786 CHAPTER 12 Statistics A second average is the median . To determine the median of a set of data, rank the data from smallest to largest, or largest to smallest, and determine the value in the middle of the set of ranked data. This value will be the median. Definition: Median The median is the value in the middle of a set of ranked data. When there are an even number of pieces of data, the median is halfway between the two middle pieces of the ranked data. In this case, to determine the median, add the two middle pieces and divide this sum by 2. Example 2 Determine the Median Determine the median of the nurses’ ages in Example 1. Solution Ranking the data from smallest to largest gives 21, 30, 37, 51, and 51. Since 37 is the value in the middle of this set of ranked data (two pieces of data above it and two pieces below it), 37 years is the median. 7 Now try Exercise 15. A third average is the mode . Timely Tip Data must be ranked before determining the median. A common error made when determining the median is neglecting to arrange the data in ascending (increasing) or in descending (decreasing) order. Example 3 Determine the Median of an Even Number of Data Determine the median of the following sets of data. a) 14, 19, 21, 22, 16, 21, 16, 17 b) 11, 12, 12, 12, 13, 14 Solution a) Ranking the data gives 14, 16, 16, 17, 19, 21, 21, 22. There are eight pieces of data. Therefore, the median will lie halfway between the two middle pieces, the 17 and the 19. The median is + 17 19 2 , or 36 2 , or 18. b) There are six pieces of data, and they are already ranked. Therefore, the median lies halfway between the two middle pieces. Both middle pieces are 12’s. The median is 12 12 2 + or 24 2 , or 12. 7 Now try Exercise 17. Definition: Mode The mode is the piece of data that occurs most frequently. MATHEMATICS TODAY Buying the American Dream One of the biggest dreams for most people is to own their own home. Yet depending on where you live, the American dream may be hard to achieve. Among major metropolitan housing markets in 2022, San Jose, California, had the highest median home price, $1,798,000, and Peoria, Illinois, had the lowest median home price, $98,000. Source: NAR.realtor m San Jose, California DenisTangneyJr/E+/Getty Images Example 4 Determine the Mode Determine the mode of the nurses’ ages in Example 1. Solution The ages are 30, 21, 51, 37, and 51. The age 51 is the mode because it occurs twice and the other values occur only once. 7 Now try Exercise 19. If each piece of data occurs only once, the set of data has no mode. For example, the set of data 1, 2, 3, 4, 5 has no mode. If two values in a set of data occur more often than all the other data, we consider both these values as modes and say that the data are bimodal , which means there are two modes for the set of data. For example, the set of data 1, 1, 2, 3, 3, 5 has two modes, 1 and 3. m Peoria, Illinois Katherine Welles/ Shutterstock
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