DSAverse

Initializing Sorting Algorithms...

Sorting

Trees

Graphs

Preparing interactive visualizations...

Back to Probability and Statistics

The Central Limit Theorem

Average a few draws from any distribution, again and again. The averages pile up into a bell curve centred on μ, narrowing like σ/√n — even when the population looks nothing like a bell.

Step 1 of 11

Speed

A lopsided population on 0 to 9: small values are common, big ones rare. Its mean is μ = 1.99 and its standard deviation σ = 2.07. Take a "sample" of size 1, 2,000 times — the histogram is just the population again.

\bar{X}_{1} \approx N\!\left(\mu, \tfrac{\sigma^2}{n}\right), \quad \mu = 1.99, \quad \frac{\sigma}{\sqrt{n}} = \frac{2.071}{\sqrt{1}} = \mathbf{2.071} \\[6pt] \text{simulated: mean } 1.992, \;\text{sd } 2.048, \;\text{skewness } 1.13, \;\text{within } \pm 2\sigma/\sqrt{n}\text{: } 96.1\%

n (sample size)1Number of samples2,000

The Central Limit Theorem

Average a few draws from any distribution, again and again. The averages pile up into a bell curve centred on μ, narrowing like σ/√n — even when the population looks nothing like a bell.

Step 1 of 11

Speed

\bar{X}_{1} \approx N\!\left(\mu, \tfrac{\sigma^2}{n}\right), \quad \mu = 1.99, \quad \frac{\sigma}{\sqrt{n}} = \frac{2.071}{\sqrt{1}} = \mathbf{2.071} \\[6pt] \text{simulated: mean } 1.992, \;\text{sd } 2.048, \;\text{skewness } 1.13, \;\text{within } \pm 2\sigma/\sqrt{n}\text{: } 96.1\%

n (sample size)1Number of samples2,000

DSAverse

The Central Limit Theorem

See also

The Central Limit Theorem

See also