Skewness And Kurtosis CalculatorFree online math calculator — instant results, no sign-up
Calculate population moment skewness and excess kurtosis.
About the Skewness And Kurtosis Calculator
The Skewness And Kurtosis Calculator measures asymmetry and tail weight using population moment skewness and excess kurtosis for a list of numerical values. For the default values 2, 4, 6, 8, 10, it returns skewness = 0 and excess kurtosis = -1.3. Both outputs are dimensionless and use moments divided by the number of values, without sample bias corrections.
- Values
- Enter the numerical observations to analyze, such as 2, 4, 6, 8, 10, all expressed in the same units or all unitless.
Formula and method
μ = (1/N) Σᵢ₌₁ᴺ xᵢ; mᵣ = (1/N) Σᵢ₌₁ᴺ (xᵢ − μ)ʳ for r = 2, 3, 4; γ₁ = m₃ / m₂^(3/2); γ₂ = m₄ / m₂² − 3.
- NNumber of entered values.
- xᵢThe value at position i in the list.
- iIndex identifying each observation, from 1 through N.
- ΣSum over all entered observations.
- μArithmetic mean of the entered values.
- rOrder of the central moment: 2, 3, or 4.
- mᵣPopulation central moment of order r; m₂ is the population variance.
- γ₁Population moment skewness, a dimensionless measure of asymmetry.
- γ₂Population moment excess kurtosis, equal to standardized fourth moment minus 3.
Compute the mean, subtract it from each value, and average the squared, cubed, and fourth-power deviations to obtain m₂, m₃, and m₄. For 2, 4, 6, 8, 10, the mean is 6 and the deviations are −4, −2, 0, 2, 4. Their moments are m₂ = 8, m₃ = 0, and m₄ = 108.8. Therefore, skewness is 0 / 8^(3/2) = 0, and excess kurtosis is 108.8 / 8² − 3 = −1.3.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Values2, 4, 6, 8, 10
Results
- Skewness0
- Excess kurtosis-1.3
How to use the Skewness And Kurtosis Calculator
- 1Enter the observations in Values; keep 2, 4, 6, 8, 10 to reproduce the default example.
- 2Check that all entries are numeric and use consistent units.
- 3Ensure at least two values differ so the variance is greater than zero.
- 4Run the calculation to obtain population moment skewness and excess kurtosis.
- 5Read both outputs together; the default list gives skewness = 0 and excess kurtosis = −1.3.
Reading your result
Skewness: positive values indicate asymmetry toward higher values, while negative values indicate asymmetry toward lower values. Zero means the third central moment is zero; the default list is symmetric about 6, so its skewness is 0. Excess kurtosis: zero matches the normal distribution's standardized fourth moment, positive values indicate a larger standardized fourth moment, and negative values indicate a smaller one. The default result of −1.3 corresponds to ordinary kurtosis of 1.7, below the normal benchmark of 3. Kurtosis is commonly interpreted in terms of tail weight and sensitivity to extreme observations, not simply peak height.
Limitations
The calculator treats the entered list as an equally weighted population and does not apply bias-corrected sample estimators. An empty list or zero variance makes these statistics undefined; zero variance occurs when every value is identical, including a single-value input. Small datasets can give unstable descriptions of a broader population, and extreme observations strongly affect both moments, especially kurtosis. Zero skewness does not prove symmetry, and zero excess kurtosis does not prove normality. Very large magnitudes or nearly identical values may also introduce numerical rounding error.
Common questions
Why do 2, 4, 6, 8, 10 give skewness of zero?+
The values are symmetric about their mean of 6. The cubed deviations −64, −8, 0, 8, and 64 cancel, so m₃ and skewness are both zero.
Why is the default excess kurtosis −1.3 rather than 1.7?+
The standardized fourth moment is 108.8 / 64 = 1.7. This calculator reports excess kurtosis, which subtracts the normal-distribution benchmark of 3, giving −1.3.
Why might a spreadsheet return different skewness or kurtosis for the same list?+
Some spreadsheet functions use bias-corrected sample formulas rather than population moment formulas. Others report ordinary kurtosis instead of excess kurtosis. Match both the population convention and the subtraction of 3 when comparing results.
Can the calculator compute skewness and excess kurtosis when all values are identical?+
No. Identical values have m₂ = 0, so the denominators in both formulas are zero. Both outputs are undefined, not zero.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.