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Chapter. 3. Numerically Summarizing Data. Section. 3.5. The Five-Number Summary and Boxplots. Objectives Compute the five-number summary Draw and interpret boxplots. Objective 1 Compute the Five-Number Summary.

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  1. Chapter 3 Numerically Summarizing Data

  2. Section 3.5 The Five-Number Summary and Boxplots

  3. Objectives • Compute the five-number summary • Draw and interpret boxplots

  4. Objective 1 • Compute the Five-Number Summary

  5. The five-number summary of a set of data consists of the smallest data value, Q1, the median, Q3, and the largest data value. We organize the five-number summary as follows:

  6. EXAMPLEObtaining the Five-Number Summary Every six months, the United States Federal Reserve Board conducts a survey of credit card plans in the U.S. The following data are the interest rates charged by 10 credit card issuers randomly selected for the July 2005 survey. Determine the five-number summary of the data.

  7. EXAMPLEObtaining the Five-Number Summary Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm

  8. EXAMPLEObtaining the Five-Number Summary First, we write the data in ascending order: 6.5%, 9.9%, 12.0%, 13.0%, 13.3%, 13.9%, 14.3%, 14.4%, 14.4%, 14.5% The smallest number is 6.5%. The largest number is 14.5%. The first quartile is 12.0%. The second quartile is 13.6%. The third quartile is 14.4%. Five-number Summary: 6.5% 12.0% 13.6% 14.4% 14.5%

  9. Objective 2 Draw and Interpret Boxplots

  10. Drawing a Boxplot Step 1 Determine the lower and upper fences. Lower Fence = Q1 – 1.5(IQR) Upper Fence = Q3 + 1.5(IQR) where IQR = Q3 – Q1 Step 2 Draw a number line long enough to include the maximum and minimum values. Insert vertical lines at Q1, M, and Q3. Enclose these vertical lines in a box. Step 3 Label the lower and upper fences.

  11. Drawing a Boxplot Step 4 Draw a line from Q1 to the smallest data value that is larger than the lower fence. Draw a line from Q3 to the largest data value that is smaller than the upper fence. These lines are called whiskers. Step 5 Any data values less than the lower fence or greater than the upper fence are outliers and are marked with an asterisk (*).

  12. EXAMPLEObtaining the Five-Number Summary Every six months, the United States Federal Reserve Board conducts a survey of credit card plans in the U.S. The following data are the interest rates charged by 10 credit card issuers randomly selected for the July 2005 survey. Construct a boxplot of the data.

  13. EXAMPLEObtaining the Five-Number Summary Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm

  14. Step 1: The interquartile range (IQR) is 14.4% - 12% = 2.4%. The lower and upper fences are: Lower Fence = Q1 – 1.5(IQR) = 12 – 1.5(2.4) = 8.4% Upper Fence = Q3 + 1.5(IQR) = 14.4 + 1.5(2.4) = 18.0% Step 2: [ ] *

  15. Use a boxplot and quartiles to describe the shape of a distribution. The interest rate boxplot indicates that the distribution is skewed left.

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