Measures of Spread for AP Statistics
Practice problems for these concepts can be found at:
- One-Variable Data Analysis Multiple Choice Practice Problems for AP Statistics
- One-Variable Data Analysis Free Response Practice Problems for AP Statistics
- One-Variable Data Analysis Review Problems for AP Statistics
- One-Variable Data Analysis Rapid Review for AP Statistics
Simply knowing about the center of a distribution doesn't tell you all you might want to know about the distribution. One group of 20 people earning $20,000 each will have the same mean and median as a group of 20 where 10 people earn $10,000 and 10 people earn $30,000. These two sets of 20 numbers differ not in terms of their center but in terms of their spread, or variability. Just as there were measures of center based on the mean and the median, we also have measures of spread based on the mean and the median.
Variance and Standard Deviation
One measure of spread based on the mean is the variance. By definition, the variance is the average squared deviation from the mean. That is, it is a measure of spread because the more distant a value is from the mean, the larger will be the square of the difference between it and the mean.
- Symbolically, the variance is defined by
Note that we average by dividing by n - 1 rather than n as you might expect. This is because there are only n - 1 independent datapoints, not n, if you know . That is, if you know n - 1 of the values and you also know , then the nth datapoint is determined.
One problem using the variance as a measure of spread is that the units for the variance won't match the units of the original data because each difference is squared. For example, if you find the variance of a set of measurements made in inches, the variance will be in square inches. To correct this, we often take the square root of the variance as our measure of spread.
The square root of the variance is known as the standard deviation. Symbolically,
As discussed earlier, it is common to leave off the indices and write:
In practice, you will rarely have to do this calculation by hand because it is one of the values returned when you use you calculator to do 1-Var Stats on a list (it's the Sx near the bottom of the first screen).
The definition of standard deviation has three useful qualities when it comes to describing the spread of a distribution:
- It is independent of the mean. Because it depends on how far datapoints are from the mean, it doesn't matter where the mean is.
- It is sensitive to the spread. The greater the spread, the larger will be the standard deviation. For two datasets with the same mean, the one with the larger standard deviation has more variability.
- It is independent of n. Because we are averaging squared distances from the mean, the standard deviation will not get larger just because we add more terms.
example: Find the standard deviation of the following 6 numbers: 3, 4, 6, 6, 7, 10.
Because it depends upon distances from the mean, it should be clear that extreme values will have a major impact on the numerical value of the standard deviation. Note also that, in practice, you will never have to do the calculation above by hand—you will rely on your calculator.
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