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VARIABILITY
 
MEASURES OF
VARIABILITY
© aSup-2007
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VARIABILITY
 
 Knowing the central value of a set of
measurement tells us much, but it does not
by any means give us the total pictures of
the sample we have measured
 Two groups of six-year-old children may
have the same average IQ of 105. One group
contain no individuals with IQs below 95 or
above 115, and that the other includes
individuals with IQs ranging from 75 to 135
 We recognize immediately that there is a
decided difference between the two groups
in variability or dispersion
© aSup-2007
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 
VARIABILITY
75
85
95
105
115
125
135
The BLUE group is decidedly more homogenous
than the RED group with respect to IQ
© aSup-2007
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VARIABILITY
 
Purpose of Measures of Variability
 To explain and to illustrate the methods of
indicating degree of variability or
dispersion by the use of single numbers
 The three customary values to indicate
variability are
○ The total range
○ The semi-interquartile range Q, and
○ The standard deviation S
© aSup-2007
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 
VARIABILITY
The TOTAL RANGE
 The total range is the easiest and most
quickly ascertained value, but it also the
most unreliable
 The BLUE group (from an IQ of 95 to one of
115) is 20 points. The range of RED group
from 75 to 135, or 60 points
 The range is given by the highest score
minus the lowest score
 The RED group has three times the range of
the BLUE group
© aSup-2007
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VARIABILITY
 
The SEMI-INTERQUARTILE RANGE Q
 The Q is one-half the range of the middle 50
percent of the cases
 First we find by interpolation the range of
the middle 50 percent, or interquartile
range, the divide this range into 2
© aSup-2007
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 
VARIABILITY
Low
Middle
Quarter
Lowest
Quarter
Q1
High
Middle
Quarter
Q2
Highest
Quarter
Q3
Q2 – Q1 Q3 – Q2
Q3 – Q1 = 2Q
Q=
© aSup-2007
Q3 – Q1
2
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 
VARIABILITY
The STANDARD DEVIATION S
 Standard deviation is by far the most
commonly used indicator of degree of
dispersion and is the most dependable
estimate of the variability in the population
from which the sample came
 The S is a kind of average of all deviation
from the mean
S=
© aSup-2007
√
∑ x2
N
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VARIABILITY
 
 As a general concept, the standard
deviation is often symbolized by SD, but
much more often by simply S
 In verbal terms, a S is the square root of
the arithmetic mean of the squared
deviations of measurements from their
means
© aSup-2007
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 
VARIABILITY
Data Illustrating
Sum of Squares,
Variance, and
Standard
Deviation
© aSup-2007
Person
A
B
C
D
E
F
G
H
I
J
Score
13
17
15
11
13
17
13
11
11
11
10
 
