Skewness and kurtosis calculator
Paste your numbers, get skewness and excess kurtosis the way Excel, SAS and SPSS count them.
How do I calculate skewness and kurtosis?
With m₂, m₃ and m₄ the average squared, cubed and fourth-power deviations from the mean (each sum divided by n): g₁ = m₃ ÷ m₂^(3/2) and g₂ = m₄ ÷ m₂² − 3. Sample versions correct these for small samples: G₁ = g₁ × √(n(n − 1)) ÷ (n − 2) and G₂ = (n − 1) ÷ ((n − 2)(n − 3)) × ((n + 1)g₂ + 6). For the data 3, 4, 5, 2, 3, 4, 5, 6, 4, 7 (n = 10): g₁ = 0.303193, G₁ = 0.359543 and G₂ = −0.1518 — the values in the Microsoft Excel documentation for SKEW.P, SKEW and KURT.
What is the difference between G1, g1 and b1?
Three conventions from Joanes and Gill (1998). g₁ and g₂ are the plain moment ratios (NIST e-Handbook, older textbooks, Excel’s SKEW.P). G₁ and G₂ are the adjusted versions that are unbiased for normal data and are used by SAS, SPSS and Excel’s SKEW and KURT. b₁ and b₂ are based on the sample standard deviation (÷ n − 1) and are used by Minitab and BMDP. The results are close for large samples and differ noticeably for small ones, so use the one your course, journal or software expects.
What do positive and negative skewness mean?
Positive skewness means the longer tail is on the right (right-skewed); negative means it is on the left (NIST e-Handbook, section 1.3.5.11). As a rule of thumb from Bulmer (1979, Principles of Statistics), skewness between −0.5 and 0.5 is approximately symmetric, between 0.5 and 1 (or −1 and −0.5) moderately skewed, and beyond ±1 highly skewed. It is a convention, not a test.
What does excess kurtosis tell me?
Excess kurtosis is kurtosis minus 3, so a normal distribution scores 0. Positive values mean heavier tails than the normal distribution, negative values lighter tails (NIST e-Handbook, section 1.3.5.11). Kurtosis is about the tails, not the “peakedness” of the middle. The calculator also shows plain kurtosis (excess + 3), the form NIST defines first.
Is my skewness large enough to matter?
Compare it with its standard error. The standard error of skewness is √(6n(n − 1) ÷ ((n − 2)(n + 1)(n + 3))) and of kurtosis 2 × SES × √((n² − 1) ÷ ((n − 3)(n + 5))). As a rule of thumb, Z = G ÷ standard error beyond ±2 suggests the population really is skewed (Stan Brown, brownmath.com). With 10 values the standard error of skewness is 0.687, so G₁ = 0.36 is well inside the noise.
How many values do I need?
At least 3 for skewness and 4 for kurtosis, and the values must not all be equal. Small samples give very unstable estimates; the standard errors above show how unstable.
Worked example: how one value changes the shape
Take the evenly spaced values 4 through 12. They are symmetric about 8, so skewness is G₁ = 0.00, and excess kurtosis G₂ = −1.20: a flat, light-tailed set, below the 0 of a normal distribution. Now take 1, 2, 2, 3, 3, 3, 4, 4, 5, 20. The long tail on the right gives G₁ = 2.90 and G₂ = 8.82, both large and positive. With n = 10 the standard error of skewness is 0.687, so Z = 4.23, beyond +2, which is the point at which the rule of thumb says the population is really skewed. Remove the 20 and the remaining nine values, which are symmetric about 3, give G₁ = 0.00. A single outlier accounts for all of the skewness here, so a large value should send you to check the data point first and the distribution second.
Mistakes to avoid
Comparing numbers from different conventions is the main one: Excel's SKEW and SKEW.P differ, as do the three forms of kurtosis, and the difference is large for small samples. Another is reading kurtosis as peakedness; it describes the weight of the tails, and a data set with a sharp center and no tails can have low kurtosis. A third is using skewness as a normality test on a small sample, where its standard error is so wide that almost anything passes. Finally, skewness is zero for symmetric data but a zero value does not prove symmetry, since the third moment can cancel out in lopsided data that has an odd mix of tails. For a decision about normality, look at a histogram or a normal probability plot first.
Sources: D. N. Joanes and C. A. Gill (1998), Comparing measures of sample skewness and kurtosis, The Statistician 47; R package e1071 documentation (skewness, kurtosis); NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.11; Microsoft Excel help for SKEW, SKEW.P and KURT; Stan Brown, Measures of Shape: Skewness and Kurtosis. More about spread: the standard deviation calculator and the standard error calculator.