Comparisons using standard deviation
Standard deviation (\sigma) is a measure of spread, which helps give us a meaningful estimate of the variability in a data set. It is a weighted average of the distance of each data point from the mean. A small standard deviation indicates that most scores are close to the mean, while a large standard deviation indicates that the scores are more spread out away from the mean value.
Note that the mean will determine where the scores are clustered, while the standard deviation tells us how tightly they are clustered. The two sets of data above have a similar mean but different standard deviations.
When comparing data sets, to find which data set had higher scores, we would choose the set with the higher mean. To find the set with the more consistent scores, we would choose the set with the lower standard deviation.
Examples
Example 1
Two machines A and B are producing chocolate bars with the following mean and standard deviation for the weight of the bars.
| Machine | Mean (g) | Standard deviation (g) |
|---|---|---|
| \text{A} | 52 | 1.5 |
| \text{B} | 56 | 0.65 |
What does a comparison of the mean of the two machines tell us?
What does a comparison of the standard deviation of the two machines tell us?
Example 2
Han, a cricketer, has made scores of 52,\,20,\,68,\,70, and 150 in all his innings this season. In his next innings, he scores no runs.
What is the change in his season batting average before and after the sixth inning?
What is the change in his standard deviation before and after his sixth innings? Give your answer correct to two decimal places.
When comparing data sets, to find which data set had higher scores, we would choose the set with the higher mean. To find the set with the more consistent scores, we would choose the set with the lower standard deviation.