Sample size and the central limit theorem

The mean of 25 values from a population with standard deviation 15 has a standard error of 15 ÷ √25 = 3.

Minji averages five rolls of a die, three hundred times over

A made-up story. Minji has a die and a lot of patience. She rolls it five times, writes down the average of the five rolls, and starts again. After 300 rounds she has 300 averages. A single roll is anything from 1 to 6 with equal chance, so she expects her averages to be spread just as evenly. They are not. About four in five of them fall between 2.5 and 4.5, and fewer than one in a hundred are below 1.5 or above 5.5. The lab below starts with the same experiment, 300 samples of size 5. Why do the averages bunch up, and by how much?

Surveys, opinion polls, quality checks on a factory line and medical trials all work the same way. Nobody measures everyone. Instead they take a sample and use the sample mean to estimate the mean of the whole population. Different samples give different means, so the useful question is how much the sample mean wobbles from one sample to the next, and how that wobble depends on the size of the sample.

One die is a flat population with mean 3.5

A fair die is a perfect small population. Its values 1 to 6 are equally likely, so its mean is μ = 3.5 and its standard deviation is σ = √(35/12), about 1.708. Its shape is flat, nothing like a bell. Now take a sample: roll the die n times and average the results. With n = 1 the sample mean is a single roll, so it is flat too. With n = 2 it is the sum of two dice divided by 2, which has the tent shape from the two-dice lesson. With n = 5 or 10 the histogram of sample means is already a smooth hill around 3.5, and it gets narrower as n grows.

Sample means pile up around μ, spread by σ/√n

OpenStax states the central limit theorem for sample means like this. Take random samples of size n from any population with mean μ and standard deviation σ. As n increases, the sample means tend to follow a normal distribution with mean μ and standard deviation σ/√n. The number σ/√n is called the standard error of the mean. OpenStax adds a point that is easy to miss: n is the number of values averaged together, not the number of times you repeat the experiment. Taking more samples fills in the histogram; taking bigger samples makes it narrower. In Minji's experiment, n is 5, not 300.

Two consequences matter in practice. First, the sample mean is right on average: the means pile up around μ, not above or below it. Second, precision improves only with the square root of n. For the die, the standard error is 1.708/√1 = 1.708 for single rolls, 1.708/√25 ≈ 0.342 for samples of 25, and 1.708/√100 ≈ 0.171 for samples of 100. Four times the sample size halves the standard error; a hundred times the sample size divides it by ten. OpenStax's own example: a population with mean 90 and standard deviation 15, samples of size 25, gives a standard error of 15/√25 = 3, so a sample mean two standard errors above the expected value is 90 + 2 × 3 = 96.

A skewed population still gives a hill of means by n = 30

The theorem does not need a bell-shaped population. Switch the scene to the skewed population, where small values are common and large ones rare (its mean is 3.06). With n = 1 the histogram has a long right tail. By n = 30 the sample means form a nearly symmetric hill around 3.06. This is why polls can report a margin of error without knowing how opinions are distributed. For a yes or no question the population is 0s and 1s with σ at most 0.5, so with 1,000 people the 95% margin is about 2 × 0.5/√1000 ≈ 0.032, the familiar "plus or minus 3 percentage points".