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Unit 4
Measurement of Skewness
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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
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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.
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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)
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Definition of Measurement of SkewnessThere are two formulas to
calculate the measurement of skewness :
Deviation Standard
Mode-Mean 1 Skewness of tCoefficien sPearson'
Deviation Standard
Median)-3(Mean 2 Skewness of tCoefficien sPearson'
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Definition of Measurement of SkewnessMeasure of the skewness is use
to determine the difference between the mean, median and mode in distribution. The following table can summarize it:
Example 17 :The following table shows the height distribution (cm) for 100 students
Height (cm)
151-155 156-160 161-165 166-170 171-175
Frequency
5 20 42 26 7a) Calculate the: i) Pearson’s Coefficient of Skewness 1 ii) Pearson’s Coefficient of Skewness 2b) 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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ClassIntervals
f Mid point, x
fx CumulativeFrequency
Location of data
x2 fx2
151-155 5 153 765 5 1-5 23409 117045156-160 20 158 3160 25 6-25 24964 499280161-165 42 163 6846 67 26-67 26569 1115898166-170 26 168 4368 93 68-93 28224 733824171-175 7 173 1211 100 94-100 29929 209503
∑f= 100
∑fx=16350
∑fx2
= 2675550
Step 2 : Find the mean by using the formula :
= 163.5
f
fx_xMean,
100
16350
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Step 3 : Identify the location of median
class by using the formula :
Location of median class = 50
2
f
2
100
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C x
fm
fm2
f
Lm X~
Median,
Step 4 : Find the median by using the
formula :
5x 42
252
100
160.5
= 160.5 + 2.98= 163.48
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Step 5 : Identify the mode class (161-165),
since this class has the highest frequency = 42Step 6 : Find the mode by using the
formula :
= 160.5+ 2.89 = 163.39
C x Δ2Δ1
Δ1Lb
^XMode,
5 x 26)(4220)(42
20)(42160.5
5x 1622
22160.5
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Step 7 : Calculate the standard deviation
using the formula :
f
fxfx
1f
1SDeviation, Standard
2
2
100
2(16350)2675550
1100
1
2673225)(267555099
1
23.49
= 4.85
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Step 8 : Calculate the Pearson’s Coefficient of Skewness 1 by using the formula : Deviation Standard
Mode-Mean 1 Skewness of tCoefficien sPearson'
4.85
163.39163.5
4.85
0.11
= 0.02
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Step 9 : Calculate the Pearson’s Coefficient of
Skewness 2 using the formula: Deviation Standard
Median)-3(Mean 2 Skewness of tCoefficien sPearson'
4.85
163.48)3(163.5
4.85
0.06
= 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.
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 = 64.6 Median = 34.3 Variance = 24432.1 Mode = 35.4