The normal approximation and continuity
Many distributions become nearly normal in large samples. Knowing when that approximation is safe saves a great deal of calculation.
Approximating the binomial
When \(np \ge 10\) and \(n(1-p) \ge 10\), a binomial count is approximately normal with mean \(np\) and standard deviation \(\sqrt{np(1-p)}\). For 400 flips of a fair coin, the count of heads is approximately \(N(200, 10)\), so getting more than 220 heads (2 standard deviations above) has probability about 2.5%.
The continuity correction
The binomial is discrete, while the normal is continuous. To approximate \(P(X \le 220)\), use \(P(Y \le 220.5)\) on the normal curve, extending by half a unit to cover the whole bar. This continuity correction improves accuracy, especially for moderate \(n\).
Sample proportions
Dividing a binomial count by \(n\) gives the sample proportion \(\hat{p}\). Its mean is \(p\) and its standard deviation is \(\sqrt{\frac{p(1-p)}{n}}\). With large samples it is approximately normal. If 60% of voters support a measure, a poll of 1,000 will give \(\hat{p}\) within about 3 percentage points of 60% roughly 95% of the time.
\sigma_{\hat p} = \sqrt{\frac{p(1-p)}{n}}Approximating the Poisson
A Poisson distribution with a large mean \(\lambda\) (say 20 or more) is approximately normal with mean \(\lambda\) and standard deviation \(\sqrt{\lambda}\). A hospital averaging 36 births a day will see more than 48 births (2 SDs above) on roughly 2.5% of days.
Worked example
A factory's defect rate is 5%. In a batch of 1,000 items, the number of defects has mean 50 and standard deviation \(\sqrt{1000(0.05)(0.95)} \approx 6.9\). The chance of more than 65 defects is about \(P(Z > \frac{65.5 - 50}{6.9}) = P(Z > 2.25) \approx 0.012\). A batch with that many defects suggests a problem in the process.
When not to approximate
If \(np\) or \(n(1-p)\) is small, the binomial is skewed and the normal approximation is poor: for \(n = 20\) and \(p = 0.05\), \(np = 1\). Use exact binomial calculations, or a Poisson approximation for rare events, instead.
Software versus approximation
Modern software computes exact binomial and Poisson probabilities instantly, so approximations matter less for calculation. They still matter for understanding: they explain why sample proportions and averages behave predictably, and they underlie the confidence intervals and tests in later units.
Common mistakes
Using the approximation when the success-failure condition fails. Forgetting the continuity correction for exact counts. Using \(\sqrt{p(1-p)}\) without dividing by \(n\) for proportions. Mixing counts and proportions in the same calculation.
Key takeaways
Large-sample binomial counts and proportions are approximately normal when \(np\) and \(n(1-p)\) are at least 10. Sample proportions have standard deviation \(\sqrt{p(1-p)/n}\). Large-mean Poisson counts are approximately normal. Check conditions before approximating.
Flashcard
np >= 10 and n(1 - p) >= 10
Check your understanding
A fair coin is flipped 100 times. The count of heads is approximately normal with mean and SD: