Skewness and Kurtosis Calculator
Which way a distribution leans and how heavy its tails are, with the standard errors that say whether either is real.
Results
What this tool does
Two sets of numbers can share a mean and a standard deviation and still look nothing alike. Skewness measures the lean: a long tail to the right, the shape of incomes and waiting times, gives a positive figure. Kurtosis measures the tails: a positive figure means extreme values happen more often than a normal curve would allow, which matters far more than it sounds when the extremes are losses. Both are reported alongside their standard errors, because in a small sample both can look dramatic and mean nothing.
Formula
skewness = m₃ ÷ s³ , kurtosis = m₄ ÷ s⁴ − 3 · with the sample corrections that Excel and SPSS use
Variables
| Symbol | Meaning | Unit |
|---|---|---|
numbers | Your numbers, one per line or separated by commas | — |
OUT | Skewness | — |
KU | Excess kurtosis | — |
DI | Which way it leans | — |
TA | The tails | — |
SS | Standard error of the skewness | — |
ZS | Skewness against its error | — |
ZK | Kurtosis against its error | — |
MN | Mean (average) | — |
SD | Standard deviation (sample) | — |
CT | Count | — |
Worked example
- Your numbers, one per line or separated by commas22, 24, 25, 25, 26, 26, 27, 27, 28, 29, 31, 34, 38, 45, 58, 72
- Skewness1.9598
- Excess kurtosis3.4527
- Which way it leansA long tail to the right
- The tailsHeavier than a normal curve
- Standard error of the skewness0.5643
- Skewness against its error3.473
- Kurtosis against its error3.165
- Mean (average)33.5625
- Standard deviation (sample)13.7984
- Count16
Limitations
- The calculation runs entirely in your browser. The values you type are never sent to a server.
Frequently asked questions
What counts as too skewed?
The figure has no fixed threshold, which is why this page also gives the skewness divided by its own standard error. That ratio behaves roughly like a z score: past about two, the lean is more than sampling noise would give. Below that, a skewness of 0.4 in a sample of twenty means nothing at all — small samples produce large-looking skew by chance, and that is exactly the trap the standard error exists to catch.
Why is the kurtosis of a normal curve zero and not three?
Because this page reports excess kurtosis, which is the raw figure minus three. The raw kurtosis of a normal distribution is exactly three, so subtracting it puts the normal curve at zero and makes the sign do the work: positive means heavier tails and a sharper peak than a normal, negative means the opposite. Almost every modern package reports it this way, but some older tables do not, so a figure near three from another source may be the same shape as a zero here.