Central Limit Theorem Calculator

What the central limit theorem calculator does

The central limit theorem describes what happens to the average of a sample as the sample gets large. This calculator takes a population's mean and standard deviation and a sample size, and returns the two numbers that describe the distribution of the sample mean: its centre and its spread.

Those two numbers are the foundation that confidence intervals and hypothesis tests are built on. Below is the idea behind the theorem and what the calculator gives you.

How to use it

1. Enter the population mean and the population standard deviation.
2. Enter the sample size, which must be at least 30.
3. Press Calculate for the mean and standard deviation of the sample mean, or Reset to clear it.

The idea behind the theorem

Imagine taking a sample from some population, working out its average, and writing that average down. Then doing it again, and again. Those sample averages form a distribution of their own, and the central limit theorem makes a remarkable promise about it: as the sample size grows, that distribution of averages becomes a normal, bell-shaped curve, no matter what shape the original population had.

This is the surprising part. The population could be skewed, lumpy, or nothing like a bell, and yet the averages of large enough samples still pile up into a tidy normal curve. A sample size of about 30 is the common rule of thumb for "large enough," which is why this calculator asks for at least that many.

How the two numbers are worked out

The distribution of the sample mean has a centre and a spread, and both follow simple rules. Its centre sits exactly on the population mean, so the averages cluster around the same value the whole population is centred on. Its spread, called the standard error, is the population standard deviation divided by the square root of the sample size:

Standard error = population standard deviation ÷ √n

The square root of n in the bottom means the spread shrinks as the sample grows, so averages from big samples cluster more tightly around the true mean than averages from small ones. That tightening is the whole reason larger samples give more reliable estimates.

Why it matters so much

The central limit theorem is quietly behind a huge amount of statistics. It is the reason you can use the normal curve to reason about a sample mean even when the underlying data is not normal, which is almost always the real situation. Without it, every method would need to know the exact shape of the population, which is rarely known.

So when a confidence interval or a hypothesis test about a mean uses the bell curve, this theorem is the licence that makes it valid. It turns the messy, unknown shape of real data into something regular enough to work with, as long as the sample is reasonably large.

A worked example

Suppose a population has a mean of 50 and a standard deviation of 10, and you take samples of size 100.

The distribution of the sample mean is centred on 50, the population mean. Its standard error is 10 ÷ √100 = 10 ÷ 10 = 1. So sample averages from 100 observations cluster around 50, typically within about a unit of it, far tighter than the spread of 10 in the population itself. Quadruple the sample to 400 and the standard error would halve to 0.5, tightening the averages further.

Entering your values

Enter the population mean, the population standard deviation, and a sample size of at least 30. The calculator returns the mean of the sample mean, which equals the population mean, and its standard deviation, the standard error, to four decimal places.

Questions people ask

What is the central limit theorem?

The result that the distribution of a sample mean becomes normal as the sample size grows, whatever the shape of the original population. It is centred on the population mean with a spread that shrinks as samples get larger.

Why does the sample size need to be at least 30?

About 30 is the common rule of thumb for the theorem's normal approximation to be good. For very skewed populations a larger sample may be needed, and for near-normal ones fewer would do.

What is the standard error?

The standard deviation of the sample mean, equal to the population standard deviation divided by the square root of the sample size. It measures how tightly sample averages cluster around the true mean.

Why is the theorem important?

Because it lets you use the normal curve to reason about a sample mean even when the data is not normal, which is what makes most confidence intervals and tests about a mean valid.

References

A quick note on where the methods here come from. The central limit theorem and the standard error of the mean are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook that covers the central limit theorem.

1. NIST/SEMATECH e-Handbook of Statistical Methods (the central limit theorem and sampling distributions). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (the central limit theorem). https://openstax.org/details/books/introductory-statistics-2e