104 CHAPTER 2 Descriptive Statistics The interquartile range (IQR) of a data set is a measure of variation that gives the range of the middle portion (about half) of the data. The IQR is the difference between the third and first quartiles. IQR = Q3 - Q1 DEFINITION In Section 2.3, an outlier was described as a data entry that is far removed from the other entries in the data set. One way to identify outliers is to use the interquartile range. Using the Interquartile Range to Identify Outliers 1. Find the first 1Q12 and third 1Q32 quartiles of the data set. 2. Find the interquartile range: IQR = Q3 - Q1. 3. Multiply IQR by 1.5: 1.51IQR2. 4. Subtract 1.51IQR2 from Q1. Any data entry less than Q1 - 1.51IQR2 is an outlier. 5. Add 1.51IQR2 to Q3. Any data entry greater than Q3 + 1.51IQR2 is an outlier. GUIDELINES Using the Interquartile Range to Identify an Outlier Find the interquartile range of the data set in Example 2. Are there any outliers? SOLUTION From Example 2, you know that Q1 = 47 and Q3 = 58.5. So, the interquartile range is IQR = Q3 - Q1 = 58.5 - 47 = 11.5. To identify any outliers, first note that 1.51IQR2 = 1.5111.52 = 17.25. There is a data entry, 19, that is less than Q1 - 1.51IQR2 = 47 - 17.25 Subtract 1.5(IQR) from Q1. = 29.75 A data entry less than 29.75 is an outlier. but there are no data entries greater than Q3 + 1.51IQR2 = 58.5 + 17.25 Add 1.5(IQR) from Q3. = 75.75. A data entry greater than 75.75 is an outlier. So, 19 is an outlier. Interpretation The costs of tuition and fees for the liberal arts colleges listed in the middle of the data set varies by at most $17,250. Notice that the outlier, 19 (or $19,000), does not affect the IQR. TRY IT YOURSELF 3 Find the interquartile range for the points scored by the 55 winning teams listed on page 39. Are there any outliers? Answer: Page A37 Another important application of quartiles is to represent data sets using box-and-whisker plots. A box-and-whisker plot (or boxplot) is an exploratory data analysis tool that highlights the important features of a data set. To graph a box-and-whisker plot, you must know the values shown at the top of the next page. EXAMPLE 3
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