normapprox() — Normal approximation

Uses a smooth bell curve to stand in for the bumpy binomial bars.

Normal(np, np(1−p)) ≈ Binomial(n, p)

Key idea

• For large n, a Binomial count (wins out of n yes/no tries) is well approximated by a Normal bell curve.
• The matching Normal uses mean (centre) np and variance (spread squared) np(1 minus p). (e.g. 100 * 0.5 = 50)
• The standard deviation (spread) is the square root of that variance. (e.g. sqrt(25) = 5)
• So Binomial(100, 0.5) behaves much like Normal(50, 25).
• The approximation works best when both np and n(1 minus p) are at least about 10.
• Continuity correction: the binomial only hits whole numbers but the normal is smooth, so nudge the cut-off by half a unit (±0.5) to match the bars. (e.g. P(X ≥ 60) → P(Normal > 59.5))
• The same idea works for the Poisson: when the rate λ is large, Poisson(λ) ≈ Normal(λ, λ) — its mean AND variance are both λ, so the sd is √λ. (e.g. Poisson(25) ≈ N(25, 25))
• Rule of thumb: the Poisson-to-Normal fit is good once λ is about 10 or more, and (being a count) it uses the same ±0.5 continuity correction.

Continuity correction — the ±0.5 fix

1. The binomial lands only on whole counts, and each count k owns a bar one unit wide — from k − 0.5 to k + 0.5.
2. The smooth normal has no bars, so to capture a whole bar you stretch the boundary out by half a unit.
3. “At least 60” (X ≥ 60) keeps the whole 60-bar, so use 59.5: P(X ≥ 60) ≈ P(Normal > 59.5).
4. “More than 60” (X > 60) drops the 60-bar, so use 60.5: P(X > 60) ≈ P(Normal > 60.5).
5. “Exactly 60” becomes the slice from 59.5 to 60.5.
6. Worked: Binomial(100, 0.5), P(X ≥ 60). z = (59.5 − 50)/5 = 1.9, so ≈ 1 − Φ(1.9) ≈ 0.029 — much closer to the exact 0.028 than the no-correction answer 0.023.

Definition

$$\text{Bin}(n,p)\approx N(np,np(1-p))$$

$n$

number of trials

$p$

chance of success each trial

$np$

the mean of the approximating normal

$np(1-p)$

its variance

$\lambda$

the Poisson rate (also its mean and variance)

In plain words

Counting every step on a tall staircase is tiring. From far away the steps look like a smooth ramp — the normal curve is that ramp.

Where you'd use it

• Flip a fair coin 100 times. Rather than summing dozens of binomial terms, normapprox uses a bell curve with mean 50 and sd 5 to estimate the chance of 60+ heads at about 3%.
• A factory batch of 1000 items has a 3% fault rate. The number of duds is roughly Normal with mean 30 and sd ≈ 5.4, so you can quickly estimate the chance of, say, more than 40 faulty items.
• In a yes/no poll of 500 people with p = 0.5, the count of “yes” answers is roughly Normal with mean 250 and sd ≈ 11.2, letting you gauge how likely a given margin is without exact binomial math.

Worked example 9

Flip a coin 20 times (p = 0.5). The binomial bars line up almost exactly under a smooth normal curve centred at 10.

Frequently asked questions

When can I approximate a binomial with a normal?

When np and n(1 − p) are both ≳ 5–10.

What is the continuity correction?

A ±0.5 tweak because you’re approximating a discrete count with a continuous curve.