EXAMPLE 4 Finding the mean and standard deviation
The numbers of home runs Barry Bonds hit in his 22 major league seasons are
| 16 | 25 | 24 | 19 | 33 | 25 | 34 | 46 | 37 | 33 | 42 |
| 40 | 37 | 34 | 49 | 73 | 46 | 45 | 45 | 5 | 26 | 28 |
280
To find the mean of these observations,
Figure 12.6 displays the data as points above the number line, with their mean marked by a vertical line. The arrow shows one of the distances from the mean. The idea behind the standard deviation is to average the 22 distances. To find the standard deviation by hand, you can use a table layout:
| Observation | Squared distance from mean |
|---|---|
| 16 | (16 − 34.6)2 = (−18.6)2 = 345.96 |
| 25 | (25 − 34.6)2 = (−9.6)2 = 92.16 |
| ⋮ | |
| 28 | (28 − 34.6)2 = (−6.6)2 = 43.56 |
| sum = 4139.12 |
The average is
Notice that we “average” by dividing by one less than the number of observations. Finally, the standard deviation is the square root of this number: