or drag a file onto the table
Type data directly or paste from Excel. Enter moves down. New rows are added automatically. Paste options under More options.

Skewness and kurtosis

Skewness measures how asymmetric a distribution is. A symmetrical distribution has a skewness of 0. Positive skewness means a longer right tail, and pulls the mean above the median; negative skewness means a longer left tail, and pulls the mean below it. Kurtosis measures the weight of the tails. Excess kurtosis subtracts 3, so a normal distribution scores 0: positive values mean heavier tails and more outliers (leptokurtic), negative values mean lighter tails (platykurtic). See the distribution shape guide for worked examples.

Which method should I choose?

The three options differ only in how they correct for sample size. They converge as n grows, but on small samples they can differ noticeably — which is why a number copied from Excel may not match one copied from R.

MethodSkewnessExcess kurtosisMatches
Type 1
(population)		g1 = m3 / m21.5g2 = m4 / m22 − 3R moments, e1071 type 1, Excel SKEW.P. Also what the normality tests use internally.
Type 2
(default)		G1 = g1 · √(n(n−1)) / (n−2)G2 = [(n+1)g2+6](n−1) / ((n−2)(n−3))Excel SKEW and KURT, SPSS, SAS, e1071 type 2. The estimator used elsewhere on this site.
Type 3b1 = g1 · ((n−1)/n)1.5b2 = (g2+3)(1−1/n)2 − 3MINITAB, BMDP, and e1071's own default.

Note that R's kurtosis() returns the raw kurtosis, not the excess.

The p-values

The normality p-values come from the D'Agostino–Pearson omnibus test, which combines a transform of the skewness (D'Agostino, 1970) with a transform of the kurtosis (Anscombe & Glynn, 1983) into a statistic that follows a chi-squared distribution with 2 degrees of freedom. Those transformations are defined on the type 1 statistics, so the tests always use type 1 no matter which method is selected for display — the same convention as R's fBasics::dagoTest(). They need at least 8 observations, and are unreliable below about 20. If you want a dedicated normality test, use the Shapiro-Wilk test or the D'Agostino-Pearson calculator.

How to use this calculator

  1. Type or paste one column of numbers into the grid, or use Import for a .csv or .xlsx file.
  2. Optionally open More options to pick the estimator, the excess-kurtosis convention and the histogram binning rule.
  3. Press Calculate.