Normal Approximation

The foundation of the normal approximation is the Central Limit Theorem: if X₁, X₂, …, Xₙ are independent random variables each drawn from the same distribution with mean μ and finite variance σ², then the sample mean X̄ = (X₁ + X₂ + ⋯ + Xₙ)/n has a distribution that approaches N(μ, σ²/n) as n → ∞. In practice, n ≥ 30 is usually enough for the approximation to be excellent, even if the population is moderately skewed.

To use the standard normal table, we convert X̄ to a Z-score: Z = (X̄ − μ) / (σ/√n). This Z-score measures how many standard errors X̄ is away from the population mean μ. The quantity σ/√n is called the standard error of the mean (SEM). Once we have Z, we can use the standard normal distribution N(0,1) to find probabilities.

For a sample proportion p̂ = X/n (where X counts successes in n independent trials each with success probability p), the CLT gives p̂ ≈ N(p, p(1−p)/n). The standard error is √(p(1−p)/n). The rule of thumb for this approximation to be valid is np ≥ 10 and n(1−p) ≥ 10. The Z-score is Z = (p̂ − p) / √(p(1−p)/n).

When approximating a discrete distribution (like the binomial) with the continuous normal curve, accuracy improves by adding a continuity correction of ±0.5. For example, P(X ≤ 15) for a binomial is approximated by P(Y ≤ 15.5) where Y ~ N(np, np(1−p)). This correction accounts for the fact that each discrete bar in a histogram has width 1, so the bar for 15 spans from 14.5 to 15.5.

After computing a Z-score, look up Φ(z) — the area under the standard normal curve to the left of z. Tables typically give Φ(z) directly. For tail probabilities: P(Z > z) = 1 − Φ(z). For two-sided probabilities: P(|Z| > z) = 2(1 − Φ(z)). These probabilities tell us how surprising a given sample result would be if the assumed population parameters were true.

• Z — Standard score (dimensionless): Number of standard errors the sample mean is from the population mean; follows N(0,1) approximately.
• \bar{X} — Sample mean (same as data): The arithmetic average of the n sampled values.
• \mu — Population mean (same as data): The true mean of the entire population being studied.
• \sigma — Population standard deviation (same as data): The spread of individual values in the population.
• n — Sample size (count): Number of observations in the sample; larger n means smaller standard error.
• \sigma/\sqrt{n} — Standard error of the mean (same as data): The standard deviation of the sampling distribution of X̄; shrinks as n grows.

Problem. A factory produces bolts whose lengths are normally distributed with mean μ = 50 mm and standard deviation σ = 4 mm. A quality inspector takes a random sample of n = 64 bolts. What is the probability that the sample mean length is between 49 mm and 51 mm?

Answer. P(49 ≤ X̄ ≤ 51) ≈ 0.9544, or about 95.44%.

• Central Limit Theorem — The CLT is the theoretical guarantee behind the normal approximation — it proves that sampling distributions approach normality as sample size grows.
• Sampling Distributions — The normal approximation describes the shape of a sampling distribution; understanding both concepts together is essential for inference.
• Standard Normal Distribution — All normal approximation probability calculations are made using the standard normal (Z) distribution and its table of areas.
• Binomial Distribution — The normal approximation with continuity correction is most commonly applied to binomial probabilities when n is large.
• Confidence Intervals — Confidence intervals for means and proportions are constructed directly using the normal approximation to the sampling distribution.