Skewness: Measure of Asymmetry
The skewed and askew are widely used terminologies that refer to something that is out of order or distorted on one side. Similarly, when referring to the shape of frequency distributions or probability distributions, the term skewness also refers to the asymmetry of that distribution. A distribution with an asymmetric tail extending out to the right is referred to as “positively skewed” or “skewed to the right”, while a distribution with an asymmetric tail extending out to the left is referred to as “negatively skewed” or “skewed to the left”. The range of skewness is from minus infinity ($-\infty$) to positive infinity ($+\infty$). In simple words, skewness (asymmetry) is a measure of symmetry or in other words, skewness is a lack of symmetry.
Karl Pearson (1857-1936) first suggested measuring skewness by standardizing the difference between the mean and the mode, such that, $\frac{\mu-mode}{\text{standard deviation}}$. Since population modes are not well estimated from sample modes, therefore Stuart and Ord, 1994 suggested that one can estimate the difference between the mean and the mode as being three times the difference between the mean and the median. Therefore, the estimate of skewness will be: $\frac{3(M-median)}{\text{standard deviation}}$. Many of the statisticians use this measure but after eliminating the ‘3’, that is, $\frac{M-Median}{\text{standard deviation}}$. This statistic ranges from $-1$ to $+1$. According to Hilderand, 1986, absolute values of skewness above 0.2 indicate great skewness.
Skewness has also been defined with respect to the third moment about the mean, that is $\gamma_1=\frac{\sum(X-\mu)^3}{n\sigma^3}$, which is simply the expected value of the distribution of cubed $Z$ scores. Skewness measured in this way is also sometimes referred to as “Fisher’s skewness”. When the deviations from the mean are greater in one direction than in the other direction, this statistic will deviate from zero in the direction of the larger deviations. From sample data, Fisher’s skewness is most often estimated by: $g_1=\frac{n\sum z^3}{(n-1)(n-2)}$. For large sample sizes ($n > 150$), $g_1$ may be distributed approximately normally, with a standard error of approximately $\sqrt{\frac{6}{n}}$. While one could use this sampling distribution to construct confidence intervals for or tests of hypotheses about $\gamma_1$, there is rarely any value in doing so.
Arthur Lyon Bowley (1869-19570, has also proposed a measure of skewness based on the median and the two quartiles. In a symmetrical distribution, the two quartiles are equidistant from the median but in an asymmetrical distribution, this will not be the case. The Bowley’s coefficient of skewness is $\frac{q_1+q_3-2\text{median}}{Q_3-Q_1}$. Its value lies between 0 and $\pm1$.
The most commonly used measures of skewness (those discussed here) may produce some surprising results, such as a negative value when the shape of the distribution appears skewed to the right.
It is important for researchers from the behavioral and business sciences to measure skewness when it appears in their data. A great amount of skewness may motivate the researcher to investigate the existence of outliers. When making decisions about which measure of the location to report and which inferential statistic to employ, one should take into consideration the estimated skewness of the population. Normal distributions have zero skewness. Of course, a distribution can be perfectly symmetric but may far away from the normal distribution. Transformations of variables under study commonly employed to reduce (positive) skewness. These transformations may include square root, log, and reciprocal of a variable.
