Central limit theorem
Average observations, then repeat the study
The population is right-skewed, with mean 0 and SD 1 in standardised units. Begin with 1,000 studies containing just one observation each.
Keep the same population and 1,000 repeated studies. Average 25 observations in each study: the means become less spread out even though individuals do not.
Watch one study become one mean
Individual observations from the population
These bars summarise individual values drawn from the selected population. Every population is standardised to mean 0 and SD 1; even the rare binary outcomes are shown in those standardised units. Compare this shape with the distribution of sample means.
Means from repeated samples
Each sample contributes one mean to these bars. The comparison curve is a normal distribution with the theoretical mean and standard error. Its parameters come from the population and sample size.
Worked comparison
Follow the calculation
What the symbols mean
x̄ is the mean of n observations. σ is the population standard deviation. The standard error, or SE, is the standard deviation of sample means across repeated samples.
Assumptions and full calculation
All four populations have mean 0 and standard deviation 1. The applet draws independent samples with replacement. Both histograms summarise simulated values. The normal comparison curve has mean 0 and variance 1/n. Each population has finite variance, as required by this form of the central limit theorem. Results vary between simulations. The same random seed and settings reproduce the same values.