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I have the following histogram created in Minitab. I am wondering whether this histogram is actually positively skewed, negatively skewed, or symmetric.

By observing the graph itself, it seems that it is negatively skewed. However, mean (58.08) > median (57.7). Does this mean that it is positively skewed, or fairly symmetric?

Any helpful answer will be highly appreciated :D

Joshua
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  • possible duplicate of [Outlier Detection on skewed Distributions](http://stats.stackexchange.com/questions/129274/outlier-detection-on-skewed-distributions) – Xi'an Mar 18 '15 at 09:01
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    Histograms have many failings, but this one is doing a good job at telling you that you have precisely one outlier at about 10. Your title has it right: The question is in essence how to assess skewness when the main body of the data indicates right skewness but there is a low outlier. One hint is to recalculate the mean and median without the outlier. – Nick Cox Mar 18 '15 at 10:22
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    It's not symmetric. (Does it have the same shape as its mirror image? Obviously not.) To determine whether it is positively or negatively skewed, you need to provide a *quantitative* definition of what you mean by "skewed." (There exist many, such as a standardized third central moment--the "skewness"--but they don't always agree on what is positive and what is negative.) The definition you have used (mean minus median) would settle the issue, right? – whuber Mar 18 '15 at 14:19
  • I don't think the suggested duplicate is actually close enough to call a duplicate. It's possible there's some other post that would be a duplicate, but I don't think this is it. – Glen_b Mar 18 '15 at 14:28

1 Answers1

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I am wondering whether this histogram is actually positively skewed, negatively skewed, or symmetric.

Skewness:

Skewness isn't a well-defined thing; it's a relatively vague notion, one that there have been numerous attempts to give specific definitions of. The different definitions are not necessarily consistent with each other. For example moment skewness might suggest the opposite direction to second Pearson skewness, (a measure based on mean minus median).

Additionally, zero skewness (by whatever single number measure) doesn't imply symmetry.

Assessing skewness from a histogram

This is also tricky.

  • it's often unclear which direction, if any, the skewness might be in

  • a small change in the bin width or even just the bin origin can completely change the impression of skewness. If you only use a few bins (as you have in your diagram), this risk is greater. So sometimes even when it looks clear, the impression may be a false one.

[Unfortunately, stats packages tend to base their bin widths off certain optimal formulae, but the thing they optimize is not especially suited for vidually judging the shape and identifying fine detail. Our eyes can smooth roughness, and default histograms tend to be much too smooth for such purposes. I typically increase the number of bins by a factor of 2 or often much more.]

See this answer assessing approximate distribution of data based on a histogram which discusses some of the issues, but I'll give some highlights here -

The difference between these two histograms boils down to a simple change of bin origin:

two histograms, the first looking somewhat left skew, the second somewhat right skew

And the difference between these two is a change of binwidth (and origin):

two histograms, the first looks symmetric, the second right skew

That is, the same data can look quite different, depending on how you set up your plot. For more details and advice, see the above-linked answer.

Looking at your plot

Keeping those caveats in mind, you might either describe that plot of yours as appearing mildly right skew but with an extreme low outlier, or possibly as appearing left skew because of the low outlier, or simply as appearing asymmetric.

I think most people might say something more like the first, but it's just a guess. The last one is a pretty safe assessment, but it's not especially helpful.

Glen_b
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