1. Section 13.5 Measures of Dispersion

2. What You Will Learn
Range
Standard Deviation

3. Measures of Dispersion
Measures of dispersion are used to indicate the spread of the data.

4. Range
The range is the difference between the highest and lowest values; it indicates the total spread of the data.
Range = highest value – lowest value

5. Example 1: Determine the Range
The amount of caffeine, in milligrams, of 10 different soft drinks is given below. Determine the range of these data.
38, 43, 26, 80, 55, 34, 40, 30, 35, 43

6. Example 1: Determine the Range
Solution
38, 43, 26, 80, 55, 34, 40, 30, 35, 43
Range = highest value – lowest value
= 80 – 26 = 54
The range of the amounts of caffeine is 54 milligrams.

7. Standard Deviation
The standard deviation measures how much the data differ from the mean. It is symbolized with s when it is calculated for a sample, and with σ (Greek letter sigma) when it is calculated for a population.

8. Standard Deviation
The standard deviation, s, of a set of data can be calculated using the following formula.

9. To Find the Standard Deviation of a Set of Data
1. Find the mean of the set of data.
2. Make a chart having three columns:
Data Data – Mean (Data – Mean)2
3. List the data vertically under the column marked Data.
4. Subtract the mean from each piece of data and place the difference in the Data – Mean column.

10. To Find the Standard Deviation of a Set of Data
5. Square the values obtained in the Data – Mean column and record these values in the (Data – Mean)2 column.
6. Determine the sum of the values in the (Data – Mean)2 column.

11. To Find the Standard Deviation of a Set of Data
7. Divide the sum obtained in Step 6 by n – 1, where n is the number of pieces of data.
8. Determine the square root of the number obtained in Step 7. This number is the standard deviation of the set of data.

12. Example 3: Determine the Standard Deviation of Stock Prices
The following are the prices of nine stocks on the New York Stock Exchange. Determine the standard deviation of the prices.
$17, $28, $32, $36, $50, $52, $66, $74, $104

13. Example 3: Determine the Standard Deviation of Stock Prices
Solution
The mean is

14. Example 3: Determine the Standard Deviation of Stock Prices

15. Example 3: Determine the Standard Deviation of Stock Prices
Solution
Use the formula
The standard deviation, to the nearest tenth, is $27.01.