Distributions Updated November 2026

Central limit theorem

The central limit theorem says that the mean of a large enough random sample is approximately normally distributed, whatever the shape of the population, with mean equal to the population mean and standard deviation sigma / sqrt(n). That is why z and t methods work for means.

Syntax

sample mean ~ Normal( mu, sigma / sqrt(n) )

Part	What it is
`mu`	the population mean
`sigma`	the population standard deviation
`n`	the sample size; the approximation improves as n grows

Examples

Population: mean 100, sd 20. Samples of n = 25
sd of the sample mean = 20 / sqrt(25)

Result: 4

P(sample mean > 106) for the same setup
z = (106 - 100) / 4 = 1.5

Result: 1 - 0.9332 = 0.0668

Excel: =1 - NORM.DIST(106, 100, 4, TRUE)

Quadrupling the sample size from 25 to 100
20 / sqrt(100)

Result: sd of the mean halves, from 4 to 2

Common mistakes

Using sigma instead of sigma / sqrt(n) when the question is about a sample mean; the mean of many values varies far less than one value.
Assuming the theorem makes the raw data normal; it is the distribution of the sample mean that becomes normal, the population keeps its shape.
Applying n >= 30 blindly to very skewed or heavy-tailed data; such populations need larger samples before the mean looks normal.

Practice Central limit theorem

These drills check your answer on live data, free, no account needed.

Sampling distribution of the meanOrder values at an online shop have mean $197 and standard deviation $2. A random sample of 16 orders is taken. What is the probability that the sample mean is above $196.10? Round to 4 decimal places.
Normal approximation to the binomial20% of shoppers use a discount code. Among 61 independent shoppers, use the normal approximation with a continuity correction to estimate the probability that at least 17 use a code. Answers within 0.002 count, so a z table with z rounded to 2 decimals works. Round to 4 decimal places.

Questions

Why is 30 the magic number?

It is a rule of thumb: for most moderately skewed populations, means of 30 or more observations are close enough to normal. Symmetric populations need fewer; very skewed ones need more.

What is the standard error?

The standard deviation of the sample mean, sigma / sqrt(n), or s / sqrt(n) when sigma is estimated from the sample. It is the quantity the central limit theorem describes.

Does the central limit theorem apply to proportions?

Yes. A sample proportion is a mean of 0s and 1s, so it is approximately normal with sd sqrt(p(1 - p) / n) when np and n(1 - p) are both at least about 10.

Standard errorThe standard error of the mean is s / sqrt(n); for a proportion it is sqrt(p(1 - p) / n). Worked examples, SE versus standard deviation and the Excel formula.
Normal distributionHow to find normal distribution probabilities with z-scores, read the standard normal table, find a percentile cutoff, and use NORM.DIST and NORM.INV.
Confidence interval for a meanHow to calculate a confidence interval for a mean with x-bar plus or minus t times s over root n, a full 95% example, when to use z, and Excel CONFIDENCE.T.
Sample sizeSample size formulas n = (z sigma / E)^2 for a mean and n = z^2 p(1 - p) / E^2 for a proportion, worked examples giving 97 and 385, and rounding rules.