Measures of Position 2.5 102 CHAPTER 2 Descriptive Statistics Quartiles Percentiles and Other Fractiles The Standard Score What You Should Learn How to find the first, second, and third quartiles of a data set, how to find the interquartile range of a data set, and how to represent a data set graphically using a box-and-whisker plot How to interpret other fractiles such as percentiles, and how to find percentiles for a specific data entry How to find and interpret the standard score (z-score) Quartiles In this section, you will learn how to use fractiles to specify the position of a data entry within a data set. Fractiles are numbers that partition, or divide, an ordered data set into equal parts (each part has the same number of data entries). For instance, the median is a fractile because it divides an ordered data set into two equal parts. The three quartiles, Q1, Q2, and Q3, divide an ordered data set into four equal parts. About one quarter of the data fall on or below the first quartile Q1. About one half of the data fall on or below the second quartileQ2 (the second quartile is the same as the median of the data set). About three quarters of the data fall on or below the third quartile Q3. DEFINITION Finding the Quartiles of a Data Set The amounts of fuel (in gallons per year) that automobile commuters waste due to traffic congestion in the 15 largest U.S. urban areas (where the populations are over 3 million) are listed. Find the first, second, and third quartiles of the data set. What do you observe? (Source: Texas A&M Transportation Institute) 34 30 31 31 25 31 25 24 38 26 39 26 38 31 35 SOLUTION First, order the data set and find the median Q2. The first quartile Q1 is the median of the data entries to the left of Q2. The third quartile Q3 is the median of the data entries to the right of Q2. Data entries to Data entries to the left of Q2 the right of Q2 24 25 25 26 26 30 31 31 31 31 34 35 38 38 39 Q1 Q2 Q3 Interpretation In about one quarter of the large urban areas, auto commuters waste 26 gallons of fuel or less, about one half waste 31 gallons or less, and about three quarters waste 35 gallons or less. TRY IT YOURSELF 1 Find the first, second, and third quartiles for the points scored by the 55 winning teams using the data set listed on page 39. What do you observe? Answer: Page A37 EXAMPLE 1
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