Central Limit Theorem Calculator

Find the standard error of the mean (SE = σ/√n) and the sampling distribution of the sample mean, then compute the probability that a sample mean falls above, below, or between chosen values.

Quick Facts

Method
Sampling distribution of the mean: X̄ ≈ Normal(μ, σ²/n), with standard error SE = σ/√n
Probabilities use the z-score z = (x − μ)/SE and the standard normal distribution.

Your Results

Calculated
Standard error (SE = σ/√n)
-
Std. deviation of the sample mean
Mean of sample means
-
Equals the population mean μ
z-score of x₁
-
(x₁ − μ) / SE
Probability
-
Sampling-distribution probability

Ready

Enter μ, σ, and n, then calculate.

The Central Limit Theorem in plain terms

The Central Limit Theorem (CLT) says that if you take many random samples of size n from any population with mean μ and finite standard deviation σ, the distribution of the sample means becomes approximately normal as n grows — even when the original population is skewed, bimodal, or otherwise non-normal. This is the reason the normal distribution shows up almost everywhere in applied statistics and why so much of inference (confidence intervals, z-tests, t-tests) rests on it.

Two facts describe that sampling distribution exactly. First, its mean equals the population mean: the average of all possible sample means is μ itself (the sample mean is an unbiased estimator). Second, its spread is smaller than the population's by a factor of √n.

The formulas this calculator uses

The standard deviation of the sampling distribution of the mean is called the standard error of the mean (SE):

SE = σ / √n

So the sampling distribution of the sample mean is approximately:

X̄ ≈ Normal( μ , σ²/n )

To find the probability that a sample mean lands above, below, or between chosen values, convert each value to a z-score and read the standard normal distribution:

z = ( x − μ ) / SE = ( x − μ ) / (σ/√n)

Note the difference from an individual observation: a single data point uses z = (x − μ)/σ, but a sample mean uses the standard error σ/√n in the denominator, which makes sample means cluster far more tightly than individual values.

A worked example

Suppose IQ-style scores have μ = 100 and σ = 15. For a sample of n = 25 people, the standard error is SE = 15/√25 = 15/5 = 3. What is the chance the sample's average score is 103 or less? The z-score is (103 − 100)/3 = 1.00, and the standard normal CDF at z = 1 is about 0.8413, so P(X̄ ≤ 103) ≈ 84.1%. By contrast, a single person scoring 103 or less has z = (103 − 100)/15 = 0.20, giving only about 57.9% — a smaller probability because individual scores vary more than averages of 25 scores.

Why √n and not n

Because variances of independent variables add, the variance of the sample mean is σ²/n. Taking the square root to get a standard deviation leaves σ/√n. The practical consequence is diminishing returns: cutting the standard error in half requires quadrupling the sample size, not doubling it. Going from n = 100 to n = 400 halves SE; going from 400 to 800 only reduces it by about 29%.

Frequently Asked Questions

What is the standard error of the mean?
It is the standard deviation of the sampling distribution of the sample mean, SE = σ/√n. It measures how much sample means bounce around the true population mean from one sample to the next. Larger samples give a smaller standard error and therefore more precise estimates of μ.
How large does n need to be for the CLT to hold?
A common rule of thumb is n ≥ 30, which works well for mildly non-normal populations. Heavily skewed or heavy-tailed populations may need considerably more, while a population that is already normal makes the sample mean exactly normal for any n. The theorem is about the sampling distribution of the mean, not the raw data.
Do I use σ or the sample standard deviation s?
This calculator assumes the population standard deviation σ is known, so the sampling distribution is normal and z-scores apply. When σ is unknown and you estimate it with the sample standard deviation s, the correct reference distribution is Student's t with n − 1 degrees of freedom, which is slightly wider for small samples. For large n the t and z results converge.
What is the difference between standard deviation and standard error?
Standard deviation σ describes the spread of individual observations in the population. Standard error σ/√n describes the spread of the sample mean across repeated samples. The standard error is always smaller (for n > 1) and shrinks as the sample grows; the population standard deviation does not change with sample size.

Related calculators