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Chapter 3: Averages and
Variation
Section 2: Measures of
Dispersion
Measure of Dispersion
• Reflects the amount • Example: Find the mean
of spread or
and median for the
following sets of data.
variability in a
collection of data.
• 71 73 74 76 77 79
– Mean & Median = 75
• 46 63 70 80 91 100
– Mean & Median = 75
A measure of central
tendency is incapable of
detecting differences in the
spread or variability in a
collection of data values.
Range
• The difference between the
highest and lowest data values.
• Range = Highest – Lowest
Example: Find the mean and range for
the following sets of data.
1
2
3 4 5
6 7
8 9 10 11
Number of Books Read by History Students
Mean = 6 Range = 10
1
2
3 4 5
6 7
8 9 10 11
Number of Books Read by Sociology Student
Mean = 6 Range = 10
A
X
A
A
B
X
38
25
34
24
26
23
24
22
20
21
20
19
16
18
14
17
6
16
2
15
B
B
The sum of the deviations from the mean always equals zero.
Variance
• The average of the
sum of the squared
deviation scores.
• Population Variance
2
 =  x 

N
• Sample Variance
 x
s2 =

x
n 1
2

2
Standard Deviation
• Square root of the variance
• Typical distance from the mean for the data
values
• Population Standard Deviation
 =
x   2

N
• Sample Standard Deviation
s=
 x  x 2
n 1
Example
• Find the population variance for the
following data values.
• 6 11 5 1 6 6 7 5 7 6
***First find the population mean
x

 =
N
Population Variance Sample
(cont)
x
6
11
5
1
6
6
7
5
7
6
x–

0
5
-1
-5
0
0
1
-1
1
0
 )2
(x –
0
25
1
25
0
0
1
1
1
0
54
Population Variance Sample
(cont)
54
x   

•  =
= 10 = 5.4
N
• Using the previous example, the
population standard deviation would be
found by:
2
2

=
5.4
= 2.32
Example Sample Variance
• Find the sample variance and sample
standard deviation for the following data
values.
4 3 7 4 2
• First find the sample mean
x
20

= 5 =4
x =
n
Sample Variance Example (Cont)
Use a table to calculate the sample variance.
x
2
4
3
7
4
x– x
-2
0
-1
3
0
(x – x )2
4
0
1
9
0
14
Sample Variance Example (Cont)
•
s2 =
2


x

x

n 1
=
14
4
= 3.5
Take the square root of the sample variance to find the sample
standard deviation.
• s=
3.5
= 1.87