Central Limit Theorem Calculator

Work out central limit theorem instantly with clear inputs, formula shown and shareable results.

About Central Limit Theorem Calculator

The central limit theorem says the sampling distribution of the mean approaches a normal distribution with the same mean and standard deviation σ/√n, whatever the shape of the population, once n is reasonably large.

Formulas

Standard error

SE = σ / √n

Standardised sample mean

z = (x̄ − μ) / (σ/√n)

Worked example

With the default values below, Central Limit Theorem Calculator returns Standard error σ/√n of 2.00000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Population mean: 70
Population standard deviationminimum 0: 12
Sample sizeminimum 1: 36
Observed sample mean: 73

Results

Standard error σ/√n: 2.00000000
z score of the sample mean: 1.50000000
P(sample mean exceeds this): 0.0668072310
Mean of the sampling distribution: 70.000000

Step by step

1Values used
2Standard error
3Standardised sample mean
4Standard error σ/√n= 2.00000000
5z score of the sample mean= 1.50000000
6P(sample mean exceeds this)= 0.0668072310
7Mean of the sampling distribution= 70.000000

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For μ = 70, σ = 12, n = 36 and x̄ = 73?

The standard error is 2, so z = 1.5 and the chance of a mean this high or higher is about 6.68%.

How large must n be?

Around 30 suffices for mildly skewed populations, but heavily skewed or heavy-tailed data may need hundreds.