2.6.1. Skewness

Skewness is often defined as the lack of symmetry in a distribution (Everitt, 2004, p. 350; Upton & Cook, 2014, p. 396; Zedeck, 2014b, p. 338). There is either no skew, left skew or right skew. The skew is where there are few. A few examples to illustrate.
Three examples of a left skewed distribution are shown in Figure 30.


Figure 30 . Three distributions with left skew

Three examples of a right skewed distribution are shown in Figure 31.

Figure 31 . Three distributions with right skew

Three examples of distributions with no skew are shown in Figure 32.

Figure 32 . Three distributions with no skew

How to put skewness in a single measurement turns out to be relatively difficult and there are a few different methods.
The first method might be to notice that in a symmetrical distribution the mode = mean, so if this is not the case, there is some asymmetry. The so-called Pearson Mode Skewness is therefore simply the difference between the mean and mode (mean – mode), divided by the standard deviation (to put everything in perspective):
Skewness = (mean – mode) / Standard deviation

A problem arises if the data is multimodal. In those cases another thing to notice about symmetrical distributions comes into play: mean = median. The difference between the mode and the mean is usually bigger in skewed distributions than the difference between the median and mean, so Mr. Pearson decided it could be good to multiply the difference by three. The so-called Pearson Median Skewness is therefore three times the mean minus the median, divided by the standard deviation:
Skewness = 3(median – mode) / Standard deviation

There are other methods as well, methods that use quartiles, methods that use other decentiles, or even a further generalization of tiles in general. Most software packages use a method based on the third moment (explained by Pearson (1895)). If you are interested in these other methods click here.

If the skew is positive the distribution is likely to be right skewed, while if it is negative it is likely to be left skewed. As a rule of thumb for interpretation of the absolute value of the skewness (Bulmer, 1979, p. 63):
0 < 0.5 => fairly symmetrical
0.5 < 1 => moderately skewed
1 or more => highly skewed
There are also tests that can be used to check if the skewness is significantly different from zero. These are often used to check if a dataset could have come from a normally distributed population. More on this will be discussed in the section on checking for normality.

For unimodal continuous distributions the mean will be often lower than the median, while for right-skewed unimodal continuous distributions the mean will be greater than the median. Note that there are exceptions to this rule of thumb. For those interested in these exceptions I highly recommend the article from Von Hippel (2005).

Skewness is often indicated by either
 

>>Next section: Kurtosis

References

Bulmer, M. G. (1979). Principles of Statistics (Dover). New York: Dover.
Everitt, B. (2004). The Cambridge dictionary of statistics (2nd ed). Cambridge: Cambridge University Press.
Pearson, K. (1895). Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material. Philosophical Transactions of the Royal Society of London. (A.), 186, 343–414. http://doi.org/10.1098/rsta.1895.0010
Upton, G. J. G., & Cook, I. (2014). Dictionary of statistics (3rd ed.). Oxford: Oxford University Press.
Von Hippel, P. T. (2005). Mean, Median, and Skew: Correcting a Textbook Rule. Journal of Statistics Education, 13(2). Retrieved from http://www.amstat.org/publications/jse/v13n2/vonhippel.html
Zedeck, S. (Ed.). (2014). APA dictionary of statistics and research methods (First edition). Washington, DC: American Psychological Association.