# Negative skew on positive data set

I have a data set of 93 points where the lowest value is 0.26 and the highest is 1.06. When I compute the sample skewness in Mathematica using the languages built in function I get a skewness of -0.752! This seems bizarre to me. Is it possible to have a data set in which every point is positive, yet the skewness is negative? I fail to see how this can happen. What's going on here, error in the language? Something else I might be missing?

Yes this is entirely possible. To quote wikipedia:

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive or negative, or undefined.

If you take a look at what this means visualy (from WhatIs.com):

Your data seems to have a tendendcy to be left skewed and thus has heavier tails on the left side of the distribution, when compared to a symetrical (normal) distribution.

• "skewness ... about its mean" is the key point here. Shifting the location of a distribution (adding a positive or negative constant to all the values) does not affect its skewness – Henry Aug 4 '17 at 14:33

I have a data set of 93 points where the lowest value is 0.26 and the highest is 1.06. ... I get a skewness of -0.752

This seems bizarre to me.

Is it possible to have a data set in which every point is positive, yet the skewness is negative?

Certainly, since skewness is unaffected by location shift. Since it seems unsusprising to have negative skewness with negative data, try this exercise ---

• Make yourself a set of data with negative skewness, any way you like -- and by any reasonable measure of skewness you prefer.

Small data sets are fine; you'll need at least 3 observations. (I'll do the same exercise, parenthetically)

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(here's my set of data:

-95  -27 -303  -21 -162  -62 -130  -95 -395  -11  -89   -6   -5 -156 -224
-71  -45  -31  -23  -71


One possible formula for moment skewness gives about -1.50. Second Pearson skewness is -0.868)

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• Now add a constant amount to every observation, enough to make all the data positive. Recompute the skewness.

This will be much more effective if you actually do the exercise. Try it!

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If you did it right, the skewness didn't change.

(My try: I add 400 to every data point. Now I get that moment skewness is -1.50 and second Pearson skewness is -0.868)

• A second exercise would be to make a data set (all positive) with positive skewness and flip it around in place -- that is, let $y_i = \max(x_i)+\min(x_i)-x_i$ for $i=1, 2,...,n$; you'll end up with negative skew and positive data.

Try this small data set (but also make one for yourself):

1  2  4  8 15


the skewness is positive (by pretty much any skewness measure you like).

Now make a new data set but subtracting those values from 15+1 = 16:

15 14 12 8 1

(leaving the min and max unchanged). You should get the same value for skewness, but with the opposite sign, so it will be negative. Again, positive data, negative skewness.

I fail to see how this can happen.

Something else I might be missing?

Well, it kind of looks like you don't quite understand what any of the usual measures of skewness is measuring but it's hard to guess what precise line of thinking you had because you didn't explain it -- it's not clear why you thought what you did.

This happens with real data -- see for example the numerous examples here: Real life examples of distributions with negative skewness

Or consider the very simple case of binomial sampling with $p$ considerably higher than one-half (roll an ordinary die ten times and count the number of times you rolled more than 1). Here's a simulated set of 1000 such counts:

As you see, the values are positive (all between 4 and 10 in this run) and yet (by inspection) the skewness will be negative.