Chapter 5: Averaging Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger

Chapter 5: Averaging

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

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


ISBN 1-86152-991-0 © Cengage


ISBN 1-86152-991-0 © Cengage

Objectives

•Describe data using a measure of average

•Calculate the mean, median and mode for various types of data

•Critically assess the use of these measures of location

Measures of location

The 3 main measures of central location are the mean, median and mode.

•The mean (simple) is an averaging of all the values.

•The median is the middle value of an ordered list

•The mode is the most frequent value




ISBN 1-86152-991-0 © Cengage


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


ISBN 1-86152-991-0 © Cengage

Using a list of numbers

Example: 5 salaries:

£6500 £6500 £6500 £6500 £10500

The mean

5

105006500650065006500

n

xx

7300£5

36500

To calculate the mean, we add to find the total and divide by the number included.




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The median

This list is already in order:

£6500 £6500 £6500 £6500 £10500

The middle one is the third value

median = £6500




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ISBN 1-86152-991-0 © Cengage

The mode

The most frequently occurring value is the salary of £6500

mode = £6500

Using tables of discrete data

No. of errors frequency fx 0 12 0 1 10 10 2 5 10 3 4 12 4 3 12 34 44

Example: errors found in pages of print




ISBN 1-86152-991-0 © Cengage


ISBN 1-86152-991-0 © Cengage

29.134

44

n

fxx

In this example we need to sum the frequency to find the total number of pages (34 pages in this case). We then multiply number of errors (per page) by frequency to find total number of errors (44 errors in total) in the last column

The mean (rounded to 2 d.p.)

2

136

2

1

nmedian

= 18½ ordered value

To find the middle value we work from cumulative frequency

The median

We need to find the position of the middle value using the formula:




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Using the example table

No. of errors frequency Cum fre (F) 0 12 12 1 10 22 2 5 27 3 4 31 4 3 34 34

modal group

The ordered position 17.5 is above 12 and below 22, so the modal group is the second group. The median = 1




ISBN 1-86152-991-0 © Cengage


ISBN 1-86152-991-0 © Cengage

The mode

The value 0 occurs more than any other (12 times). The mode = 0

Using tables of continuous data

Monthly commission (£’s) f x fx

100 but less than 200 10 150 1500

200 but less than 300 28 250 7000

300 but less than 400 42 350 14700

400 but less than 600 50 500 25000

600 but less than 1000 20 800 16000

150 64200

Example: Monthly commission paid to 150 sales personnel




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ISBN 1-86152-991-0 © Cengage

428£150

64200

n

fxx

In this case, frequencies are summed to find the number of sales personnel and midpoints are used to find the last column totals.

The mean


ISBN 1-86152-991-0 © Cengage

The median

In this case the position of the median is found using n/2. Given the number of sales personnel of 150, the median corresponds to the 75th person on this continuous scale. We are essentially cutting the distribution in two (not working with an ordered list).

Weekly income (£’s) f Cum fre (F)

100 but less than 200 10 10



400 but less than 600 50 130

600 but less than 1000 20 150

150

median group

Two methods to find median - graphical and the use of formula




ISBN 1-86152-991-0 © Cengage


ISBN 1-86152-991-0 © Cengage

Cumulative frequency plot

0

20

40

60

80

100

120

140

160

0 200 400 600 800 1000 1200

Monthly commission (£'s)

Cu

mu

lati

ve

fre

qu

en

cy

The median using the graphical method

n/2=75

Median = £388

The median using the formula:

f

Fnilmedian

2/

where l is the lower boundary of the median group

i is the width of the median group

F is the cum freq up to the median group

f is the freq in the median group




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42

382/75100300median

median = £388.10




ISBN 1-86152-991-0 © Cengage

The mode

To find the mode you look for the highest point on the histogram (or you can look at a more advanced book for the method of calculation)

You will need to construct a histogram and scale the height of the blocks where necessary




ISBN 1-86152-991-0 © Cengage


ISBN 1-86152-991-0 © Cengage

mode = £345 from histogram

To find the highest point on the histogram:


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Conclusions

•You have seen the calculation of the mean, median and mode for different types of data

•The mean is the most widely accepted measure of average

•The differences between the mean, median and mode will tell you something about the distribution of the data

Documents

Chapter 5: Averaging Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger