Measures of variability

Measures of Variability
Gautam Kumar
Assistant Professor
University Department of Teacher Education
Utkal University, Bhubaneswar

VARIABILITY OR DISPERSION?
Test Score of group – A of Class X
40, 38, 36, 17, 20, 19, 18, 3, 5, 4
Mean =
40+38+36+17+20+19+18+3+5+4
10
= 20
Test Score of group – B of Class X
19, 20, 22, 18, 21, 23, 17, 20, 22, 18
Mean =
19+20+22+18+21+23+17+20+22+18
10
= 20
Wednesday, 29 July 2020 GAUTAM I UUDTE I UTKAL UNIVERSITY 2

VARIABILITY OR DISPERSION?
• Measures of central Tendency – Mean, Median, Mode provides central value
or typical representation of a set of scores as a whole. However, there is
tendency for data to be dispersed, scattered or to show variability around the
average.
• The tendency of the attributes of a group to deviate from the average or
central value is known as dispersion of variability.

Types of Measures of Variability
• Range
• Quartile Deviation (Q)
• Average Deviation (AD)
• Standard deviation (SD)

RANGE
Range is the simplest measure of variability or dispersion. It is calculated by
subtracting the lowest in the series from the highest.
Test Score of group – A of Class X
40, 38, 36, 17, 20, 19, 18, 3, 5, 4
Range = 40-3 = 37
Test Score of group – B of Class X
19, 20, 22, 18, 21, 23, 17, 20, 22, 18
Range = 23-17 = 6

USE OF RANGE
• When we need simply the highest and lowest scores of the total spread of
data.
• The group or distribution is too small.
• Require variability within a group within no time.

Quartile Deviation (Q)
Quartile Deviation is calculated by the following formula
Q =
𝑄3 − 𝑄1
2
Where Q1 is first quartile
Q3 id third quartile

USE OF Quartile Deviation (Q)
• The distribution is skewed, contains a few extreme scores.
• The measure of central tendency is available in median.
• When we have to determine the concentration around the middle 50 % of the
cases.
• Percentile and quartile is already available.

Average Deviation (AD)
Average Deviation is the mean of deviations of all separate scores in the series
from their mean (Garrett, 1971).
Computation of Average Deviation from ungrouped data
AD =
𝒙
𝑵
15, 10, 6, 8, 11
Mean of the data is
15+10+6+8+11
5
= 10

AD =
12
5
= 2.4
Scores
(X)
Deviation from the mean
(X-M = x)
𝒙
15 15-10=5 5
10 10-10=0 0
6 6-10=-4 4
8 8-10=-2 2
11 11-10=1 1
N=5 𝒙 = 12

AD =
𝒇𝒙
𝑵
Scores f Mid-point
(X)
fX x=(X-M) fx |𝒇𝒙|
110-114 4 112 448 11.94 44.76 44.76
105-109 4 107 428 6.94 27.76 27.76
100-104 3 102 306 1.94 5.82 5.82
95-99 0 97 0 -3.06 0 0
90-94 3 92 276 -8.06 -24.18 24.18
85-89 3 87 261 -13.06 -39.18 39.18
80-84 1 82 82 -18.06 -18.06 18.06
N=18 𝒇𝒙=1801 𝒇𝒙 =162.76

Mean =
𝑓𝑋
𝑁
=
1801
18
= 100.06
AD =
𝒇𝒙
𝑵
=
𝟏𝟔𝟐.𝟕𝟔
𝟏𝟖
= 9.04

USE OF Average Deviation (AD)
• When the distribution is normal or near to normal
• If it is needed to weigh all deviations from the mean according to their size.
• A less reliable measures of variability can be employed.

Standard Deviation (SD)
• Standard deviation is defined as the square root of the average of the squares of the
deviations of each score from the mean.
SD (𝜎)=
𝑋−𝑀 2
𝑁
=
𝑥2
𝑁
Where
X= Individual Score
M= Mean of the give set of scores
N= Total no of the scores
x=Deviation of each from the mean

Computation of Standard Deviation (SD)
Computation of Standard deviation from Ungrouped Data
52, 50, 56, 68, 65, 62, 57, 70
SD (𝜎)=
𝑥2
𝑁
Score
(X)
Deviation from the
mean (X – M) or x
x2
52 -8 64
50 -10 100
56 -4 16
68 8 64
65 5 25
62 2 4
57 -3 9
70 10 100
𝑥2 =
382
SD (𝜎)=
𝑥2
𝑁
=
382
8
= 47.75
= 6.91

Computation of Standard deviation from Grouped Datac
Score f X fX M x = X – M x2 fx2
127-129 1 128 128 115 13 169 169
124-126 2 125 250 115 10 100 200
121-123 3 122 366 115 7 49 147
118-120 1 119 119 115 4 16 16
115-117 6 116 696 115 1 1 6
112-114 4 113 452 115 -2 4 16
109-111 3 110 330 115 -5 25 75
106-108 2 107 214 115 -8 64 128
103-105 1 104 104 115 -11 121 121
100-102 1 101 101 115 -14 196 196
N=24 𝒇𝑿=27
60
𝑓𝑥2=1074
Computation of
Mean
Mean =
𝒇𝑿
𝑵
=
𝟐𝟕𝟔𝟎
𝟐𝟒
= 115

Computation of Standard deviation from Grouped Data
SD (𝜎)=
𝑓𝑥2
𝑁
=
1074
24
= 44.75
=6.69

Computation of Standard deviation from Grouped Data (Short-cut Method)
Score f X x' =
𝑿−𝑨
𝒊
fx' fx'2
127-129 1 128 4 4 16
124-126 2 125 3 6 18
121-123 3 122 2 6 12
118-120 1 119 1 1 1
115-117 6 116 0 0 0
112-114 4 113 -1 -4 4
109-111 3 110 -2 -6 12
106-108 2 107 -3 -6 18
103-105 1 104 -4 -4 16
100-102 1 101 -5 -5 25
N=24 fx’ = -8 fx’2 = 122
𝝈 = 𝒊
fx’ 2
𝑵
−
fx’
𝑵
𝟐
Where
i = size of class interval
N = total no of frequencies
X = mid point of the class
A = assumed mean

Computation of Standard deviation from Grouped Data (Short-cut Method)
𝝈 = 𝒊
fx’ 2
𝑵
−
fx’
𝑵
𝟐
= 𝟑
122
𝟐𝟒
−
−8
𝟐𝟒
𝟐
= 𝟑
122
𝟐𝟒
−
64
𝟐𝟒 𝐱 𝟐𝟒
𝟐
=
𝟑
𝟐𝟒
𝟏𝟐𝟐 𝐱 𝟐𝟒 − 𝟔𝟒
=
𝟏
𝟖
𝟐𝟖𝟔𝟒 =
53−52
8
= 6.69

USE OF Standard Deviation (AD)
• More reliable measure of variability is required.
• If computation of the correlation coefficients, significance of difference between means is
required.
• If measure of central tendency is available in the form Mean.
• The distribution is normal or near to normal.

Measures of variability

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