33 Standard Deviation (1 of 4)
Learning Objectives
- Use mean and standard deviation to describe a distribution.
Introduction
In the section “Distributions for Quantitative Data,” we discussed the spread of a distribution in terms of a typical range of values. In “Quantifying Variability Relative to the Median,” we made this idea more precise with the interquartile range, IQR. The IQR gives us a measure of spread about the median. We defined a typical range of values about the median as the values between the first and third quartiles.
Now we want to develop a numerical measure of spread that we can use with the mean. In constructing a measure of spread about the mean, we want to compute how far a “typical” number is away from the mean.
Measuring Spread about the Mean
Let’s consider the sample data set 2, 2, 4, 5, 6, 7, 9. The mean of this data set is
[latex]\stackrel{¯}{x}=\frac{2+2+4+5+6+7+9\text{}}{7}\text{}=\text{}\frac{35}{7}=5[/latex]
Here is a dotplot of this data set with the mean marked by the vertical blue line.
We can see that some data is close to the mean and some data is further from the mean.
Since we want to see how the data points deviate from the mean, we determine how far each point is from the mean. We compute the difference between each of these values and the mean. These differences are called the deviations from the mean for each point.
| 2 − 5 = −3 |
| 2 − 5 = −3 |
| 4 − 5 = −1 |
| 5 − 5 = 0 |
| 6 − 5 = 1 |
| 7 − 5 = 2 |
| 9 − 5 = 4 |
When visualized on a dotplot, these differences are viewed as distances between each point and the mean. A negative difference indicates that the data point is to the left of the mean (shown in blue on the graph below). A positive difference indicates that the data point is to the right of the mean (shown in green on the graph below).
Our goal is to develop a single measurement that summarizes a typical distance from the mean. Before we continue, let’s practice determining the distance of a single data point from the mean.
Learn By Doing
The two questions below refer to the following dotplot. The mean is 9 and it is marked by the vertical blue line.
Since we want to determine how far a typical number is away from the mean, we might try to average these numbers. However, if we add them all up, we will get 0 (try it). Getting 0 with this procedure (finding differences from mean and adding them all together) is no accident – it always produces 0. We have to overcome this problem.
Recall that we are trying to find the typical distance between data points and the mean. It therefore makes sense to take the absolute value of each of these differences.
| | 2 − 5 | = | −3 | = 3 |
| | 2 − 5 | = | −3 | = 3 |
| | 4 − 5 | = | −1 | = 1 |
| | 5 − 5 | = | 0 | = 0 |
| | 6 − 5 | = | 1 | = 1 |
| | 7 − 5 | = | 2 | = 2 |
| | 9 − 5 | = | 4 | = 4 |
Now we can compute the average of these deviations. There are seven data points, so we add these seven distances and divide by 7. The result is a measure of spread about