Unit 4
Measurement of
Skewness
1
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
2
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.
3
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 exceeds Mean and
Median. Distribution is
Skewed to the left
(negative)
Mode<Median<Mean
Mean exceeds Mode and
Median. Distribution is
Skewed to the right
(positive)
Mean=Median=Mode
Distribution is
Symmetrical (0)
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Definition of Measurement of
Skewness

There are two formulas to calculate the
measurement of skewness :
Pearson' s Coefficien t of Skewness 1 
Mean - Mode
Standard Deviation
Pearson' s Coefficien t of Skewness 2 
3(Mean - Median)
Standard Deviation
5
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:
6

Example 17 :
The following table shows the height distribution (cm)
for 100 students
Height (cm) 151-155
Frequency
5
156-160
20
161-165
42
166-170
26
171-175
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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.
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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.
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Class
Intervals
151-155
156-160
161-165
166-170
171-175
f
5
20
42
26
7
∑f=
100

Mid
point, x
153
158
163
168
173
Cumulative Location
x2
fx2
Frequency of data
765
5
1-5
23409 117045
3160
25
6-25
24964 499280
6846
67
26-67
26569 1115898
4368
93
68-93
28224 733824
1211
100
94-100 29929 209503
∑fx2 =
∑fx=
267555
16350
0
fx
Step 2 : Find the mean by using the formula :
_  fx
Mean, x 
f

16350
100
= 163.5
9
Step 3 : Identify the location of median
class by using the formula :
Location of median class 
f
2

100
2
= 50
10
Step 4 : Find the median by using the
formula :
 f

f
~
m
 2
Median, X  L  
xC
m
 fm 


 100

 2  25 
 160.5  
x 5
 42 


= 160.5 + 2.98
= 163.48
11
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 :
^
 Δ1 
Mode, X  Lb  
xC

 Δ1 Δ2 
(42  20)


 160.5  
x5

 (42  20)  (42  26) 
 22 
 160.5  
x5

 22  16 
= 160.5+ 2.89
= 163.39
12
Step 7 : Calculate the standard deviation
using the formula :
Standard Deviation, S 
2



fx
1 

2
fx 


 f  1
f




2

(16350) 
 2675550 


100  1 
100 
1

1
(2675550  2673225)
99
 23.49
= 4.85
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Step 8 : Calculate the Pearson’s Coefficient
of Skewness 1 by using the formula :
Pearson' s Coefficien t of Skewness 1 


Mean - Mode
Standard Deviation
163.5  163.39
4.85
0.11
4.85
= 0.02
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Step 9 : Calculate the Pearson’s Coefficient of
Skewness 2 using the formula:
Pearson' s Coefficien t of Skewness 2 

3(Mean - Median)
Standard Deviation
3(163.5  163.48)
4.85
0.06

4.85
= 0.01
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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
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Exercise :
1. The following data was collected from an
analysis conducted by a student.
Average = 64.6
Variance = 24432.1
Median = 34.3
Mode = 35.4
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)
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