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
| Value | Skewness means | Excess kurtosis means |
|---|---|---|
| < −1 | Long left tail | Very light tails |
| −0.5 to 0.5 | About symmetric | About normal tails |
| > 1 | Long right tail | Heavy 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])
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)))
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.
Frequently asked questions
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.