12.4 Measures of Dispersion 799 to the mean and therefore have a smaller standard deviation. You will soon be able to verify such relationships yourself. Sometimes only a very small standard deviation is desirable or acceptable. Consider a cereal box that is to contain 8 oz of cereal. If the amount of cereal put into the boxes varies too much—sometimes underfilling, sometimes overfilling— the manufacturer will soon be in trouble with consumer groups and government agencies. At other times, a larger spread of data is desirable or expected. For example, intelligence quotients (IQs) are expected to exhibit a considerable spread about the mean because everyone is different. The following procedure explains how we determine the standard deviation of a set of data. DETERMINING THE STANDARD DEVIATION OF A SET OF DATA FROM A SAMPLE 1. Determine 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. 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. 7. Divide the sum obtained in Step 6 by − n 1, where n is the number of pieces of data in the sample. 8. Determine the square root of the number obtained in Step 7. This number is the standard deviation of the set of data. PROCEDURE Example 2 illustrates the procedure to follow to determine the standard deviation of a set of data. Example 2 Determine the Standard Deviation A veterinarian in an animal hospital recorded the following life spans of selected Labrador retrievers (to the nearest year): 7, 9, 11, 15, 18, 12 Determine the standard deviation of the life spans. Solution First determine the mean: = Σ = + + + + + = = x x n 7 911151812 6 72 6 12 Next construct a table with three columns, as illustrated in Table 12.5, and list the data in the first column (it is often helpful to list the data in ascending or descending order). Complete the second column by subtracting the mean, 12 in this case, from each piece of data in the first column.
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