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Kurtosis Calculator

Enter a list of numbers to compute kurtosis.

How it is calculated

The list needs at least 4 numbers and not all equal. Separate them with spaces, semicolons, or commas. A list with spaces or semicolons and no comma-plus-space may use decimal commas, so 2,5 3,5 4 9 is read as 2.5, 3.5, 4, 9, while 1,2,3,4 and 1, 2, 3 use the comma as a separator. The tool takes the mean, then the second and fourth central moments: m₂ = Σ(x − mean)² ÷ n and m₄ = Σ(x − mean)⁴ ÷ n. Kurtosis is m₄ ÷ m₂², and excess kurtosis is that minus 3, so a normal distribution reads 0. This is the population version, dividing by n, not the sample-corrected version spreadsheets often report, so small samples give different numbers. Kurtosis does not depend on scale, so the data are rescaled internally and very large values do not overflow. Fewer than 4 values, text, or identical values give no result.

Worked example

Data: 2, 4, 4, 4, 5, 5, 7, 9, 30, so n = 9 and the mean is 70 ÷ 9 ≈ 7.78. The squared deviations average to m₂ ≈ 65.28, and the fourth powers average to m₄ ≈ 27,301.4. Kurtosis is 27,301.4 ÷ 65.28² ≈ 6.41. Excess kurtosis is 6.41 − 3 = 3.41. The positive value comes mostly from the single extreme value 30, which creates a heavy right tail.

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Common mistakes

  • Mixing separators in one list, such as 2,5, 3,5, 4. A comma followed by a space is always a separator, so decimal commas only work in a list separated by spaces or semicolons alone; use a decimal point when unsure.
  • Comparing the result with a spreadsheet's KURT value. That function applies a small-sample correction, while this tool divides by n, so the two disagree for short lists.
  • Reading excess kurtosis as outlier count. A single extreme value can dominate it, as the value 30 does in the example.
  • Using fewer than 4 values or a list of identical numbers. The first is rejected, and identical numbers have zero spread, so there is nothing to measure.
  • Confusing kurtosis with excess kurtosis. Normal data has kurtosis near 3 but excess kurtosis near 0, so check which line you are quoting.

About this calculator

Kurtosis measures how heavy or light the tails of a distribution are compared to its center — in other words, how likely extreme values are relative to a normal distribution. A normal (Gaussian) distribution has a kurtosis of exactly 3, so statisticians usually report excess kurtosis, which is kurtosis minus 3, so that a normal distribution reads as 0. Positive excess kurtosis (leptokurtic) means heavier tails and more outliers than normal, while negative excess kurtosis (platykurtic) means lighter tails and fewer extreme values.

Financial analysts watch excess kurtosis in return distributions because heavy tails mean extreme gains or losses are more likely than a normal-distribution model would predict — a key input for risk management and stress-testing beyond what standard deviation alone captures. Quality-control engineers and researchers fitting statistical models to real data use it the same way, as a check on whether a normal-distribution assumption is safe to use.

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