What Is Central Limit Theorem?
The central limit theorem says that the average of many independent observations is approximately normally distributed, even when the individual values are not. Its mean is the population mean and its standard deviation is σ ÷ √n.
Choose a population shape and a sample size. The calculator draws 2,000 samples, plots a histogram of their means, reports the standard error and compares the normal-approximation probability with the simulated one.
Try the exponential population at n = 2, then n = 30. The histogram moves from a skewed shape to a nearly symmetric one.
How to Use the Central Limit Theorem Calculator
- Pick a population shape.
- Enter the sample size n.
- Enter the sample mean x you want a probability for.
- Read the standard error and compare the CLT probability with the simulation.
Central Limit Theorem Formula
Mean of the sampling distribution
Standard error
z for a sample mean
Central Limit Theorem Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| μ | Population mean | |
| σ | Population standard deviation | |
| n | Sample size | |
| σ ÷ √n | Standard error of the sample mean |
Central Limit Theorem Worked Example
An exponential population has μ = 1 and σ = 1. For n = 30 the standard error is 1 ÷ √30 = 0.1826. For x = 1.2, z = (1.2 − 1) ÷ 0.1826 = 1.095 and P(x̄ ≤ 1.2) ≈ 0.863. The simulation gives a close figure, a little different because the sampling distribution of a skewed population is still slightly skewed at n = 30.
With the example values loaded in the calculator above, the steps are:
- Population: Exponential with mean 1, with μ = 1 and σ = 1.
- The central limit theorem says the sample mean x̄ is approximately normal with mean μ and standard error σ ÷ √n = 1 ÷ √30 = 0.182574.
- z = (1.2 − 1) ÷ 0.182574 = 1.095, so P(x̄ ≤ 1.2) ≈ 0.86334.
- We then drew 2000 samples of size 30 from the same population and averaged each one. The simulated mean (0.99407) and SD (0.17886) should sit close to μ and σ ÷ √n.
Result: Mean of the sampling distribution = 1.
How to Read the Simulation
Each of the 2,000 simulated samples contributes one sample mean, and the histogram shows how those means spread out. For an exponential population with mean 1 and SD 1 and n = 30, the standard error is 1 ÷ √30 = 0.1826, and the histogram should centre near 1 with about that spread.
The normal-approximation probability and the simulated one will rarely match exactly. The simulation has its own sampling noise, and for a skewed population the sample mean at n = 30 is still slightly skewed. Raise n and the two figures move closer.
Where Central Limit Theorem Is Used
Seeing the theorem
Watch sample means from skewed and two-humped populations settle into a bell shape as n grows.
Standard error
See how σ ÷ √n shrinks as the sample size increases.
Probability of a sample mean
Find P(sample mean ≤ x) and compare the normal approximation with the simulation.
Teaching
Compare four population shapes at the same sample size.
Common Central Limit Theorem Mistakes to Avoid
- The theorem concerns the sampling distribution of the mean, not the raw data.
- Observations must be independent.
- Heavier skew needs a larger n before the means look normal.
- The simulation uses a fixed seed, so the same inputs always give the same picture.
Central Limit Theorem Terms and Definitions
- Sampling distribution
- The distribution of a statistic, such as the mean, across all possible samples of the same size.
- Standard error
- The standard deviation of the sampling distribution.
Central Limit Theorem FAQs
What does the central limit theorem say?
For independent samples of size n from a population with mean μ and finite standard deviation σ, the distribution of the sample mean approaches a normal distribution with mean μ and standard deviation σ ÷ √n as n grows.
How large must n be?
There is no single cutoff. Around 30 is a common rule of thumb for moderately skewed data, but very skewed populations need more. The histogram above shows how close the shape is.
Does the population have to be normal?
No. That is the point of the theorem. If the population is already normal, the sample mean is exactly normal for any n.
Why does the simulation differ slightly from the formula?
The simulation draws a finite number of samples, so it has its own small random error. The standard error formula is exact.