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Standard Deviation

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Standard Deviation

The most common summary statistic used to measure the spread is the standard deviation (s). It is found by taking the average of the squared deviations from the mean, and then finding the square root.

The deviation of a score from the mean is the difference between the score and the mean.

σn is the standard deviation of a population and σn - 1 is the standard deviation of a sample.

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Example 1: Find the standard deviation of Esma's 9 golf scores with a mean of 5.

3 6 5 4 6 4 83 6

Standard deviation = √22 + 12 + 02 + 12 + 12 + 12 + 32 + 22 + 12

9 = √4 + 1 + 0 + 1 + 1 + 1 + 9 + 4 + 1

9 = √22

9 = 1.56

Add the squared deviations from the mean, divide by the total number of scores, then take the square root.

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Example 2: Using a calculatorUsing a calculator, find the mean and standard deviation of Esma's golf scores.

3 6 5 4 6 4 8 36

1. Enter scores 3 M+ 6 M+ 5 M+ 4 M+ ....... 6 M+Some calculators have a SAC key instead of M+ .

2. Check that n = 9.

3. Press to get the mean of 5.

4. Press σn to get the standard deviation of 1.56.

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1. Using your calculator, find the mean and standard deviation (correct to 1 decimal place) for each of these sets of data. a) 42, 35, 63, 70, 81, 80, 85b) $300, $400, $600, $440, $300, $700, $250, $580, $260c) 37.4℉, 38.2 ℉, 39.0℉, 36.8℉, 38.5℉, 38.0℉, 36.8℉, 40.5℉d) 165 kg, 146 kg, 178 kg, 190 kg, 158 kg, 147 kge) 23, 18, 24, 16, 17, 20, 15, 22, 19

2. Joshua swam a kilometre each morning for 10 days in preparation for a swimming carnival. His times (in minutes) were:

28 24 22 24 25 24 26 26 2427

a) What is his median swim time? b) What is his mean swim time?c) What is his range of swim times?d) What is the interquartile range of his times?e) What is the standard deviation of his times (correct to 1 decimal place)f) If Joshua asked you to tell him the most appropriate measures of location and spread for these times, which two would you choose? Justify your answer.

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3. Ted and Julie were paid by piecework for making T-shirts. The number made each day over an 8-day period were:

Ted: 18 25 19 19 26 24 15 22Julie: 16 20 21 28 12 26 18 19

a) For each person find: (i) the number of T-shirts made in the 8-day period(ii) the interquartile range of T-shirts made(iii) the mean number of T-shirts made(iv) the standard deviation of T-shirts made (correct to 1

decimal place)b) Comment on the statement that Ted is a more consistent worker than Julie by comparing their means and standard deviations. Give reasons for your answer.

4. This stem-and-leaf plot shows the waiting times in a medical centre (in minutes). a) Find the mean waiting time (1 d.p.)b) Find the standard deviation of waiting times (correct to 1 d.p.)

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5. Numbers of motor accidents per week over a 9-week period at a busy intersection were:

4 3 6 0 4 9 2 3 5a) What is the median number of accidents per week?b) What is the mean number of accidents per week?c) Does the mean or median best describe the centre of this data set? Give reasons. d) What is the range of the data?e) What is the interquartile range?f) Find the standard deviation for the data (correct to 1 d.p.)g) Comment on the statement: 'The number of accidents per week is fairly consistent'. Justify your answer.

6. The percentage marks for 250 students in a Business Studies examination are listed.

a) What is the mean mark (to the nearest whole number)?b) What is the standard deviation of the marks (correct to 1 d.p.)?c) Write a comment to the school principal describing the results of these students.