Central Limit Theorem Calculator
Enter the population mean, standard deviation, and sample size to get the sampling distribution of the mean, its standard error, and normal-approximation probabilities for a sample mean you choose.
Example: with Population mean (μ) 100 · Population standard deviation (σ) 15 · Sample size (n) 25 · Sample mean to test (x̄) 105 → Sampling distribution of x̄: Mean 100, SE 3 (approx. Normal).
- Standard error (SE)3
- z for your sample mean1.667
- P(x̄ ≤ value)0.9522 (95.2%)
Computed by the calculator below using its default values. Change any input to see your own numbers.
Sample means cluster as Normal(μ, σ/√n) once n is large enough — the heart of the CLT.
Why the sample mean goes normal
The central limit theorem says that if you take samples of size n and average each one, those sample means pile up into a normal, bell-shaped distribution, no matter what the original population looks like, as long as n is large enough. The averaging washes out skew and odd shapes. The sampling distribution centers on the same mean as the population, but it is tighter, because averages vary less than individual values.
Standard error and the normal shortcut
How much tighter? The spread of the sample means is the standard error, SE = σ/√n. Quadruple the sample size and the standard error halves. Once you have the SE, you can treat the sample mean as normal and answer probability questions with a z-score, z = (x̄ - μ)/SE. This tool uses that normal approximation, which is excellent for large n and rougher for small n or very skewed populations.
How it’s calculated
The sampling distribution of the mean has mean μ and standard error SE = σ/√n. By the central limit theorem it is approximately normal for large n. Probabilities use z = (x̄ - μ)/SE and the standard normal CDF, computed with the Abramowitz and Stegun 26.2.17 approximation (accuracy about 7.5e-8).
The normal approximation is exact only if the population is normal; for other shapes it improves as n grows (n ≥ 30 is a common rule of thumb). Requires a known population standard deviation σ.
Normal-model reference (from the CLT)
| z = (x̄ - μ)/SE | P(below) | P(above) |
|---|---|---|
| -1.96 | 2.5% | 97.5% |
| -1.00 | 15.9% | 84.1% |
| 0.00 | 50% | 50% |
| 1.00 | 84.1% | 15.9% |
| 1.96 | 97.5% | 2.5% |
Standard normal probabilities; valid for the sample mean once n is large enough for the CLT to apply.
Common mistakes
- Dividing σ by n instead of by the square root of n. The standard error is σ/√n.
- Applying the normal approximation to a tiny, skewed sample. The CLT needs a large enough n to take hold.
- Confusing the population standard deviation with the standard error. SE describes the spread of sample means, not individual observations.
Frequently asked questions
What does the central limit theorem calculator do?
It computes the sampling distribution of the sample mean: its mean (equal to μ), its standard error σ/√n, and normal-approximation probabilities for a sample mean you specify, via z = (x̄ - μ)/SE.
What is the standard error formula?
SE = σ/√n, the population standard deviation divided by the square root of the sample size. It measures how much sample means vary around the true mean.
How large does n need to be?
It depends on the population's shape. For roughly symmetric populations, small n works; for skewed ones, n of 30 or more is a common rule of thumb before the normal approximation is trustworthy.
Why is the sampling distribution narrower than the population?
Averaging cancels out extremes, so sample means cluster more tightly than individual values. The larger the sample, the more cancellation, and the smaller the standard error.
Does the population have to be normal?
No, that is the point of the theorem. The distribution of sample means approaches normal even when the population is not, provided the sample size is large enough.