1. 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

2. 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

3. 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

4. The types of data to be considered: A simple list of numbers Tables of discrete data Tables of continuous data

5. 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

6. The mean 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

7. 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

8. The mode The most frequently occurring value is the salary of £6500 mode = £6500

9. Using tables of discrete data 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

10. 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.)

11. = 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

12. Using the example table 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

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

14. Using tables of continuous data 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

15. 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

16. 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 75 th person on this continuous scale. We are essentially cutting the distribution in two (not working with an ordered list).

17. 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

18. The median using the graphical method n/2=75 Median = £388

19. The median using the formula: wherel 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

20. 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

21. 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

22. mode = £345 from histogram To find the highest point on the histogram:

23. 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