# How to judge skewness based on the mean and range?

Is there any rule of thumb to judging skewness of data based on its mean and range (max-min)? I found such implication in one of the papers I'm reading and I can't see why it would be obvious.

The level of the mean and the range (maximum – minimum) suggests that the price levels are right-skewed. - http://essay.utwente.nl/60867/, page 48.

• Strange. Can you provide the title or a link to the paper? – Jon Nov 20 '16 at 20:26
• They probably mean just that the mean lies to the left of the centerpoint of the range interval. – kjetil b halvorsen Nov 21 '16 at 3:24
• Presumably like @kjetilbhalvorsen I guess that the implication is that the maximum and minimum are known. They are not just looking at its difference. It's the ambiguity, which doesn't often bite, between (a) the range as the interval from the minimum to the maximum and (b) the range as the difference between the maximum and the minimum. Remember that many non-mathematicians (and I'm one of them too) are liable to write the minus sign as equivalent to as dash and meaning "and". I've not followed up the link, as questions here should be self-contained. – Nick Cox Nov 21 '16 at 10:59
• Thank you all! It seems that there's nothing deep in this statement and it's mainly about where the mean is compared to min and max. – Paula Nov 21 '16 at 19:09
• The main problem is that in any symmetric (non-skewed) continuous distribution, the mean is certain not to lie midway between the min and the max. Even worse, with many distributions having tapering tails--such as Normal, Gamma, lognormal, Weibull, and many more forms commonly encountered in data--one or both of the min and max is so extremely variable that using it to draw any kind of conclusion is about the riskiest way one can find to analyze the data. – whuber Nov 21 '16 at 23:33

There is a way that this would make some kind of sense

For a variable that's non-negative the minimum must be between 0 and the mean -- consequently if the range is many times as large as the mean then it would suggest that the maximum is much further above the mean than the minimum could be below it, which at least suggests that the variable may be right skew.

[I think that if I were trying to make the judgement, it would make more sense to look at the minimum and maximum compared to the mean separately, and since the standard deviation is available in the table, to also consider that - if the standard deviation is similar in size to, or larger than the mean, it suggests skewness in a non-negative variable. I think there's generally consistent information coming from these in the table, but even just the fact that the variable is bounded on the left would lead us to anticipate a likely tendency to right skewness in the first place.]

It's not obvious to me, either. The skewness of a random variable is its third central moment divided by its standard deviation. If you wanted to know the sample skewness, you would presumably compute it directly instead of doing something with the mean and extrema. Extrema in particular aren't constrained much by moments.

• "Skewness" is often meant in a broader sense of being an asymmetric distribution--and referring to some amount of departure from symmetry. The standardized third central moment merely is one way to measure some forms in which skewness is manifest. – whuber Nov 21 '16 at 23:30

This is a heuristic for approximating skewness of the long-tail in non-negative valued distributions. It's used to signal changes in the tails.

Without a minimum specified, you'll find it relevant to latency, time-to-resolve, and any log-normal distributions. With only non-negative values, and an assumed minimum of zero, the observed maximum is also the range of the sample set.

Long tail metrics are generally interested in the long right tail. The bigger the difference between maximum and average, the longer the right tail.

That's not to say that ”in the middle” is preferred, better, or relevant. Many distribution have a mean that will be observed with a long tail. It's the change in the metric that is of interest.

Typically you may see these heuristics as a percentage (avg/max) because the scaling is more standardized. But as others have pointed out it's reckless on it's own to conclude anything. The value is in observing the change of it as a time series.

It's also cheap. Skew heuristics based on maximum and average require the least memory - 3 longs of storage for online calculation (sum, count, max) and are not constrained to any predefined observation range.