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Chapter 6: Measures of
Difference
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Objectives
•Describe the differences in data using
appropriate statistics
•Calculate the range and standard deviation for
various types of data
•Know when to use these measures of difference
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
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ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Measures of dispersion
The main measures of dispersion are the
standard deviation, range and quartiles.
The standard deviation is an average of
differences about the mean.
The range is the difference between the largest
and smallest values
The quartiles are quarter way and three quarter
way markers
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
The mean is generally accepted as the mostly
commonly used statistics. It is useful to know what
is typical and the mean will allow us to answer
questions like ‘what is the average age’ and ‘what
was the average amount spent’. However, in most
research we also want to know about the differences
and answer questions like ‘what age differences are
there’ and ‘how do spending patterns vary’.
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
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Learning 2004
The types of data to be considered:
•A simple list of numbers
•Tables of discrete data
•Tables of continuous data
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Using a list of numbers
Example: 5 salaries:
£6500 £6500 £6500 £6500 £10500
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
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ISBN 1-86152-991-0
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The mean is £7300
x 6500  6500  6500  6500  10500

x

n
5
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
The standard deviation is:
•Regarded as the most important measure of difference
•Involves finding the difference about the mean
•Looks at the average size of these differences
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Standard deviation is given by
s
 (x  x)
2
n
Steps:
1. calculate the mean
x
2. Calculate the deviations from the mean
(x  x)
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
3. square deviations
(x  x)
4. sum squared deviations
(x  x)
5. average squared deviations
 (x  x)
n
which give variance
6. square root for standard deviation
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
2
2
2
Example
x  £7300
x
(x  x)
( x  x )2
6500
-800
640000
6500
-800
640000
6500
-800
640000
6500
-800
640000
10500
3200
10240000
12800000
12800000
s
 1600
5
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
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COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
The range is just the difference between the
largest and smallest values
It is useful in giving an overview of the spread in the
data but it is very dependent on the two most extreme
(and often most unreliable) values
Range = £10500 - £6500 = £4000
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Using tables of discrete data
Example: errors found in pages of print
No. of errors
0
1
2
3
4
frequency
12
10
5
4
3
34
fx
0
10
10
12
12
44
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
To calculate the mean we divide the total obtained
from the last column and divided by the total number
of pages.
(the mean has been rounded to 2 d.p.)
fx 44

x

 1.29
n
34
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
s
No. errors
0
1
2
3
4
 f (x  x)
2
n
f (x  x) ( x  x )2 f ( x  x ) 2
12 -1.29 1.6641 19.9692
10 -0.29 0.0841
0.8410
5
0.71 0.5041
2.5205
4
1.71 2.9241 11.6964
3
2.71 7.3441 22.0323
34
57.0594
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
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ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
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Learning 2004
57.0594
s
 1.30
34
to 2 d.p.
Rule of thumb
most observations (95% making some assumptions)
are within +/- 2 standard deviations from the mean
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Using tables of continuous data
When the tables give interval ranges, it is necessary to
use mid-points. Let x take the mid-point values.
s
 f (x  x)
2
n
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
x
150
250
350
500
800
f
10
28
42
50
20
150
fx ( x  x ) ( x  x ) 2 f ( x  x ) 2
1500 -278 77284
772840
7000 -178 31684
887152
14700
-78
6084
255528
25000
72
5184
259200
16000
372 138384 2767680
64200
4942400
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
The mean
64200
x
 428
150
The standard deviation
4942400
s
 181.52
150
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Alternatively
s
 fx
n
2
  fx 

 

n


2
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
x
150
250
350
500
800
f
10
28
42
50
20
150
fx 2
225000
1750000
5145000
12500000
12800000
32420000
fx
1500
7000
14700
25000
16000
64200
2
32420000  64200 
s

  181.52
150
 150 
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Conclusions
•It is useful to have statistics that give both a measure
of location (typically the mean) and a measure of
spread (typically the standard deviation
•You have seen the calculation of the standard
deviation for different types of data
•A spreadsheet or other computer package would
typically be used for such calculation
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004