▶Center and spread
The mean locates a distribution and the variance measures the typical squared departure from that center.
Moments and Non-Gaussian Distributions
Reading shape beyond the mean and variance
Moments summarize a distribution's center, spread, asymmetry, and tail behavior. They reveal why two distributions with similar centers can still behave very differently.
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The essential idea: mean and variance describe the first layer; skewness and kurtosis expose asymmetry and heavy tails.
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What to notice
- Raw moments use powers of X.
- Central moments measure deviations from the mean.
- Higher moments distinguish non-Gaussian shapes.
Four summaries, four questions
Asymmetry and tails
Standardized third and fourth moments compare shape without depending on physical units.
Can the same mean and variance hide a different shape?
Compare a Gaussian with a Student t or Gamma distribution. Change the shape parameter, then read the moments and tails together.
selected distributionmatching Gaussian
Gaussian distributions are fixed by mean and variance. Non-Gaussian distributions need higher-order information to describe asymmetry and tail risk.