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SECTION 5.6 Mean, Median, Mode 221 Textbook Reference Section 12.1 5.6 MEAN, MEDIAN, MODE CLAST OBJECTIVES " Determine the mean, median, and mode " Recognize properties and interrelations among the mean, median, and mode " Infer relations and make accurate predictions from studying statistical data " Interpret real world data involving frequency and cumulative frequency tables Mean, median, and mode are called measures of central tendency because each describes the center of the data sample. Mean (Average) The mean or average is a calculation and is found by dividing the sum of the data values by the number of data values. Sum of data values Mean = Number of data values Median The median is the middle position when data values are ranked from low to high. To find the median rank the data in ascending order. If there is an odd number of data values, then the median is the value that lies in the middle position. If there is an even number of data value, then the median is the average of the two middle values. Mode The mode is the most frequently occurring (popular) value. Example a) Find the mean, median, and mode of the following set: {5, 20, 35, 20, 50, 45, 20, 10, 25, 45} Solution There are 10 elements in the set. Mean: 5 + 20 + 35 + 20 + 50 + 45 + 20 + 10 + 25 + 45 10 = 275 = 27.5 10 Median: 5, 10, 20, 20, 20, 25, 35, 45, 45, 50 Since we have 10 data values, the middle positions are the 5 th and 6 th positions. Note there are four data values below the 5 th position and 4 data values above 6 th positions. 20 and 25 are in the 5 th and 6 th positions. When we average 20 and 20 + 25 45 25, we get = = 22.5 5 2 Thus, the median is 22.5. Mode: The mode is 20 as it appears most frequently. © Houghton Mifflin Company. All rights reserved. 222 CHAPTER 5 Probability and Statistics Check Your Progress 5.6 1. Consider the scores of nine students: {84, 88, 52, 87, 70, 66, 95, 84, 71}. Find the mean, median, and mode. 2. Consider the following set of babies’ birth-weight (in pounds): {6.5, 7, 10, 8, 7, 7.5, 8, 7}. Find the mean, median, and mode. Comparing Mean, Median, and Mode Mean Median Mode Consider three cases. Case 1: Symmetric 1 2 3 Case 2: Right - Skewed 1 2 4 5 6 7 8 Median Mode 3 4 9 Mean 5 6 Mode Case 3: Left - Skewed Median Mean 1 2 3 4 5 6 © Houghton Mifflin Company. All rights reserved. SECTION 5.6 Examples b) The graph below shows the number of hours that college students spend in the game room per week with the years spent in college. Compare the mean, median, and mode. Mean, Median, Mode 223 Solutions Hours The distribution is symmetric. Thus the mean, median, and mode are equal. 1 2 3 Years in College 4 5 c) The graph below shows the distribution of scores on an algebra quiz. Compare the mean, median, and mode. The distribution is right-skewed. This indicates that the mode is less than the median, which is less than the mean. Frequency 25 20 15 10 5 0 65 70 75 80 85 90 95 Algebra Quiz Scores d) Half of the students scored 90 on an English test. Most of the remaining students scored 70 and the last few scored 30. Compare the mean, median, and mode. The distribution is skewed left. This indicates that the mean is less than the median, which is less than the mode. 30 70 90 © Houghton Mifflin Company. All rights reserved. 224 CHAPTER 5 Probability and Statistics Number of Babies Check Your Progress 5.6 3. The ounces of milk consumed per day by 6-month old babies is shown in the graph below. Compare the mean, median, and mode. 6 5 4 3 2 1 0 22 24 26 28 30' 32 34 Ounces of Milk 4. The distribution of placement exam scores are shown below. Discuss the mean, median, and mode. 15 10 5 0 95 90 85 80 75 70 65 60 Placement Test Scores 5. More than half of the shoes in a store are priced at $34.00. Most of the others are priced at $45.00, and a few are priced at $61.00. Discuss the mean, median, and mode. 6. Half of the scores on a History test were 95. Most of the remaining scores were 75. A few were 40. Discuss the mean, median, and mode. Applications of Mean, Median, and Mode Mode: The category with the highest frequency, percent, or proportion. Mean: Multiply each category by its proportion and add. Median: Organize categories in ascending order in one column. Write the corresponding frequencies or proportions in an adjacent column. Starting at the top, add the frequencies or proportions up to the fifty-percent point. The corresponding category will be the median. © Houghton Mifflin Company. All rights reserved. SECTION 5.6 Examples e) The table below shows the proportion of students taking a number of college-level classes this semester. Find the mean, median, and mode. No. of classes 0 1 2 3 4 5 Proportion 0.02 0.02 0.11 0.23 0.54 0.08 Mean, Median, Mode 225 Solutions Mean: = 0(0.02) + 1(0.02) + 2(0.11) + 3(0.23) + 4(0.54) + 5(0.08) = 0 + 0.02 + 0.22 + 0.69 + 2.16 + 0.4 = 3.49 Median: (Find the 50 % mark) 0.02 + 0.02 + 0.11 + 0.23 = 0.38 Å 38 % 0.38 + 0.54 = 0.92 Å 92 % This indicates that the 50 % - mark falls in the “ 4 Classes ” category. Thus the median is 4 classes. Mode: The mode is 4 classes because the “ 4 Classes” category has the largest proportion. f) In the table below, the scores on a Spanish test are listed with the corresponding percentile rank. What percent of students scored between 60 and 80? What percent scored above 80? What percent scored below 60? Score 90 80 70 60 40 50 Percentile Rank Between 60 and 80: 93 % – 34 % = 59 % 99 93 72 34 16 5 Above 80: 99 % – 93 % = 6 % Below 60: 34 % Check Your Progress 5.6 For Questions 7 and 8, use the information found in the table below. The table shows the results of a survey in which people were asked the number of times per week that they shop at a grocery store. No. of Trips to Grocery Store 0 1 2 3 4 Percent Men 18 45 12 17 8 Percent Women 7 12 53 13 15 7. Find the median number of trips to the grocery store for men. 8. Find the mean number of trips to the grocery store for women. © Houghton Mifflin Company. All rights reserved. 226 CHAPTER 5 Probability and Statistics For Questions 9 and 10, use the information in the table below, which shows the distribution of the total family income for a certain county for the year 2000. Income Level ($) 0 – 9,999 10,000 – 19,999 20,000 – 29,999 30,000 – 39,999 40,000 – 49,999 50,000 – 59,999 60,000 – 69,999 70,000 and over Percent of Families 12 14 16 13 13 9 9 14 9. What percentage of families has incomes of at least $30,000? 10. What percentage of families has incomes between $20,000 and $49,999? See If You Remember SECTIONS 5.1 – 5.5 1. How many 5-digit numbers can be formed from the digits 0 – 9, if digits cannot be repeated? 2. Seven candidates are running for three city council seats. How many different groups can be formed? 3. What are the odds in favor of randomly drawing a green ball from a box that contains 2 yellow, 3 red, 5 green, and 6 white balls? 4. A fair die is rolled three times. What is the probability of getting a three, then a two, and then a one? © Houghton Mifflin Company. All rights reserved.