There are four measures of skewness, each divided into absolute and relative measures. The relative measure is known as the coefficient of skewness and is more frequently used than the absolute measure of skewness. Further, when a comparison between two or more distributions is involved, it is the relative measure of skewness, which is used.

The measures of skewness are:

1. Karl Pearson's measure,
2. Bowley’s measure,
3. Kelly’s measure, and
4. Moment’s measure.

These measures are discussed briefly below:

## 1. Karl Pearson's Measure

The formula for measuring skewness as given by Karl Pearson is as follows: Skewness = Mean - Mode The direction of skewness is determined by ascertaining whether the mean is greater than the mode or less than the mode. If it is greater than the mode, then skewness is positive. But when the mean is less than the mode, it is negative. The difference between the mean and mode indicates the extent of departure from symmetry. It is measured in standard deviation units, which provide a measure independent of the unit of measurement. It may be recalled that this observation was made in the preceding chapter while discussing standard deviation. The value of coefficient of skewness is zero, when the distribution is symmetrical. Normally, this coefficient of skewness lies between +1. If the mean is greater than the mode, then the coefficient of skewness will be positive, otherwise negative.

Example 1: Given the following data, calculate the Karl Pearson's coefficient of skewness: Solution: This shows that there is a positive skewness though the extent of skewness is marginal.

Example 2: From the following data, calculate the measure of skewness using the mean, median and standard deviation:

 X 10 - 20 20 - 30 30 - 40 40 - 50 50-60 60 - 70 70 - 80 f 18 30 40 55 38 20 16

Solution:

 x MVx dx f fdx 2 fdX cf 10 - 20 15 -3 18 -54 162 18 20 - 30 25 -2 30 -60 120 48 30 - 40 35 -1 40 -40 40 88 40-50 45=a 0 55 0 0 143 50 - 60 55 1 38 38 38 181 60 - 70 65 2 20 40 80 201 70 - 80 75 3 16 48 144 217 Total 217 -28 584

a = Assumed mean = 45, cf = Cumulative frequency, dx = Deviation from assumed mean, and i = 10   The result shows that the distribution is negatively skewed, but the extent of skewness is extremely negligible.

## 2. Bowley's Measure

Bowley developed a measure of skewness, which is based on quartile values. The formula for measuring skewness is:

Skewness = Where Q3 and Q1 are upper and lower quartiles and M is the median. The value of this skewness varies between +1. In the case of open-ended distribution as well as where extreme values are found in the series, this measure is particularly useful. In a symmetrical distribution, skewness is zero. This means that Q3 and Q1 are positioned equidistantly from Q2 that is, the median. In symbols, Q3 - Q2 = Q2 Q1' In contrast, when the distribution is skewed, then Q3 - Q2 will be different from Q2 Q1' When Q3 - Q2 exceeds Q2 Q1' then skewness is positive.

As against this; when Q3 - Q2 is less than Q2 Q1' then skewness is negative. Bowley’s measure of skewness can- be written as:

Skewness = (Q3 - Q2) - (Q2 Q1 )      or        Q3 - Q2 - Q2 + Q1

Or        Q3 + Q1 - 2Q2 (2Q2 is 2M)

However, this is an absolute measure of skewness. As such, it cannot be used while comparing two distributions where the units of measurement are different. In view of this limitation, Bowley suggested a relative measure of skewness as given below: Example 3: For a distribution, Bowley’s coefficient of skewness is - 0.56,

Q1=16.4 and Median=24.2. What is the coefficient of quartile deviation?

Solution:  Example 4: Calculate an appropriate measure of skewness from the following data:

 Value in Rs Frequency Less than 50 40 50 - 100 80 100 - 150 130 150 – 200 60 200 and above 30

Solution: It should be noted that the series given in the question is an open-ended series. As such, Bowley's coefficient of skewness, which is based on quartiles, would be the most appropriate measure of skewness in this case. In order to calculate the quartiles and the median, we have to use the cumulative frequency. The table is reproduced below with the cumulative frequency.

 Value in Rs Frequency Cumulative Frequency Less than 50 40 40 50 - 100 80 120 100 - 150 130 250 150 - 200 60 310 200 and above 30 340  This shows that there is a negative skewness, which has a very negligible magnitude.

## 3. Kelly's Measure

Kelly developed another measure of skewness, which is based on percentiles. The formula for measuring skewness is as follows: Where P and D stand for percentile and decile respectively. In order to calculate the coefficient of skewness by this formula, we have to ascertain the values of 10th, 50th and 90th percentiles. Somehow, this measure of skewness is seldom used. All the same, we give an example to show how it can be calculated.

Example 5: Use Kelly's measure to calculate skewness.

 Class Intervals f cf 10 - 20 18 18 20 - 30 30 48
 30- 40 40 88 40- 50 55 143 50 - 60 38 181 60 – 70 20 201 70 - 80. 16 217

Solution: Now we have to calculate P10 P30 and P90.  This shows that the series is positively skewed though the extent of skewness is extremely negligible. It may be recalled that if there is a perfectly symmetrical distribution, then the skewness will be zero. One can see that the above answer is very close to zero.