Skewness and Kurtosis Calculator

How to Use the Skewness and Kurtosis Calculator

1. 1. Paste your data. Separate the numbers with commas, spaces or new lines. Skewness needs at least three values with some spread; kurtosis needs four.
2. 2. Read G1 and G2. These are the sample-adjusted values SPSS, SAS and Excel report, and the ones most journals expect. The cards show them first.
3. 3. Check the tests. Each statistic is divided by its standard error to give a z-score and a p-value. A significant z means the data are unlikely to come from a normal population on that measure.
4. 4. Compare the formulas. If you need to match another program, the table lists g, G and b side by side for the same data, and the working shows each formula with your numbers.

What Skewness and Kurtosis Measure

Skewness measures asymmetry. It is zero for a symmetric distribution, positive when the long tail points to the right (a few unusually high values pull the mean above the median), and negative when it points left. A common rule of thumb (Bulmer, 1979) calls |skewness| below 0.5 approximately symmetric, 0.5–1 moderately skewed and above 1 highly skewed.

Kurtosis measures how much of the variance comes from extreme values. The normal distribution has a kurtosis of 3, so software usually reports excess kurtosis, kurtosis − 3, which is zero for a normal curve. Positive excess kurtosis (leptokurtic) means heavier tails and more outliers than a bell curve would produce; negative (platykurtic) means lighter tails. Despite the textbook picture, kurtosis is about the tails, not the peak (Westfall, 2014): a distribution can be flat-topped and still have heavy tails.

Why There Are Three Kinds of Skewness and Kurtosis

Programs disagree because they use different formulas. Joanes and Gill (1998) named the three families, and R's e1071 package exposes them as types 1, 2 and 3. With m₂, m₃ and m₄ the average squared, cubed and fourth-power deviations from the mean and n the sample size:

Standard Errors and the z-Test

Under a normal population the standard error of G1 is √(6n(n − 1) ÷ ((n − 2)(n + 1)(n + 3))), and the standard error of G2 is 2 × SE(G1) × √((n² − 1) ÷ ((n − 3)(n + 5))). These are the values SPSS prints — for example 0.512 and 0.992 for n = 20. Dividing the statistic by its standard error gives z, and the calculator reports the two-sided normal p-value and a 95% interval, value ± 1.96 × SE.

How to read z depends on sample size (Kim, 2013). With fewer than 50 values, |z| above 1.96 suggests non-normality; for 50–299 values use 3.29, because the z-test becomes oversensitive; with 300 or more, look at the histogram and the size of the statistics instead, with absolute skewness above 2 or kurtosis above 7 as warning signs. For a formal normality test, use the Shapiro-Wilk calculator.

A Worked Example: Right-Skewed Data

Click "Try a skewed example" to load 12 values: 2.1, 2.4, 2.6, 2.7, 2.9, 3, 3.2, 3.5, 3.9, 4.6, 5.8 and 8.2. The mean (3.7417) is well above the median (3.1), a sign of right skew. The three skewness values are g1 = 1.5696, G1 = 1.8033 and b1 = 1.3775 — all "highly skewed", though G1 is about 30% larger than b1. The excess kurtosis values differ even more: g2 = 1.6109, G2 = 3.2928 and b2 = 0.8744.

With n = 12, SE(G1) = 0.6373 and SE(G2) = 1.2322, so z = 2.8296 for skewness (p = 0.0047) and z = 2.6722 for kurtosis (p = 0.0075). Both are significant at 5%, and the 95% interval for G1 is 0.5542 to 3.0524. In a report you would write: skewness = 1.80 (SE = 0.64), kurtosis = 3.29 (SE = 1.23). Taking natural logs of the same 12 values reduces G1 to 1.058, which is the usual first remedy for right skew.

The symmetric example is the opposite case: 16 values in a bell-like pile around 4 give G1 = 0 and G2 = −0.4582, slightly lighter-tailed than normal, with a standard error of 1.0908 that shows how little a sample this small can tell you.

Frequently Asked Questions

What is a good skewness and kurtosis for a normal distribution?

Skewness near 0 and excess kurtosis near 0 (a raw kurtosis near 3). In practice, absolute skewness below 1 and excess kurtosis between −1 and 1 are often treated as acceptable, and many journals accept |skewness| < 2 and |excess kurtosis| < 7 for large samples. Always check the histogram as well; the numbers alone can hide a bimodal shape.

Why does SPSS give a different skewness from R or Minitab?

SPSS reports G1 and G2, R's e1071 package defaults to type 3 (b1 and b2, the same as Minitab), and the moments package returns g1 and raw kurtosis. They are all valid estimators with different small-sample corrections, so the values differ for small n and converge for large n. Pick the one your software or course uses, and name it in your methods section.

What is the difference between kurtosis and excess kurtosis?

Kurtosis (Pearson's) is m₄ ÷ m₂², which equals 3 for a normal distribution. Excess kurtosis subtracts 3 so that normal is zero. SPSS, Excel's KURT and SciPy report excess kurtosis by default; Stata and the moments package report raw kurtosis. The table above shows both.

How do I interpret positive and negative skewness?

Positive skewness means the right tail is longer: most values are low and a few are very high, as with incomes or reaction times, and the mean exceeds the median. Negative skewness means the left tail is longer, as with scores on an easy test with a ceiling, and the mean falls below the median.

Does kurtosis measure how peaked a distribution is?

Not really. Kurtosis is driven by the tails: it rises when a few values lie far from the mean. A distribution can look flat in the middle and still have high kurtosis, so the old description of kurtosis as peakedness is misleading (Westfall, 2014).

How many data points do I need?

Three for skewness and four for kurtosis are the mathematical minimum, but both statistics are very noisy in small samples; the standard error of G1 is 0.64 even at n = 12. Treat results from fewer than 20 values as rough, and rely on the histogram and the z-test together.