How to use the skewness and kurtosis calculator
- Paste at least four values, separated by commas, spaces or new lines.
- Read the adjusted skewness and kurtosis (G1 and G2) in the first two rows. The population versions g1 and g2 are listed below them for comparison with other software.
- Divide each statistic by its standard error, shown for you, to see how many standard errors away from zero it is.
- The Jarque-Bera statistic and p value combine both measures into one test of whether the shape matches a normal curve.
Formula
g1 = m3 / m2^1.5 g2 = m4 / m2^2 - 3
G1 = sqrt(n (n - 1)) / (n - 2) x g1
G2 = (n - 1) / ((n - 2)(n - 3)) x ((n + 1) g2 + 6)
Jarque-Bera = n / 6 x (g1^2 + g2^2 / 4), p = exp(-JB / 2)
m2, m3 and m4 are the average of the squared, cubed and fourth-power distances from the mean. A normal distribution has skewness 0 and excess kurtosis 0. The p value is the chi-square tail for two degrees of freedom, which has the simple form exp(-JB / 2).
Thirteen values with a long right tail
The default values are 2.1, 2.4, 2.4, 2.9, 3.0, 3.1, 3.3, 3.6, 3.8, 4.4, 5.2, 6.9 and 9.8 with a mean of 4.0692. The population skewness is 1.6432 and the adjusted skewness 1.8658, with a standard error of 0.6163, so it sits 3.03 standard errors above zero.
The excess kurtosis is 1.8845 (adjusted 3.5327, standard error 1.1909). The Jarque-Bera statistic is 13 / 6 x (1.6432^2 + 1.8845^2 / 4) = 7.7739 and the p value is exp(-3.887) = 0.0205, so these values differ from a normal shape more than chance would usually allow.
Reading skewness and kurtosis
Skewness measures lopsidedness. A positive value means a longer tail on the right, as with incomes or waiting times, and a negative value a longer tail on the left. Excess kurtosis measures how heavy the tails are compared with a normal curve. A positive value means more extreme values than a normal curve would produce.
Both statistics are noisy for small samples. A single extreme value can move them a lot, so look at a chart of the data too, such as the Box Plot Maker or the Histogram Maker.
Limits of the Jarque-Bera test
The test compares your sample with a normal shape by using only skewness and kurtosis. Its p value rests on a large-sample approximation, so it is rough below a few hundred values. A small p value says the shape differs from normal; it does not say the difference matters for your analysis, and a large p value from a small sample does not prove normality.
More statistics tools
To go further with a data set, try the Kaplan-Meier Estimator, Coupon Collector Calculator and Markov Chain Calculator.
Assumptions and limits
- The values are independent observations from one group.
- At least four values are needed, and the values must not all be equal.
- Adjusted (G1, G2) and population (g1, g2) versions differ for small samples; use the one your field or software expects.
- The Jarque-Bera p value is approximate for small samples.
Frequently asked questions
Which skewness does a spreadsheet report?
The SKEW function gives the adjusted value G1, and KURT gives the adjusted excess kurtosis G2. Other packages report the population versions g1 and g2, so both are shown here.
What is excess kurtosis?
It is kurtosis minus 3, so that a normal curve scores 0. Without that subtraction the plain kurtosis of a normal curve is 3.
Does a significant Jarque-Bera result mean I cannot use a t test?
Not necessarily. Many methods tolerate moderate departures from normality, especially with larger samples. Look at how large the skewness and kurtosis are, not only at the p value.
Why are there only four values required?
The adjusted kurtosis formula divides by (n - 2) and (n - 3), so it is not defined for fewer than four values.