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Chapter 6: Measures of Difference. 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. Measures of dispersion. - PowerPoint PPT Presentation
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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Chapter 6: Measures of Difference
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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 Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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 Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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 Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
The types of data to be considered:
•A simple list of numbers
•Tables of discrete data
•Tables of continuous data
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Using a list of numbers
Example: 5 salaries:
£6500 £6500 £6500 £6500 £10500
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
The mean is £7300
5
105006500650065006500
n
xx
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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 Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Standard deviation is given by
n
xxs
2)(
xSteps:
1. calculate the mean
2. Calculate the deviations from the mean
)( xx
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
3. square deviations2)( xx
4. sum squared deviations 2)( xx
5. average squared deviations
n
xx 2)(
6. square root for standard deviation
which give variance
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Example
x )( xx 2)( xx
6 5 0 0 - 8 0 0 6 4 0 0 0 0
6 5 0 0 - 8 0 0 6 4 0 0 0 0
6 5 0 0 - 8 0 0 6 4 0 0 0 0
6 5 0 0 - 8 0 0 6 4 0 0 0 0
1 0 5 0 0 3 2 0 0 1 0 2 4 0 0 0 0
1 2 8 0 0 0 0 0
7300£x
16005
12800000s
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Range = £10500 - £6500 = £4000
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Using tables of discrete data
Example: errors found in pages of print
No. of errors frequency fx 0 12 0 1 10 10 2 5 10 3 4 12 4 3 12 34 44
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
29.134
44
n
fxx
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.)
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
N o . e r r o r s f )( xx 2)( xx
2)( xxf 0 1 2 - 1 .2 9 1 .6 6 4 1 1 9 .9 6 9 2 1 1 0 - 0 .2 9 0 .0 8 4 1 0 .8 4 1 0 2 5 0 .7 1 0 .5 0 4 1 2 .5 2 0 5 3 4 1 .7 1 2 .9 2 4 1 1 1 .6 9 6 4 4 3 2 .7 1 7 .3 4 4 1 2 2 .0 3 2 3
3 4 5 7 .0 5 9 4
n
xxfs
2)(
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
30.134
0594.57s
Rule of thumb
most observations (95% making some assumptions) are within +/- 2 standard deviations from the mean
to 2 d.p.
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
n
xxfs
2)(
When the tables give interval ranges, it is necessary to use mid-points. Let x take the mid-point values.
Using tables of continuous data
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
x f f x )( xx 2)( xx 2)( xxf 1 5 0 1 0 1 5 0 0 - 2 7 8 7 7 2 8 4 7 7 2 8 4 02 5 0 2 8 7 0 0 0 - 1 7 8 3 1 6 8 4 8 8 7 1 5 23 5 0 4 2 1 4 7 0 0 - 7 8 6 0 8 4 2 5 5 5 2 85 0 0 5 0 2 5 0 0 0 7 2 5 1 8 4 2 5 9 2 0 08 0 0 2 0 1 6 0 0 0 3 7 2 1 3 8 3 8 4 2 7 6 7 6 8 0
1 5 0 6 4 2 0 0 4 9 4 2 4 0 0
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
The mean428
150
64200x
The standard deviation
52.181150
4942400s
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Alternatively
22
n
fx
n
fxs
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
x f fx 2fx150 10 1500 225000250 28 7000 1750000350 42 14700 5145000500 50 25000 12500000800 20 16000 12800000
150 64200 32420000
52.181150
64200
150
324200002
s
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Thomson Learning 2004
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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