# Skew and Kurtosis are so similar?

I've been taking the histogram of the optical flow in videos and plotting the kurtosis and skewness of each frame. At the end of the video, I noticed that the skewness and kurtosis follow each other - ie, when the skewness goes up so does the kurtosis and same when it goes down. In fact, the kurtosis almost looks just like a scaled version of the skewness. I know skewness and kurtosis are supposed to be different concepts entirely, since they are different moments (the graphs of x^3 and x^4 do not look similar at all ) but it just makes me wonder why take both when they both seem to be similar? How are they useful separately? Also would it be due to my distributions? For what distributions would this kind of behaviour occur? For reference the functions I am using are scipy.stats.skew and scipy.stats.kurtosis so I'm assuming that they are using sample skew and kurtosis ## 2 Answers

A simple and direct answer is that skewness and kurtosis are both defined in terms of the $$Z-$$values, $$Z = (X-\mu)/\sigma$$ : Skewness = $$E(Z^3)$$ and Kurtosis = $$E(Z^4)$$. When talking about a data set, you can just replace the expectation operator "$$E$$" with the ordinary average.

Since raising a number to either the third or fourth power amplifies values that are greater than 1 (in absolute value) and diminishes values that are less than 1 (in absolute value), both skewness and kurtosis are primarily measures of the heaviness of the tails of the distribution of $$X$$.

Note that the $$Z$$-values than are less than one in absolute value correspond to $$X$$-values that are less than one standard deviation from the mean; greater than one correspond to values that are more than one standard deviation from the mean. Hence, the averages of the $$Z^3$$ and $$Z^4$$ values are highly influenced by the $$X$$-values that are many standard deviations from the mean; i.e., by the tails or outliers.

Consider skewness first. By definition of skewness, and by definition of center of gravity, the graph of the probability distribution of $$Z^3$$ balances (i.e., has a center of gravity) at the skewness of $$X$$. Hence, if the tails are heavier on the left side of the distribution than on the right, then skewness is negative. Conversely, if the tails are heavier on the right side of the distribution than on the left, then skewness is positive.

Now consider kurtosis. The graph of the probability distribution of $$Z^4$$ balances (i.e., has a center of gravity) at the kurtosis of $$X$$. Note that the kurtosis of the normal distribution is exactly 3.0. Hence, if the tails are heavier on either or both sides of the distribution than the tails of the normal distribution, then the kurtosis is greater than 3.0, and the probability distribution of $$Z^4$$ falls to the right when a fulcrum is placed at 3.0. Conversely, if the tails are lighter on either or both sides of the distribution than the tails of the normal distribution, then the kurtosis is less than 3.0, and the probability distribution of $$Z^4$$ falls to the left when a fulcrum is placed at 3.0

Hence, skewness and kurtosis really do not measure things that differently - they are both measures of tail heaviness. The biggest difference is that skewness considers relative heaviness in one tail versus the other. The Wikipedia entry on kurtosis displays both lower and upper bounds for kurtosis in terms of skewness.

There is a mistaken but remarkably persistent meme out there that kurtosis measures "peakedness or flatness" that might be a source of confusion. Your work shows (yet again) that this characterization is false. See here for a clear explanation of why the "peakedness/flatness" characterization is wrong.

Skewness and Kurtosis are similar, but different.

Skewness is the degree of distortion from the symmetrical bell curve or the normal distribution. It measures the lack of symmetry in data distribution. It differentiates extreme values in one versus the other tail. A symmetrical distribution will have a skewness of 0. Kurtosis is all about the tails of the distribution — not the peakedness or flatness. It is used to describe the extreme values in one versus the other tail. It is actually the measure of outliers present in the distribution.

High kurtosis in a data set is an indicator that data has heavy tails or outliers. If there is a high kurtosis, then, we need to investigate why do we have so many outliers. It indicates a lot of things, maybe wrong data entry or other things. Investigate! Low kurtosis in a data set is an indicator that data has light tails or lack of outliers. If we get low kurtosis(too good to be true), then also we need to investigate and trim the dataset of unwanted results. Mesokurtic: This distribution has kurtosis statistic similar to that of the normal distribution.

Leptokurtic (Kurtosis > 3): Distribution is longer, tails are fatter. Peak is higher and sharper than Mesokurtic, which means that data are heavy-tailed or profusion of outliers.

Platykurtic: (Kurtosis < 3): Distribution is shorter, tails are thinner than the normal distribution. The peak is lower and broader than Mesokurtic, which means that data are light-tailed or lack of outliers.

Hope that helps.