Central Limit Theorem Calculator (standard error & probability for a sample mean)

Understanding the central limit theorem calculator — concept and principle

This calculator applies the central limit theorem to a sample mean. From the population mean and standard deviation and the sample size, it gives the standard error and the probability that the sample mean falls between two values, below the lower one or above the upper one. It also gives the mean and spread of the sample sum.

Sample means cluster around μ with spread σ/√n, close to a normal curve.

How it is calculated

Standard error = σ ÷ √n

z = (x̄ − μ) ÷ (σ ÷ √n)

P(lower ≤ x̄ ≤ upper) = Φ(z_upper) − Φ(z_lower), with Φ the standard normal CDF

Sum of the sample: mean nμ, standard deviation √n × σ

n ≥ 30 is a common rule of thumb for the approximation to be good; strongly skewed populations need more.

The principle

Averages are less variable than single values, because highs and lows tend to cancel. The central limit theorem says more: whatever the shape of the population, the distribution of the sample mean gets closer to a normal curve as n grows, centred on μ with spread σ ÷ √n. The square root is why precision improves slowly: to halve the standard error you need four times the sample. If the population itself is normal, the sample mean is exactly normal for any n.

Worked example

Default inputs: μ = 50, σ = 12, n = 36, limits 48 and 53.

Standard error = 12 ÷ √36 = 12 ÷ 6 = 2; the mean of the sample means is 50.
z for 48 = (48 − 50) ÷ 2 = −1; z for 53 = (53 − 50) ÷ 2 = 1.5.
P(x̄ < 48) = Φ(−1) = 0.1587; P(x̄ > 53) = 1 − Φ(1.5) = 0.06681.
P(48 ≤ x̄ ≤ 53) = 1 − 0.1587 − 0.0668 = 0.7745.
Sum of 36 values: mean 36 × 50 = 1,800, SD √36 × 12 = 72. Sample size check: n ≥ 30.

Each point is the calculator re-run with only “Population mean μ” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Population mean μ	P(lower ≤ x̄ ≤ upper)
12.5	0
25	0
37.5	7.605e-8
50	0.7745
62.5	0.000001017
75	1.911e-28
100	2.039e-122

How to read the result

The probability describes how sample means behave from sample to sample, not where individual values fall: individual values spread with σ = 12, but means of 36 values spread with only 2. A value of x̄ far into the tails, with |z| above about 2, would be unusual if the stated μ and σ are right.

Common mistakes

Using σ instead of σ ÷ √n for a sample mean.
Applying the theorem to individual observations.
Trusting the approximation for tiny samples from very skewed populations.
Using the sample standard deviation without switching to a t-distribution for small n.

Keep in mind: The results assume a random sample of independent observations and known σ. Real data that are clustered or biased need other methods.

Frequently asked questions

What does the central limit theorem say?

For large enough samples, the sample mean is approximately normal with mean μ and standard deviation σ ÷ √n.

Why n ≥ 30?

It is a rule of thumb that works for many populations; skewed ones need larger samples.

What is the standard error?

The standard deviation of the sample mean, σ ÷ √n.

Sources: OpenStax, Introductory Statistics 2e (2023), §7.1 and §7.2 The Central Limit Theorem

Checked 9 Nov 2026. OpenStax, Introductory Statistics 2e (2023), Chapter 7 Formula Review — §7.1 The Central Limit Theorem for Sample Means (x̄ ~ N(μ, σ/√n), standard error σ/√n, z = (x̄ − μ) ÷ (σ/√n)) and §7.2 for Sums (ΣX ~ N(nμ, √n·σ)).

Your numbers