Standard Deviation Consider the following sets of data: 3,9,12,15,19,20 20 Range = 20-3 = 17 Mean = (3+9+12+15+19+20)/6=13

Range = 15-11 = 4 Mean = (11+12+13+13+14+15)/6 = 13 11,12,13,13,14,15 20 Range = 15-11 = 4 Mean = (11+12+13+13+14+15)/6 = 13 Note: The two sets of data have the same mean i.e. 13 but are very different. 20 20

A measure of spread which uses all the data is the standard deviation. When the standard deviation is low it means the scores are close to the mean. 20 Mean When the standard deviation is high it means the scores are spread out from the mean 20 Mean

Exercise 1 Look at the three sets of scores below and place the standard deviations for these scores in order, low to high 1 2 3 Mean Mean Mean

Calculating the Standard Deviation of a set of scores The standard deviation or “root, mean, square deviation” is a measure of how far all the scores differ from the mean. It can be calculated from first principles or by the application of a formula. Consider the scores listed earlier: 3,9,12,15,19,20 Mean =(3+9+12+15+19+20) / 6 = 13 We now construct a table to see how far each score differs from the mean.

Score Deviation (Deviation )2 3 (3-13) = -10 100 9 (9-13) = -4 16 12 (12-13) = -1 1 15 (15-13) = 2 4 19 (19-13) = 6 36 20 (20-13) = 7 49 The mean square deviation is 206 6 =34.7 Total 206 The standard deviation is

Standard Deviation by Formula All of the values can be found using a scientific calculator. You do not have to learn this formula as it is given on the exam paper cover.

0 - 2 Using the Sharp EL-531VH You need to be in STAT mode. DRG MODE 2nd F The calculator display shows MODE ? 0 - 2 Press 1

You are now in STATS mode and the calculator display shows S t a t x The next task is to enter the data. We will use the example already covered: We will enter the numbers: 3,9,12,15,19,20 All the STATS keys are in green on the calculator. 3 9 12 15 19 20 M+ We can now pick out any values from the keyboard using RCL

The values we require for the formula are laid out in the following key positions on the calculator: 7 8 9 x sx 4 5 6 1 2 3 n x x2 7 +/- These values are obtained by pressing first. RCL

The values can now be obtained and entered into the formula: x2 = 1220. x = 78. n = 6.

Note: There is a slight difference in answer between the two methods. The formula uses n-1 instead of n. This is because the data is treated as a sample.