Central Limit Theorem

  • CLT Animation
    How to use
    • Drag the Population Shape slider to change what the underlying data looks like.
    • Sample Size (n).
    • Hit Draw Sample to get the mean.
    • Watch that mean drop into the histogram at the bottom.
    Population Shape Right-Skewed
    UniformNormalRight-SkewedLeft-Skewed
    Sample Size (n) 30
    Draws per Click 10
    Population
    Distribution of the Sample Mean
  • CLT in Words

    The Central Limit Theorem (CLT) says: if you repeatedly take random samples of size n from any population and calculate each sample's mean, those sample means will be approximately normally distributed, centered on the true population mean, as long as n is reasonably large. It doesn't matter whether the original population is uniform, skewed, or anything else.

    Conditions
    • 1 come from any distribution
    • 2 Each observation is independent
    • 3
    Then
    Why this matters, in practice:
    • A political poll reports "52% ± 3%", that little margin of error only works because of the CLT.
    • A factory checks the average weight of each batch against a target, those checks only make sense because batch averages settle into a predictable bell curve.
    • Comparing two versions of a website or product to see which one actually performs better relies on the CLT to tell a real difference from random luck.