximating a binomial with a normal

When n is large and p not too extreme, approximating B(n, p) by N(np, np(1−p)) — and applying the continuity correction (±0.5) when moving from the discrete binomial to the continuous normal.

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Statistics · Statistical distributions

Approximating a binomial with a normal

When n is large and p not too extreme, approximating B(n, p) by N(np, np(1−p)) — and applying the continuity correction (±0.5) when moving from the discrete binomial to the continuous normal.

Why it works

When the bar chart becomes a bell

Estimate

P(X > 55)

for

X \sim B(100, 0.5)

, done exactly, means summing forty-five separate binomial terms. But draw the bar chart of a large-

n

binomial and something remarkable appears: the bars settle into a bell shape — the normal distribution's silhouette. So a normal can stand in for the binomial, and the forty-five-term sum becomes one area under a curve.

The approximation matches the binomial's mean and variance:

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

That is, mean

\mu = np

and variance

\sigma^2 = np(1-p)

(so

\sigma = \sqrt{np(1-p)}

). It works well when $n$ is large and $p$ is not too close to $0$ or $1$ — a common rule of thumb is

np > 5

and

n(1-p) > 5

, so the bell isn't cut off at an end.

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