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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 Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
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
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
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 © 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.
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 median
This list is already in order:
£6500 £6500 £6500 £6500 £10500
The middle one is the third value
median = £6500
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
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
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
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:
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 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
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
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
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
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
200 but less than 300 28 38
300 but less than 400 42 80
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
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
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
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
42
382/75100300median
median = £388.10
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 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
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
mode = £345 from histogram
To find the highest point on the histogram:
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
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