Download Chap_3_Measures_of_Variability

PXGZ6102 BASIC
STATISTICS FOR
RESEARCH IN
EDUCATION
Chap 3 - Measures of Variability –
Standard Deviation, Variance
Measures of Variability
Range
 Mean Deviation
 Variance
 Standard Deviation

Range
Refers to the overall span of the scores
 Eg. 18, 34, 44, 56, 78

The range is 78 – 18 = 60
Mean
Mean Deviation
1
2
Eg. Scores 1,2,3
 Mean = 2
 Mean deviation = |1-2 | + | 2-2 | + | 3-2 |
3
= 1 + 0 + 1 = 0.67
3

3
Mean Deviation
Eg. Shoes sizes in Ali’s home:
11,12,13,14,15,16,17 the mean is 14
 In Ahmad’s home: 5,8,11,14,17,20, 23 the
mean is also 14
 But the distribution in Ahmad’s home is
greater

Calculation of Mean Deviation
Ali’s
Scores
11
12
13
14
15
16
17
N=7
Mean
14
14
14
14
14
14
14
(x - mean)
-3
-2
-1
0
1
2
3
|x – mean|
3
2
1
0
1
2
3
|x – Mean| =12
Σ |x – Mean|
12
Mean Deviation = ---------------------- = ------- = 1.71
n
7
Calculate the Mean Deviation of
Ahmad’s data

5,8,11,14,17,20, 23
Variance
Variance of a distribution is the average of
the squared deviations
 The formula for the variance of a
population is slightly different than the
formula for a sample variance

Population & Sample Variance
Population Variance
σ2 = Σ (x - µ)2
N
Where σ2
x
µ
N
= the symbol for the population variance
= a raw score
= the population mean
= the number of scores in the population
Sample Variance
s2 
2
(
x

x
)

n 1
Where s2 = the symbol for sample variance
x = the sample mean
n = the number of scores in the distribution
Why divide by (n-1) for the sample variance to
estimate the population variance?

Variances of samples taken from
population tend to be smaller than the
population variance. Dividing the sample
formula with n-1 gives the correction and
the actual population variance
Population Variability
Population
Distribution
Sample Scores
x
x
Sample A
Sample C
Sample B
Sample scores are not as spread out as the population distribution. Thus the variance
of sample tend to underestimate the variance of population. Placing n -1 in the
denominator increases the value of the sample variance and provides a better estimate
of the population variance
Example: Calculate the variance of
the following sample scores, s2

A) 3, 4, 6, 8, 9
For sample
s2 
2
(
x

x
)

n 1
= 26 = 6.50
4
x
x
x  x ( x  x)
3
6
-3
9
4
6
-2
4
6
6
0
0
8
6
2
4
9
6
3
9
2
Example: Calculate the variance of
2
the following population scores, 

A) 3, 4, 6, 8, 9
For Population
2 
2
(
x


)

N
= 26 = 5. 2
5
X
Xm
X - Xm
(X – Xm)2
3
6
-3
9
4
6
-2
4
6
6
0
0
8
6
2
4
9
6
3
9
Exercise 1
1) Find the sample variance and the
population variance of the following
distributions
 A) 2, 4, 5, 7, 9
 B) 22, 32, 21, 20, 19, 15, 23
 C) 23, 67, 89, 112, 134, 156, 122, 45

Other forms of the Variance formula

Deviation score, D = X - µ (for population)
or d = X – Xm (for sample)
Xm = sample mean
Square of deviation = (X - µ)2
Sum of Squares of Deviation, SS = Σ (X - µ)2
Population (N)
Mean Deviation
Variance
σ2 = Σ (X - µ)2
(Definitional Formula)
(Deviational
Formula)
(Sum of Squares
Formula)
(Computational
Formula)
Xm = Mean
|X - µ|
N
N
Sample (n)
| X – Xm|
n -1
s2 = Σ (X – Xm)2
n-1
σ2 = Σ D2
N
s 2 = Σ d2
n-1
σ2 = SS
N
s2 = SS
n-1
σ2 = ΣX2 – [(ΣX)2 /N]
N
s2 = ΣX2 – [(ΣX)2 /n]
n -1
If you use the Computational Formula – no need to
find the Mean
X
4
7
9
5
8
3
X2
ΣX
ΣX2
Is (ΣX)2 = ΣX2 ?
Exercise 2
Calculate the mean, variance and Standard
deviation for the following distribution for sample
x
4
7
9
5
8
3
f
2
3
4
1
5
2
σ2 = Σ f(X - µ)2
N
s2 = Σ f(X – Xm)2
n-1
Xm = Mean
Standard Deviation
Is the square root of the variance
 Is the square root of the deviational
formula, sum of squares formula and
computational formula of variance
 Example: What is the Standard Deviation
of the distributions in Exercise 1?

Exercise 3
Find the Standard Deviation of the
following sample distributions
 A) 3, 4, 5, 7, 8, 9
 B) 12, 23, 34, 56, 13, 24

Exercise 4

What are the Standard Deviations of the
distributions in Exercise 2 were those of
the population?
Exercise 5


1) Calculate the range, mean deviation, variance
and standard deviation of this sample of scores
13, 18, 3, 23, 6, 12, 34
2) The distribution of Maths marks for class 4A is
as follows: 34, 45, 23, 47, 12, 67, 89.
Calculate the range, mean deviation, variance
and standard deviation using both the
definitional formula and the computational
formula
Standard Deviation for Grouped Data – Example for
Population
σ = ΣfD2 – ΣfD
N
N
Class Interval
f
Midpoint
(md)
2
fmd
Deviation
fD
D = md - Xm
D2
fD2
60 – 64
1
62
62
-9
-9
81
81
65 – 69
2
67
134
-4
-8
16
32
70 – 74
5
72
360
1
5
1
5
75 - 79
2
77
154
6
12
36
72
N = 10
Σ fD = 0
Xm = 710 = 71
10
σ =
Xm = Grouped mean
190 - 0
10
10
2
= √19 = 4.36
Σ fD2 = 190
Exercise 6

Find the mean, variance and standard
deviation of the following grouped data
(population):
Class Interval
0 -4
5 -9
10 – 14
15 – 19
Frequency
1
1
5
3
Exercise 7

Find the mean, variance and standard
deviation of the following distribution using
the grouped calculation method
78, 74, 80, 65, 63, 74, 67, 58, 74, 65,
65, 63, 74, 86, 80, 74, 67, 50, 78, 89
Exercise 8

Find the mean, variance and standard
deviation of the following distribution
64
64
50
50
82
70
75
70
80
60
64
42
64
64
54
35
70
75
90
97
60
70
70
12
60
46
48
34
70
75
60
46
To display variability – Box-andwhisker plot (refer to Chapter 1)
End of Chapter 3 – Measures of
Variability