Section 3.3 Measures of Position 127 Procedure for Computing the Percentile Corresponding to a Given Data Value Step 1: Arrange the data in increasing order. Step 2: Let x be the data value whose percentile is to be computed. Use the following formula to compute the percentile: Percentile = 100 · (Number of values less than x) + 0.5 Number of values in the data set Round the result to the nearest whole number. EXAMPLE 3.24 Computing the percentile corresponding to a given data value In 1989, the rainfall in Los Angeles during the month of February was 1.90. What percentile does this correspond to? Solution Step 1: Table 3.10 presents the data in increasing order. Step 2: There are 42 values in the data set. There are 17 values less than 1.90. Therefore, Percentile = 100 · 17 + 0.5 42 = 41.7 We round the result to 42. The value 1.90 corresponds to the 42nd percentile. Objective 3 Compute the quartiles of a data set Quartiles There are three percentiles, the 25th, the 50th, and the 75th, that are used more often than the others. These percentiles divide the data into four parts, each of which contains approximately one quarter of the data. For this reason, these three percentiles are called quartiles. DEFINITION Every data set has three quartiles: • The first quartile, denoted Q1, is the 25th percentile. Q1 separates the lowest 25% of the data from the highest 75%. • The second quartile, denoted Q2, is the 50th percentile. Q2 separates the lower 50% of the data from the upper 50%. Q2 is the same as the median. • The third quartile, denoted Q3, is the 75th percentile. Q3 separates the lowest 75% of the data from the highest 25%. Explain It Again The second quartile is the same as the median: The second quartile, Q2, divides the data in half. Therefore, Q2 is the same as the median. To compute the first and third quartiles, simply compute the 25th and 75th percentiles as explained previously. The easiest way to compute the second quartile is to compute the median as explained in Section 3.1. EXAMPLE 3.25 Computing quartiles Find the first and third quartiles for the Los Angeles rainfall data, presented in Table 3.9. Solution The first quartile is the same as the 25th percentile and the third quartile is the same as the 75th percentile. We follow the steps to compute the data values corresponding to the 25th and 75th percentiles.
navidi_monk_essential_statistics_1e_ch1_3
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