# Skewness and Kurtosis Calculator URL: https://speedycalculator.com/calculators/statistics/skewness/ Last updated: 2026-10-30 > The skewness calculator measures the asymmetry of a data set with the adjusted sample skewness (as in Excel's SKEW), the population skewness and Pearson's median skewness, and also gives the excess kurtosis, standard error of skewness, a significance z-score and a plain-language description of the shape. ## Inputs - Data ## Results - Skewness (adjusted sample, G₁) - Shape - Excess kurtosis (G₂) - Tails - Population skewness (g₁) - Pearson median skewness 3(x̄ − median) ÷ s - Standard error of skewness - z = skewness ÷ SE - Mean - Median - Standard deviation - Number of values ## Formula ``` G₁ = n ÷ ((n − 1)(n − 2)) × Σ((x − x̄) ÷ s)³ · g₁ = (1/n)Σ((x − x̄) ÷ σ)³ · G₂ = n(n + 1)Σ(x − x̄)⁴ ÷ ((n − 1)(n − 2)(n − 3)s⁴) − 3(n − 1)² ÷ ((n − 2)(n − 3)) · SES = √(6n(n − 1) ÷ ((n − 2)(n + 1)(n + 3))) ``` ## Explanation Skewness describes whether a distribution leans to one side. In a symmetric distribution, such as the normal curve, the mean and median coincide and skewness is zero. When a long tail stretches to the right – incomes, house prices, reaction times – skewness is positive and the mean is pulled above the median. A long left tail, as in exam scores near a ceiling or the age at death in rich countries, gives negative skewness. Kurtosis describes the tails. Excess kurtosis compares them with a normal distribution: positive values (leptokurtic) mean heavier tails and more extreme values than a normal curve, negative values (platykurtic) mean lighter tails and a flatter shape, and values near zero are mesokurtic. The calculator uses the same adjusted formulas as Excel's SKEW and KURT and most statistics packages, needs at least 3 values for skewness and 4 for kurtosis, and divides skewness by its standard error to tell you whether the asymmetry is larger than you would expect by chance in a sample of that size from a normal population. ### How to use the Skewness and Kurtosis Calculator - **Data** Results update as you type, and everything is calculated in your browser. ### How it works Paste your data. The calculator standardizes every value with the mean and sample standard deviation, averages the cubes (with the small-sample adjustment) for skewness and the fourth powers for kurtosis, and compares the skewness with its standard error. ### Worked example For 2, 3, 3, 4, 5, 6, 8, 12 and 15 the mean is 6.444, the median 5 and the sample standard deviation 4.447. The adjusted skewness is G₁ = 1.122 – highly skewed to the right, because 12 and 15 form a long right tail – and the population version is g₁ = 0.926. Pearson's median skewness is 3 × (6.444 − 5) ÷ 4.447 = 0.974. Excess kurtosis is 0.241, close to normal. The standard error of skewness for n = 9 is 0.717, so z = 1.57: with so few values the asymmetry is not statistically significant. ### Reading skewness and kurtosis ValueSkewness meansExcess kurtosis means< −1Long left tailVery light tails−0.5 to 0.5About symmetricAbout normal tails> 1Long right tailHeavy tails, outlier-prone ### What each result means - **Skewness (adjusted sample, G₁)** = skew([Data]) - **Population skewness (g₁)** = pskew([Data]) - **Pearson median skewness 3(x̄ − median) ÷ s** = 3×(mean([Data]) - median([Data]))÷stdev([Data]) - **Standard error of skewness** = sqrt(6×count([Data])×(count([Data]) - 1)÷((count([Data]) - 2)×(count([Data]) + 1)×(count([Data]) + 3))) - **z = skewness ÷ SE** = [Skewness (adjusted sample, G₁)]÷[Standard error of skewness] - **Mean** = mean([Data]) - **Median** = median([Data]) - **Standard deviation** = stdev([Data]) - **Number of values** = count([Data]) ## Worked example Inputs — Data: 2, 3, 3, 4, 5, 6, 8, 12, 15. Results — **Skewness (adjusted sample, G₁) = 1.1224**; Shape = Highly skewed to the right (long right tail); Excess kurtosis (G₂) = 0.2413; Tails = Close to normal (mesokurtic); Population skewness (g₁) = 0.9259; Pearson median skewness 3(x̄ − median) ÷ s = 0.9744; Standard error of skewness = 0.7171; z = skewness ÷ SE = 1.565; Mean = 6.4444; Median = 5; Standard deviation = 4.4472; Number of values = 9. ## Key terms - **Skewness**: A measure of the asymmetry of a distribution around its mean. - **Excess kurtosis**: Kurtosis minus 3, so that the normal distribution scores 0. - **Leptokurtic**: Heavier tails and a sharper peak than a normal distribution. - **Platykurtic**: Lighter tails and a flatter top than a normal distribution. ## FAQ **What is a normal skewness value?** Between about −0.5 and 0.5 is approximately symmetric; between ±0.5 and ±1 moderately skewed; beyond ±1 highly skewed. **How do I know if skewness is significant?** Divide it by its standard error. |z| above about 2 suggests real asymmetry. With 9 values SE ≈ 0.72, so even a skewness of 1.12 gives only z ≈ 1.57. **What does positive skewness mean?** A longer right tail: most values are low or moderate and a few are much larger, so the mean exceeds the median. Incomes are the classic example. **Why does my result differ from another program?** There are three common formulas. This calculator's main value matches Excel's SKEW and SPSS (G₁); the population version g₁ is also shown. For large samples they agree. **What is kurtosis?** A measure of tail weight. Excess kurtosis is 0 for a normal distribution, positive for heavy tails (e.g. the t distribution), negative for light tails (e.g. the uniform distribution, −1.2). **How many data points do I need?** At least 3 for skewness and 4 for kurtosis, but estimates are very unstable below about 20–30 values. **How can I reduce skewness?** Right-skewed positive data often become symmetric after a log or square-root transformation. ## Sources - [NIST/SEMATECH e-Handbook – Measures of skewness and kurtosis](https://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm) - [OpenStax – Introductory Statistics 2e](https://openstax.org/details/books/introductory-statistics-2e) - [Joanes & Gill (1998), Comparing measures of sample skewness and kurtosis, The Statistician](https://doi.org/10.1111/1467-9884.00122)