90 Chapter 3 Numerical Summaries of Data Objective 2 Compute the median of a data set The Median The basic idea behind the median is simple:We try to find a number that splits the data set in half, so that half of the data values are less than the median and half of the data values are greater than the median. The procedure for computing the median differs, depending on whether the number of observations in the data set is even or odd. Explain It Again Finding the middle number: For large data sets, it may not be easy to find the middle number just by looking. If n is odd, the middle number is in position n + 1 , and if n is even, the two 2 middle numbers are in positions n n and + 1. 2 2 Procedure for Computing the Median Step 1: Arrange the data values in increasing order. Step 2: Determine the number of data values, n. Step 3: If n is odd: The median is the middle number. In other words, the median is the number in position n + 1 2 . If n is even: The median is the average of the middle two numbers. These are the numbers in positions n 2 and n 2 + 1. In Example 3.1, we found the mean of five exam scores. In Example 3.2, we will find the median. EXAMPLE 3.2 Computing the median Find the median of the exam scores 78, 83, 92, 68, and 85. Solution Step 1: We arrange the data values in increasing order to obtain 68 78 83 85 92 Step 2: There are n = 5 values in the data set, so n is odd. Step 3: The middle number is 83, so the median is 83. EXAMPLE 3.3 Computing the median One of the goals of medical research is to develop treatments that reduce the time spent in recovery. Eight patients undergo a new surgical procedure, and the number of days spent in recovery for each is as follows. 20 15 12 27 13 19 13 21 Find the median time spent in recovery. Solution Step 1: We arrange the numbers in increasing order to obtain 12 13 13 15 19 20 21 27 Step 2: There are n = 8 numbers in the data set, so n is even. Step 3: The middle two numbers are 15 and 19. The median is the average of these two numbers. Median = 15 + 19 2 = 17 The median time spent in recovery is 17 days. Using technology to compute the mean and median In practice, technology is often used to compute means and medians, as Example 3.4 shows.
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