Skewness Calculator

What the skewness calculator does

Skewness measures how lopsided a data set is. A perfectly symmetric set has a skewness of zero, and the further it leans to one side, the further the skewness moves from zero. This calculator works it out from your data, along with the mean, the standard deviation, and, for four or more values, the kurtosis.

Skewness is the number behind that gut feeling that data is bunched at one end with a long tail trailing off the other. Below is how it is measured and how to read it.

How to use it

1. Enter your numbers, one per box, starting with the first three and adding more with the buttons, up to thirty.
2. Press Calculate for the skewness and the summary statistics, or Reset to clear it.

You need at least three values to measure skewness, and at least four for the kurtosis to appear.

How the skewness is worked out

Skewness looks at how far each value sits from the mean, but it cubes those distances rather than squaring them. Cubing keeps the direction: a value above the mean stays positive, one below stays negative. So if the big distances are mostly on the high side, the cubes add up to a positive total, and if they are on the low side, a negative one.

That total is then scaled by the standard deviation cubed and adjusted for the number of values, which puts skewness on a standard, unitless scale you can compare across data sets. The particular formula here is the adjusted Fisher-Pearson coefficient, the same one Excel's SKEW function uses, so its results line up with that.

Reading the sign and the size

The sign tells you the direction. A positive skewness means the tail stretches to the right, toward the high values, which is common for things like incomes, with most values modest and a few very large. A negative skewness means the tail stretches to the left, toward the low values. A skewness near zero means the data is roughly symmetric.

The size tells you how strong the lean is. As a rough guide, a skewness between minus 0.5 and 0.5 is fairly symmetric, between 0.5 and 1 either way is moderately skewed, and beyond 1 or minus 1 is highly skewed. There is a quick sanity check too: when data is skewed right the mean usually sits above the median, and when it is skewed left the mean usually sits below.

The kurtosis it also gives you

Where skewness is about which side the tail is on, kurtosis is about the tails and the peak together, how much of the data lives out in the extremes versus bunched in the middle. The calculator reports excess kurtosis, measured against the normal bell curve, so a value near zero behaves like a normal distribution.

A positive excess kurtosis means heavier tails and a sharper peak than normal, with more extreme values than a bell curve would predict. A negative one means lighter tails and a flatter peak. Like the skewness here, it uses the same adjusted formula as Excel's KURT function.

A worked example

Take the five numbers 1, 2, 3, 4, 10. The mean is 4, and the sample standard deviation is about 3.54.

Four of the values sit close together while the 10 trails off to the right, so you would expect a positive skewness, and that is what comes out: about 1.70. Being well above 1, it signals a strongly right-skewed set, which matches the picture, most of the data low with one value stretching the tail out to the high side.

Entering your data, and the rounding

Enter each value in its own box, adding boxes as you need them, up to thirty. Every box you use needs a number, and the order does not matter. The skewness and kurtosis are shown to four decimal places with trailing zeros trimmed. Because both are sample-based formulas that adjust for the number of values, they are most reliable once you have a reasonable amount of data rather than just the minimum.

Questions people ask

What is skewness?

A measure of how lopsided a data set is. Zero means symmetric, a positive value means a tail to the right, and a negative value means a tail to the left.

What does positive skewness mean?

The tail stretches toward the high values, with most of the data on the low side and a few large values pulling out to the right. Incomes are a classic example.

How much skew is a lot?

As a rough guide, within half a point of zero is fairly symmetric, between 0.5 and 1 either way is moderate, and beyond 1 is highly skewed.

What is the kurtosis for?

It describes the tails and peak. Measured against the normal curve, a positive value means heavier tails and more extreme values, and a negative value means lighter tails and a flatter shape.

References

A quick note on where the methods here come from. The definitions of skewness and kurtosis are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. The particular sample formulas used here are the adjusted coefficients also implemented by Excel's SKEW and KURT functions, documented by Microsoft.

1. NIST/SEMATECH e-Handbook of Statistical Methods (skewness and kurtosis). https://www.itl.nist.gov/div898/handbook/
2. Microsoft, SKEW and KURT function documentation (adjusted Fisher-Pearson coefficients). https://support.microsoft.com/en-us/office/skew-function-bdf49d86-b1ef-4804-a046-28eaea69c9fa