# The Measurement of Adolescents, II

## CONTENTS OF CURRICULUM UNIT 86.05.03

## AN INTRODUCTION TO ELEMENTARY STATISTICS

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## Measures of Disperseion

*range*of a set of numbers is the difference between the highest and lowest numbers of the set. To find the range of a set of numbers, use the following formula:

Range = Highest Number—Lowest number

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Example
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What is the range of the following set of numbers? 3, 1, 6, 12, 9, 8, 10, 15
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Solution
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The highest number in the set is 15. The lowest number in the set is 1.
15Ð1 = 14. The range of the set is 14.

Another way of indicating the dispersion of scores is in terms of their deviations from the mean. This method is known as
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standard
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deviation
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and tells how scores tend to scatter about the mean of a set of data. If the standard deviation is small the scores tend to cluster closely about the mean. If the standard deviation is large, there is a wide scattering of scores about the mean. Standard deviation is represented by the symbol s and may be computed by the formula:

Standard Deviation = s=

(figure available in print form)

where x is a score, x is the mean, n is the number of scores, and means “the sum of”.

Six steps are used to find standard deviation:

- 1.) List each score (x) in the set of data.
- 2.) Compute the mean (x) for the data.
- 3.) Subtract the mean from each score (xÐx). The result is the deviation of each score from the mean.
- 4.) Square the deviations.
- 5.) Find the average of the squares of the deviations by dividing the sum of the squares of the deviations by the number of scores in the distribution.
- 6.) Take the square root of this average. The result is the standard deviation.

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Example
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Compute the standard deviation for the scores: 2, 3, 4, 5, 6, 7, 8
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Solution
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(figure available in print form)
The standard deviation is a number that is used to compare scores in a distribution. If the mean of a group of test scores is 75, and the standard deviation is 10, a person who receives a score of 85 is one standard deviation above the mean. If the mean of another group of test scores is 80, and the standard deviation is 3, a person who receives a score of 83 is one standard deviation above the mean. This person has done equally well, with respect to the other class members, as the person who received 85 on the first test.

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Exercises
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a.) 24, 15, 19, 29, 24, 22

- 1.) Compute the range for the following sets of scores:

b.) 113, 98, 107, 102, 123, 110

c.) 72.9, 75.6, 74.3, 86.1, 80, 82.7

d.) 56, 72, 98, 64, 87, 91, 22

a.) 26, 18, 19, 29, 20, 26

- 2.) Compute the standard deviations for the following sets of scores:

b.) 111, 98, 107, 103, 126

c.) 72.9, 75.6, 74.3, 86.1, 80, 82.7

- 3.) On an arithmetic test the mean was 78 and the standard deviation was 8. How many standard deviations from the mean was each of the following scores? 86, 74, 94, 80, 98, 70, 62