Central Limit Theorem Simulator: Sampling Distribution Calculator
Draw repeated samples from ten populations, compare two sample sizes or statistics, and read simulated values beside exact theory.
At a glance
- Computes
- How a statistic behaves across repeated samples: centre, spread, shape and tails
- You supply
- A population, a sample size, a statistic and how many samples to draw
- Use when
- You want to see sampling behaviour or test whether a sample size suffices
- Not for
- An interval from one observed sample Bootstrap Confidence Interval Calculator
Runs in your browser. The mean's exact theoretical value is shown for every population; other statistics show an exact value, a stated form, or an honest "not available" where none is claimed.
Pick a population whose parameters are known exactly, set a sample size, and watch what the sample mean does. This page draws the samples, plots the sampling distribution as it fills in, and prints the exact theoretical values next to the simulated ones so the two can be compared. It builds no interval. Reading the spread of a sampling distribution and constructing an interval around one estimate are different operations, and confidence intervals is where the second one lives. When the population is unknown and all you have is one sample, the bootstrap instrument resamples it instead: that distribution centers on your sample statistic, while this one centers on the population parameter. The comparison of the two distributions below sets them out row by row.
Pane A: mean, n = 2
Count by sample mean; 11 bins of width 0.5 (fd rule)
| Quantity | Simulated | Exact theory |
|---|---|---|
| Samples drawn | 36 | Exact enumeration |
| Mean of the statistic | 3.5000000 | 3.5 |
| SD of the statistic | 1.20761473 | 1.20761473 |
| Skewness g1 | 0.0000 | 0.0000 |
| Excess kurtosis g2 | -0.634286 | -0.634286 |
Pane A spread 1.20761473, pane B spread 0.986013297. Both means equal 3.5 exactly.
Pane B: mean, n = 3
Count by sample mean; 16 bins of width 0.333 (fd rule)
| Quantity | Simulated | Exact theory |
|---|---|---|
| Samples drawn | 216 | Exact enumeration |
| Mean of the statistic | 3.5000000 | 3.5 |
| SD of the statistic | 0.986013297 | 0.986013297 |
| Skewness g1 | -0.0000 | 0.0000 |
| Excess kurtosis g2 | -0.422857 | -0.422857 |
Shown without JavaScript: the exact sampling distribution of the mean for a fair die, enumerated over all 36 and all 216 samples. Enable JavaScript to draw from ten populations at sample sizes up to 200.
What a sampling distribution is
A sampling distribution is the distribution of a statistic across repeated samples from the same population. It is not the distribution of your data and it is not the population distribution.
The three properties, and only one depends on n
- The mean of the sampling distribution of the mean equals the population mean for every n.
- Its standard deviation equals the population SD divided by the square root of n for every n.
- Its shape approaches a normal distribution as n grows, when the population has finite variance.
Testing the n greater than or equal to 30 rule
There is no universal threshold. The lognormal preset remains visibly skewed well past n = 30, while a normal population starts normal at n = 1.
When the theorem does not apply
A Cauchy mean stays Cauchy with the same scale for every n. A centered and scaled maximum approaches an extreme-value family, not a normal distribution.
The sampling distribution and the bootstrap distribution are centered on different things
Source of draws: this page uses a known population; bootstrap uses the observed sample.
Center: this page centers on a population parameter; bootstrap centers on the sample statistic.
Readout: this page shows shape and spread; bootstrap reads an interval.
If you have one sample and want an interval, use the bootstrap instrument.
E(xbar) = mu; SD(xbar) = sigma / sqrt(n)
How?
How this is calculated
Exact finite enumeration is used when the population is finite and the ordered sample count stays at or below 300,000. Simulation uses a fixed-edge accumulator with running moments.
Formula: E(xbar) = mu; SD(xbar) = sigma / sqrt(n)
Assumptions, limits, and privacy
Samples use replacement by default. Named finite populations use their full list of units and population-denominator moments. Runs stay in this browser. Simulated values are retained only up to the most recent 100,000 for export.
Sources
- Simulation of the Sampling Distribution of the Mean Can Mislead. Journal of Statistics Education. Retrieved .
- It's Time To Retire the n >= 30 Rule. JSM Proceedings. Retrieved .
- R datasets: Michelson Speed of Light Data. R Core Team. Retrieved .