Measures of Skewness

Define Skewness:
Consider the data given: 3, 4, 4, 5, 5, 5, 5, 6, 6,7. When we graph this data, the histogram represents a symmetric distribution. A distribution is said to be symmetric when a vertical line drawn at a particular point in the histogram forms a mirror image of the shape on either side of the vertical line.  Let us consider another data; 3,4,4,4,4,5,5,6,7. When we construct a histogram to the given data, from the data we can see that the distribution is not symmetric and is more towards the lower values; skewed to the right. Let us consider one more data;3,4,5,5,6,6,6,6,7. In this case, when we construct a histogram, we can see that the distribution is not symmetric and is towards the higher values; skewed to the left. Some distributions of data have more observations on one side of the graph than the other side. We can define skeweness as asymmetry in the frequency distribution of sample data values.

In a positively skewed distribution, the mode is smaller than the median which is smaller than the mean. In case of negative skewness the distribution is asymmetrical and points in the negative direction. For example, in a class test, if the given test is an easy one then almost all the students would get good marks and only a few would perform poorly. In such case we get a negatively skewed distribution

Formula for skewness:
Given a data in single variable, y1,y2,…yn;
Formula for skewness = sigma [yi – y(bar)]3/ [(n-1)s3]    (where i is 1 to n)
(yi)= y1, y2, y3…..yn,  x(bar) is the mean of the data, s is the standard deviation and n is the number of data items in the given data

Let us learn to calculate skewness. The following steps are used in calculating skewness:
Step1: First list out the given raw data in a column (y)
Step2: find the mean and standard deviation of the raw data
Step3: Next subtract the mean from each data item and list out the difference, which would be the Deviations from the Mean in the next column [yi – y(bar)].
Step4: Raise each of these deviations from the mean to the third power and tabulate them as the third column. At the end of the column, find the sum of these values which would be the sum of third moment deviations
Step5: Finally, calculate the skewness which is the sum of the deviations from the mean, raised to the third power , divided by the product of number of data items minus 1 and the standard deviation raised to the third power.  Skewness Formula = sigma [yi – y(bar)]3 divided by [(n-1)s3]    (where i  is 1 to n)