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:
 Karl Pearson's measure,
 Bowley’s measure,
 Kelly’s measure, and
 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 
5060 
60  70 
70  80 
f 
18 
30 
40 
55 
38 
20 
16 
Solution:
x 
MVx 
d_{x} 
f 
fd_{x} 
2 fd_{X} 
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 
4050 
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 Q_{3} and Q_{1} are upper and lower quartiles and M is the median. The value of this skewness varies between +1. In the case of openended 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 Q_{3} and Q_{1} are positioned equidistantly from Q_{2} that is, the median. In symbols, Q_{3}  Q_{2} = Q_{2} – Q_{1}' In contrast, when the distribution is skewed, then Q_{3}  Q_{2} will be different from Q_{2} – Q_{1}' When Q_{3 } Q_{2} exceeds Q_{2} – Q_{1}' then skewness is positive.
As against this; when Q_{3}  Q_{2} is less than Q_{2} – Q_{1}' then skewness is negative. Bowley’s measure of skewness can be written as:
Skewness = (Q_{3}  Q_{2})  (Q_{2} – Q_{1} ) or Q_{3}  Q_{2}  Q_{2} + Q_{1}
Or Q_{3} + Q_{1}  2Q_{2} (2Q_{2} 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,
Q_{1}=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 openended 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 P_{10} P_{30} and P_{90}.
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.