About the Skewness & Kurtosis Calculator
This skewness and kurtosis calculator describes the shape of a data set. Paste your numbers and it returns the sample skewness G₁ (the adjusted Fisher-Pearson coefficient used by Excel’s SKEW, SPSS and SAS), the population skewness g₁ (Excel’s SKEW.P), the sample excess kurtosis G₂ (Excel’s KURT), the population excess kurtosis g₂ and Pearson’s kurtosis (g₂ + 3), along with their standard errors, z-statistics, Pearson’s median skewness and a histogram.
Skewness tells you whether the data have a longer tail to the right (positive) or to the left (negative); kurtosis tells you whether the tails are heavier or lighter than a normal distribution’s. Analysts use these numbers to check the normality assumption before a t-test, ANOVA or regression, to decide whether a log transformation is needed, and to describe the risk of returns in finance, where fat tails (positive excess kurtosis) matter.
Different software uses different formulas, which is the most common reason results disagree. The calculator shows both the sample (bias-adjusted) and population versions side by side. Sample skewness needs at least 3 values and sample kurtosis at least 4; with small samples, both estimates are very noisy, so judge them against their standard errors.
With the default inputs, the sample skewness g₁ (excel skew) is 1.5979. Change any value above to recalculate instantly.
How to use the skewness & kurtosis calculator
- 1Paste or type your data into the box, separated by commas, spaces or new lines.
- 2Read the sample skewness G₁ — positive means a longer right tail.
- 3Read the sample excess kurtosis G₂ — positive means heavier tails than a normal distribution.
- 4Compare each value with its standard error, and check the histogram for the overall shape.
Formula and method
Skewness and kurtosis are built from central moments: m₂ is the average squared deviation from the mean, m₃ the average cubed deviation and m₄ the average fourth-power deviation. The population (moment) skewness g₁ standardises m₃ by m₂^1.5, and the excess kurtosis g₂ standardises m₄ by m₂² and subtracts 3 so that a normal distribution scores 0. Pearson’s kurtosis is g₂ + 3, the version some textbooks call simply “kurtosis”.
G₁ and G₂ are the bias-adjusted sample versions used by Excel (SKEW, KURT), SPSS, SAS and most textbooks; they are the better choice when your data are a sample. The standard errors SES = √(6n(n − 1) / ((n − 2)(n + 1)(n + 3))) and SEK = 2·SES·√((n² − 1) / ((n − 3)(n + 5))) let you test for non-normality: a ratio beyond about ±1.96 is significant at the 5% level.
- n
- Number of values
- x̄
- Mean of the data
- m₂, m₃, m₄
- Second, third and fourth central moments (divided by n)
- g₁, G₁
- Population and sample (adjusted) skewness
- g₂, G₂
- Population and sample excess kurtosis
Worked examples
Right-skewed household incomes ($ thousands)
One high income of 120 stretches the right tail: the mean (58.92) is well above the median (53.5). The sample skewness G₁ ≈ 1.60 is about 2.5 times its standard error of 0.64, and the sample excess kurtosis G₂ ≈ 2.96 shows a heavy right tail.
Symmetric data with light tails
Evenly spaced values are perfectly symmetric, so skewness is 0. Their flat, uniform-like shape gives a population excess kurtosis of −1.25 (Pearson kurtosis 1.75) and sample excess kurtosis −1.2, the typical signature of light tails.
Left-skewed test scores
Most students scored in the 70s to 90s but one scored 45, pulling the mean (78.25) below the median (81). The sample skewness of about −1.72 confirms the longer left tail, and the single low score also makes the tails heavy (G₂ ≈ 3.82).
Frequently asked questions
What is a good skewness value?+
Values between −0.5 and 0.5 indicate an approximately symmetric distribution, 0.5 to 1 in absolute value is moderate skew and beyond ±1 is high skew. For normality checks, many analysts accept skewness between −2 and 2 together with excess kurtosis between −7 and 7 for large samples.
What is the difference between kurtosis and excess kurtosis?+
Pearson kurtosis equals 3 for a normal distribution. Excess kurtosis subtracts 3 so the normal distribution scores 0. Excel’s KURT, SPSS and this calculator’s G₂ report excess kurtosis.
Why does my software give a different skewness?+
Programs use different formulas. Excel SKEW, SPSS and SAS use the adjusted sample statistic G₁; Excel SKEW.P and NumPy/SciPy by default use the population moment version g₁; Minitab uses a third variant. The calculator shows both main versions.
Does high kurtosis mean a sharp peak?+
Not really. Kurtosis mainly measures tail weight — how often extreme values occur — rather than peakedness. A distribution can have a flat top and still have high kurtosis if it has heavy tails.
How many data points do I need?+
The formulas need at least 3 values for skewness and 4 for kurtosis, but estimates are unstable below about 20–30 observations. Use the standard errors to judge whether the values are meaningfully different from zero.