1. Measurement of Skewness
Unit 4
Measurement of Skewness

2. Objective :
At the end of the course, student should be Able to : i) Define the measurement of skewness ii) Identify the measures of skewness a) Pearson’s Coefficient of Skewness 1 b) Pearson’s Coefficient of Skewness 2 iii) Sketch the data distribution based on the value of PCS 1 and 2

3. Definition of Measurement of Skewness
is a measurement that shows the forms of data distribution and the direction of the frequency distribution; whether skewness to the left, right or symmetrical.

4. Definition of Measurement of Skewness
The concept of skewness helps us to understand the relationship between three measures; mean, median and mode as illustrated below:
Mean<Median<Mode
Mode<Median<Mean
Mean=Median=Mode
Mode exceeds Mean and Median. Distribution is Skewed to the left (negative)
Mean exceeds Mode and Median. Distribution is Skewed to the right (positive)
Distribution is Symmetrical (0)

5. Definition of Measurement of Skewness
There are two formulas to calculate the measurement of skewness :

6. Definition of Measurement of Skewness
Measure of the skewness is use to determine the difference between the mean, median and mode in distribution. The following table can summarize it:

7. Example 17 :
The following table shows the height distribution (cm) for 100 students
Height (cm)
Frequency 5 20 42 26 7
a) Calculate the:
i) Pearson’s Coefficient of Skewness 1
ii) Pearson’s Coefficient of Skewness 2
b) Sketch the distribution’s form based on
answers in question (b)
c) Give conclusion based on the sketch.

8. Step 1 : Obtain the midpoint, fx
Solution:
Step 1 : Obtain the midpoint, fx
( to calculate the mean), cumulative frequency and location of data ( to calculate the median) , x2, fx2( to calculate the variance and standard deviation) in a frequency distribution table.

9. Step 2 : Find the mean by using the formula :
Class
Intervals f
Mid point, x fx
Cumulative
Frequency
Location of data x2
fx2 5
153
765
1-5
23409
117045 20
158
3160 25
6-25
24964
499280 42
163
6846 67
26-67
26569 26
168
4368 93
68-93
28224
733824 7
173
1211
100
94-100
29929
209503
∑f= 100
∑fx=
16350
∑fx2 =
Step 2 : Find the mean by using the formula :
= 163.5

10. Step 3 : Identify the location of median class by using the formula : Location of median class = 50

11. Step 4 : Find the median by using the formula : = =

12. Step 5 : Identify the mode class (161-165),
since this class has the highest
frequency = 42
Step 6 : Find the mode by using the formula : = =

13. Step 7 : Calculate the standard deviation using the formula :
= 4.85

14. Step 8 : Calculate the Pearson’s Coefficient of Skewness 1 by using the formula :
= 0.02

15. Step 9 : Calculate the Pearson’s Coefficient of
Skewness 2 using the formula:
= 0.01

16. Step 10 : Sketch the distribution’s form based on answers in Step 9 or 10
Mode<Median<Mean
The conclusion is the distribution is skewed to the right or positive skewed

18. Exercise :
1. The following data was collected from an analysis conducted by a student. a) Find the value of Pearson’s Coefficient of Skewness 1 and 2. b) Determine the type of skewness for the answer in question (a)
Average = Median = 34.3
Variance = Mode = 35.4