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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