Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE

ISBN 1-86152-991-0 © Thomson Learning 2004


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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




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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




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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’.




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The types of data to be considered:

•A simple list of numbers

•Tables of discrete data

•Tables of continuous data




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Using a list of numbers

Example: 5 salaries:

£6500 £6500 £6500 £6500 £10500




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The mean is £7300

5

105006500650065006500

n

xx




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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




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Standard deviation is given by

n

xxs

2)(

xSteps:

1. calculate the mean

2. Calculate the deviations from the mean

)( xx




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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




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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




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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




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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




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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.)




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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)(




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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.




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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




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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




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The mean428

150

64200x

The standard deviation

52.181150

4942400s




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Alternatively

22

n

fx

n

fxs




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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




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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

Documents

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE