SECTION 2.4 Measures of Variation 87 Entry x Deviation x − M Squares 1x − M2 2 1 -3 9 3 -1 1 5 1 1 7 3 9 Interpreting Standard Deviation When interpreting the standard deviation, remember that it is a measure of the typical amount an entry deviates from the mean. The more the entries are spread out, the greater the standard deviation. Data entry Frequency 1 3 2 4 6 8 7 5 1 2 3 4 5 6 7 8 9 s = 0 x = 5 Data entry Frequency 1 3 2 4 6 8 7 5 s ≈ 1.2 x = 5 1 2 3 4 5 6 7 8 9 Data entry Frequency 1 3 2 4 6 8 7 5 s ≈ 3.0 x = 5 1 2 3 4 5 6 7 8 9 Estimating Standard Deviation Without calculating, estimate the population standard deviation of each data set. 1. Data entry Frequency 1 3 2 4 6 8 7 5 0 1 2 3 4 5 6 7 N = 8 = 4 μ 2. Frequency 1 3 2 4 6 8 7 5 0 1 2 3 4 5 6 7 Data entry N = 8 = 4 μ 3. Data entry Frequency 1 3 2 4 6 8 7 5 0 1 2 3 4 5 6 7 N = 8 = 4 μ SOLUTION 1. Each of the eight entries is 4. The deviation of each entry is 0, so s = 0. Standard deviation 2. Each of the eight entries has a deviation of ±1. So, the population standard deviation should be 1. By calculating, you can see that s = 1. Standard deviation 3. Each of the eight entries has a deviation of ±1 or ±3. So, the population standard deviation should be about 2. By calculating, you can see that s is greater than 2, with s ≈ 2.2. Standard deviation TRY IT YOURSELF 5 Write a data set that has 10 entries, a mean of 10, and a population standard deviation that is approximately 3. (There are many correct answers.) Answer: Page A37 Data entries that lie more than two standard deviations from the mean are considered unusual, while those that lie more than three standard deviations from the mean are very unusual. Unusual and very unusual entries have a greater influence on the standard deviation than entries closer to the mean. This happens because the deviations are squared. Consider the data entries from Example 5, part 3 (see table at the left). The squares of the deviations of the entries farther from the mean (1 and 7) have a greater influence on the value of the standard deviation than those closer to the mean (3 and 5). EXAMPLE 5 To explore this topic further, see Activity 2.4 on page 100. 2.4 Study Tip You can use standard deviation to compare variation in data sets that use the same units of measure and have means that are about the same. For instance, in the data sets with x = 5 shown at the right, the data set with s ≈ 3.0 is more spread out than the other data sets. Not all data sets, however, use the same units of measure or have approximately equal means. To compare variation in these data sets, use the coefficient of variation, which is discussed later in this section.
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