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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