Transcript Chapter 5 - Cengage Learning

Chapter 5: Averaging
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
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
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 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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
The mean
To calculate the mean, we add to find the total and
divide by the number included.
x 6500  6500  6500  6500  10500

x

n
5
36500

 £7300
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 median
This list is already in order:
£6500 £6500 £6500 £6500 £10500
The middle one is the third value
median = £6500
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 mode
The most frequently occurring value is the salary of
£6500
mode = £6500
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
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
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.)
fx 44

x

 1.29
n
34
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
The median
We need to find the position of the middle value
using the formula:
n  1 36  1
median 

2
2
= 18½ ordered value
To find the middle value we work from cumulative
frequency
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 the example table
No. of errors frequency
0
12
1
10
2
5
3
4
4
3
34
Cum fre (F)
12
22
27
31
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
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 mode
The value 0 occurs more than any other (12 times).
The mode = 0
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Using tables of continuous data
Example: Monthly commission paid to 150 sales
personnel
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
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
In this case, frequencies are summed to find the
number of sales personnel and midpoints are used to
find the last column totals.
fx 64200

x

 £428
n
150
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).
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
median
group
150
Two methods to find median - graphical and the
use of formula
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 median using the graphical method
Cumulative frequency plot
160
Cumulative frequency
140
120
100
n/2=75
80
60
40
20
Median = £388
0
0
200
400
600
800
1000
Monthly commission (£'s)
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
1200
The median using the formula:
 n/2 F 

median  l  i
f


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
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
 75 / 2  38 
median  300  100

42


median = £388.10
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 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
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 find the highest point on the histogram:
mode = £345 from 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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage