Kurtosis Calculator

Kurtosis and excess kurtosis of a data set, with a read on tail weight.

What it does

Kurtosis describes the tails of a distribution, the far reaches where unusual values live. A high kurtosis means the tails are heavy, so extreme values turn up more often than a normal curve would suggest. A low kurtosis means the tails are light and extreme values are rare.

The tool above takes a list of numbers, returns the sample kurtosis and the excess kurtosis, and labels the tail weight. A normal distribution is the reference point, with a kurtosis of 3. Because that 3 surprises people, statisticians often report excess kurtosis instead, which is the kurtosis minus 3, so a normal curve reads as 0.

Heavy tails matter in practice. A return series with high kurtosis will produce large moves more often than a normal model predicts, which is why risk models that assume normality can underestimate the chance of a crash. Light tails are the opposite, with fewer surprises than the normal curve implies.

Kurtosis is a fourth moment, so it reacts strongly to outliers. A single distant value can lift the number by a wide margin, which is either the signal you want or a warning that the estimate is unstable. Always read it next to a plot and beside the standard deviation and skewness.

The value here is the population style kurtosis that divides by n, matching the source definition, so it will differ slightly from software that reports the bias corrected sample kurtosis. The difference shrinks as the sample grows and matters most for small lists.

How to use

1. Type or paste your numbers, separated by commas or spaces.
2. Use at least four values, since the formula needs a stable spread.
3. Press calculate to see the kurtosis and the excess kurtosis.
4. Compare the kurtosis with 3, or the excess with 0.
5. Check the tail label for a plain reading.
6. Look at a histogram to confirm the tails are real, not one freak value.

Formula

n

Number of values in the sample.

m2

Second central moment, the average squared deviation.

m4

Fourth central moment, the average fourth power of deviations.

excess

Kurtosis minus 3, so a normal curve reads 0.

The central moments divide by n, which is the population style definition used by the source. Some software instead reports the bias corrected sample kurtosis with an extra correction term. At least four values are needed, and a nonzero spread is required, so identical values have no defined kurtosis.

Worked example

Reference table

Limits

• Needs at least four values, and more for a stable figure.
• Very sensitive to a single extreme value.
• Describes tails only, not the lean, which is skewness.
• The definition differs from some software, so values can differ slightly.

Common mistakes

• Reading a kurtosis near 3 as wrong, when it means close to normal.
• Comparing this figure with software that uses the corrected sample form.
• Running it on fewer than four values.
• Letting one outlier drive the whole conclusion.
• Confusing heavy tails with high variance, which are different ideas.

Try it yourself

Quick checks. Answers sit tucked under each question.

Questions

Sources

• NIST: measures of skewness and kurtosis

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