Chapter 6 Measures of Difference Jon Curwin and

Chapter 6: Measures of Difference Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

Objectives • Describe the differences in data using appropriate statistics • Calculate the range and standard deviation for various types of data • Know when to use these measures of difference Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

Measures of dispersion The main measures of dispersion are the standard deviation, range and quartiles. The standard deviation is an average of differences about the mean. The range is the difference between the largest and smallest values The quartiles are quarter way and three quarter way markers Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

The mean is generally accepted as the mostly commonly used statistics. It is useful to know what is typical and the mean will allow us to answer questions like ‘what is the average age’ and ‘what was the average amount spent’. However, in most research we also want to know about the differences and answer questions like ‘what age differences are there’ and ‘how do spending patterns vary’. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE 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 A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

Using a list of numbers Example: 5 salaries: £ 6500 £ 10500 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

The mean is £ 7300 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

The standard deviation is: • Regarded as the most important measure of difference • Involves finding the difference about the mean • Looks at the average size of these differences Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

Standard deviation is given by Steps: 1. calculate the mean 2. Calculate the deviations from the mean Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

3. square deviations 4. sum squared deviations 5. average squared deviations which give variance 6. square root for standard deviation Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

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

The range is just the difference between the largest and smallest values It is useful in giving an overview of the spread in the data but it is very dependent on the two most extreme (and often most unreliable) values Range = £ 10500 - £ 6500 = £ 4000 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

Using tables of discrete data Example: errors found in pages of print Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

To calculate the mean we divide the total obtained from the last column and divided by the total number of pages. (the mean has been rounded to 2 d. p. ) Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

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

to 2 d. p. Rule of thumb most observations (95% making some assumptions) are within +/- 2 standard deviations from the mean Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

Using tables of continuous data When the tables give interval ranges, it is necessary to use mid-points. Let x take the mid-point values. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

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

The mean The standard deviation Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004

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

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

Conclusions • It is useful to have statistics that give both a measure of location (typically the mean) and a measure of spread (typically the standard deviation • You have seen the calculation of the standard deviation for different types of data • A spreadsheet or other computer package would typically be used for such calculation Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A A SHORT COURSE ISBN 1 -86152 -991 -0 © Thomson © Cengage Learning 2004