Normal approximation of the binomial distribution: continuity, validity and z-scores

What this dot point is asking

NESA wants you to recognise when the binomial distribution can be approximated by a normal distribution, write down the approximating normal $N(n p, n p (1 - p))$, apply the continuity correction, and compute approximate probabilities using z-scores.

The answer

The result

If $X \sim B(n, p)$ with $n$ "large" (rule of thumb: both $n p \ge 5$ and $n (1 - p) \ge 5$), then

This is a consequence of the central limit theorem: the binomial is a sum of $n$ independent identically distributed Bernoulli trials, and sums of many i.i.d. random variables tend to a normal distribution.

Watching the bell curve appear, stage by stage

The approximation is not a coincidence: as the number of trials grows, the binomial bars settle into the shape of a bell curve with the same mean and standard deviation. These panels overlay the matching normal $N(np,\, np(1 - p))$ on the bars of $B(n, 0.5)$ for growing $n$.

Stage 1, a small $n$ is already close. With $n = 10$ the bars are wide and chunky, yet the bell curve $N(5, 2.5)$ already passes through their tops. The fit is rough in the tails but the centre is good.

Stage 2, more trials, narrower bars. At $n = 30$ (mean $15$, standard deviation $2.74$) there are more, thinner bars and they hug the curve much more closely. This is the central limit theorem taking hold: summing more Bernoulli trials pulls the shape toward the normal.

Stage 3, a close fit you can compute with. By $n = 50$ (mean $25$, standard deviation $3.54$) the bars and the curve are almost indistinguishable, so the area under the normal curve is a reliable stand-in for a sum of binomial probabilities. This is exactly when replacing a long pmf sum with one or two $z$-lookups is justified.

When the approximation works

The approximation is good when:

• $n$ is large.
• $p$ is not too close to $0$ or $1$ (which makes the distribution very skewed).

The HSC rule of thumb is $n p \ge 5$ and $n (1 - p) \ge 5$. With $n = 100$, this works for $0.05 \le p \le 0.95$. With $n = 20$, it works for roughly $0.25 \le p \le 0.75$.

When $p$ is very small and $n$ is large, the Poisson approximation is more appropriate, but that is beyond HSC Extension 1.

Continuity correction

The binomial is discrete; the normal is continuous. To improve the approximation, adjust the boundary by $\pm 0.5$.

For $X \sim B(n, p)$ and $k$ an integer,

The half-unit shift accounts for the fact that the binomial $X = k$ corresponds to the interval $[k - 0.5, k + 0.5]$ in the continuous picture. The diagram below zooms in on the bars near $k = 55$: the bar for $X = 55$ stretches across the continuous interval $[54.5, 55.5]$, so $P(X \le 55)$ keeps the whole of that bar and the boundary the normal curve should use is $55.5$, not $55$.

The direction is the part to get right: for an upper bound ("at most $k$", "$\le k$") push the boundary out to $k + 0.5$; for a lower bound ("at least $k$", "$\ge k$") push it out to $k - 0.5$. Either way you are growing the interval to include the whole of the boundary bar.

Standardising

For a normally distributed $X$ with mean $\mu$ and standard deviation $\sigma$, the standardised value is

Look up the probability in a standard normal table (or use $\arcsin$-style estimates for HSC if a table is not provided).

When to use the approximation

For HSC Extension 1, use it when:

• $n$ is large enough that summing pmf values is tedious.
• The question asks for a numerical answer (not exact).
• The question explicitly says "using the normal approximation".

Otherwise, compute the binomial directly.

How exam questions ask about the normal approximation

These questions are heavily signposted, so the marks are in following the standard sequence without skipping a step:

• "Using the normal approximation, estimate $P(\ldots)$": the explicit instruction to approximate. Quote $\mu = np$ and $\sigma = \sqrt{np(1 - p)}$, apply the continuity correction, standardise, and look up the $z$-value.
• "Explain why the normal approximation is appropriate / valid": check and state both conditions, $np \ge 5$ and $n(1 - p) \ge 5$. This is usually a 1 mark justification.
• "A coin is tossed $100$ times ... probability of at least $60$ heads": a large-$n$ "at least" or "at most" probability that would need a huge pmf sum, the classic cue to approximate. Use the continuity-corrected boundary ($59.5$ here for $X \ge 60$).
• "State the mean and standard deviation of [the count]": just $np$ and $\sqrt{np(1 - p)}$; sometimes the lead-in to a later approximation part.
• A given table value such as "use $P(Z \le 1.9) \approx 0.9713$": a strong hint about which $z$-score you should be standardising to; if your $z$ does not match, recheck the continuity correction and the arithmetic.