VARIABILITY
Data Illustrating Sum of Squares,
Variance, and Standard Deviation
Person
A
Score
13
B
C
D
17
15
11
E
F
G
13
17
13
H
I
11
11
J
11
© aSup-2007
Deviation x
x2
S=
√
∑ x2
N
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 
VARIABILITY
Data Illustrating Sum of Squares,
Variance, and Standard Deviation
Person
A
Score
13
Deviation x
-0,2
x2
0,04
B
C
D
17
15
11
+3,8
+1,8
-2,2
14,44
3,24
4,84
E
F
G
13
17
13
-0,2
+3,8
-0,2
0,04
14,44
0,04
H
I
11
11
-2,2
-2,2
4,84
4,84
J
11
-2,2
4,84
© aSup-2007
S=
√
∑ x2
N
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 
VARIABILITY
Data Illustrating Standard Deviation
Person
A
Score
13
Deviation x
-0,2
x2
0,04
B
C
D
E
17
15
11
13
+3,8
+1,8
-2,2
-0,2
14,44
3,24
4,84
0,04
F
G
H
17
13
11
+3,8
-0,2
-2,2
14,44
0,04
4,84
I
11
-2,2
4,84
J
11
-2,2
4,84
© aSup-2007
S=
√
∑ x2
N
51,6
10
√
= √ 5,16
=
S = 2,27
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 
VARIABILITY
Computing a S in Grouped Data
Scores f
55 – 59 1
50 – 54 1
45 – 49 3
40 – 44 4
35 – 39 6
30 – 34 7
25 - 29 12
20 – 24 6
15 – 19 8
10 - 14 2
© aSup-2007
S=
√
∑f x2
N
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 
VARIABILITY
Scores
f
Xc
55 – 59 1 57
50 – 54 1 52
45 – 49 3 47
40 – 44 4 42
35 – 39 6 37
30 – 34 7 32
25 - 29 12 27
20 – 24 6 22
15 – 19 8 17
10 - 14 2 12
© aSup-2007
x
Xc - Mean
x2
fx2
Computing a
S in Grouped
Data
S=
√
∑f x2
N
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 
VARIABILITY
Scores
f
Xc
55 – 59 1 57
50 – 54 1 52
45 – 49 3 47
40 – 44 4 42
35 – 39 6 37
30 – 34 7 32
25 - 29 12 27
20 – 24 6 22
15 – 19 8 17
10 - 14 2 12
© aSup-2007
x
Xc - Mean
+27,4
+22,4
+17,4
+12,4
+7,4
+2,4
-2,6
-7,6
-12,6
-17,6
x2
fx2
Computing a
S in Grouped
Data
S=
√
∑f x2
N
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 
VARIABILITY
Scores
f
Xc
55 – 59 1 57
50 – 54 1 52
45 – 49 3 47
40 – 44 4 42
35 – 39 6 37
30 – 34 7 32
25 - 29 12 27
20 – 24 6 22
15 – 19 8 17
10 - 14 2 12
© aSup-2007
x
Xc - Mean
+27,4
+22,4
+17,4
+12,4
+7,4
+2,4
-2,6
-7,6
-12,6
-17,6
x2
750,76
501,76
302,76
153,76
54,76
5,76
6,76
57,76
158,76
309,76
fx2
Computing a
S in Grouped
Data
S=
√
∑f x2
N
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 
VARIABILITY
Scores
f
Xc
55 – 59 1 57
50 – 54 1 52
45 – 49 3 47
40 – 44 4 42
35 – 39 6 37
30 – 34 7 32
25 - 29 12 27
20 – 24 6 22
15 – 19 8 17
10 - 14 2 12
© aSup-2007
x
Xc - Mean
+27,4
+22,4
+17,4
+12,4
+7,4
+2,4
-2,6
-7,6
-12,6
-17,6
x2
fx2
Computing a
S in Grouped
Data
S=
√
∑f x2
N
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 
VARIABILITY
Computing a S in Grouped Data
Score X
s
c
55 –
59
50 –
54
45 –
49
40 –
44
35 –
39
© aSup-2007
x
f
57 1
Xc Mean
+27,4
x2
750,76
fx2
S=
750,76
52 1
+22,4
501,76
501,76
47 3
+17,4
302,76 908,28
42 4
+12,4
153,76 615,04
37 6
+7,4
54,76 328,58
√
∑f x2
N
5462
50
√
= √ 109,24
=
S = 10.45
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VARIABILITY
 
Interpretation of a Standard Deviation
 The usual and most accepted interpretation
of a S is in percentage of cases included
within the range from one S below the
mean to one S above the mean
 In a normal distribution the range from -1σ
to +1σ contains 68,27 percent of the cases
 If the mean = 29,6 and S = 10,45; we say
about two-third of the cases lies from 19,15
to 40,05
© aSup-2007
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VARIABILITY
 
Interpretation of a Standard Deviation
 One of the most common source of variance
in statistical data is individual differences,
where each measurement comes from a
different person
© aSup-2007
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VARIABILITY
 
Interpretation of a Standard Deviation
 Giving a test of n items to a group of person
Before the first item is given to the group, as
far as any information from this test is
concerned, the individuals are all alike.
There is no variance
 Now administer the first item to the group.
Some pass it and some fail. Some now have
score of 1, and some have scores of zero
 There are two groups of individuals. There
is much variation, this much variance
© aSup-2007
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VARIABILITY
 
Interpretation of a Standard Deviation
 Give a second item. Of those who passed the first,
some will past the second and some will fail it.
Etc.
 There are now three possible scores : 0, 1, and 2.
 More variance has been introduced
 Carry the illustration further, adding item by item
 The differences between scores will keep
increasing, and also, by computation, the variance
and variability
© aSup-2007
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VARIABILITY
 
 Another rough check is to compare the S
obtained with the total range of
measurement
 In very large samples (N=500 or more) the S
is about one-sixth of the total range
 In other word, the total range is about six S
 In smaller samples the ratio of range to S can
be expected to be smaller (see Guilford p.71)
© aSup-2007
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