Analyzing Mean, Median, Mode, and Range

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1 Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Updated: 05/15/10 Analyzing Mean, Median, Mode, and Range Objective: Students will analyze mean, median, mode and range to determine if results are favorable or unfavorable in given situations. Connections to Previous Learning: Students should be able to determine mean, median, mode, and range. Connections to AP*: AP Statistics Topic: Distributions - Measures of Center, Shape, and Variability Materials: Student Activity pages Teacher Notes: This student activity helps students learn how to analyze mean, median, and mode when those pieces of information are given without the accompanying data. Begin by posing the following situation to the class. An employer states that The mean salary for all employees at my software company is $300,000. I ll hire you right out of high school, no college is needed. I ll also give you a 5% cost of living pay increase each year. Would you like to come to work for my company? After the class agrees to work for the employer (and most will), explain the actual salaries. She s the CEO and design engineer with an annual salary of $1,000,000. Her mother (husband, sister, best friend,...) is the CFO and salesperson making $740,000 annually. The company has a receptionist, secretary, maintenance engineer, and a packing specialist. Each earns $15,000 a year. Ask the class if the employer was dishonest with them. This question should lead to a discussion of how statistics can be used to distort or mislead. Questions 1 3 should be completed with the entire class. Students need to understand what it means for a set of data to be favorable to a situation. For example, it would be favorable to a student to use the mode to determine their final grade in a class if the grades were 100, 100, 100, 32, 20, 0, and 0. It would be unfavorable to use the mode if the student s grades were 100, 99, 98, 0, and 0. After students have completed question 3, discuss the answers with the entire class before assigning the remaining questions. Assign questions 4 6 to be completed individually, either as homework or as an assessment. *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 1

2 Analyzing Mean, Median, Mode, and Range For questions 1 3, consider the situations below. 1. A student has six test grades in algebra. Her average is the median of her test grades. Create a set of data that is favorable to the student, and then create a set of data that is unfavorable to the student. 2. At professional skating competitions, skaters are given scores ranging from 1 to 10 by ten judges. The high and the low scores are thrown out, and the skater then receives the mean of the other eight scores. Create a set of data that is favorable to the skater, and then create a set of data that is unfavorable to the skater. 3. You own a small company with eleven employees. You entice new employees by telling them that most of the people in your company make $100,000 a year. Create a set of data where using the mode is favorable to entry level employees. Create a set of data where using the mode is unfavorable to an entry level employee. Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 2

3 4. The July 2002 median resale price of a home in Dallas, Texas was $130,800. The per capita income was $32,406. In San Antonio, the median resale price of a home was $151,100 with a per capita income of $22,500. Out on the coast, the median resale price of a home in San Francisco, California was $571,000 while the per capita income was $40,357. Use these statistics to answer the following questions. a) Use one of these cities to explain the median resale price of a home. b) What is per capita income? c) Choose a city in which to live based on the data provided. State reasons for your selection based on the data. d) What generalizations can you make from this data? Do you think that they are always true? Why or Why not? Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 3

4 5. The Center for Disease Control gives statistics relating to the leading cause of death by age group. The number given is the percentage of total deaths. The following data gives the 6 leading causes of death for two age groups, and Use it to answer the questions below. Leading Causes of Death ages Leading Causes of Death ages Unintentional Injuries 39.6 Unintentional Injuries 48.5 Cancer 12.2 Homicide 15.2 Homicide 6 Suicide 11.7 Suicide 5.9 Cancer 5.4 Congenital Defects 5.4 Heart Disease 3.4 Heart Disease 3.9 Congenital Defects 1.6 a) The mode for each age group is unintentional injuries. Are mean and median useful ways to look at this data? Why or why not? b) In what other types of real world data is mode the most useful? c) While the six leading causes of death are the same in both age groups, there are some marked differences in the two groups. Name three differences and provide some possible educated guesses as to why these differences may exist. 6. Attached is a list of Major League Baseball s mean attendance by team, as reported on September 1, a) Round the mean home attendance to the nearest Make a dotplot, and use it to calculate the mean, median, and mode of the rounded data. Be sure to explain how you used the dotplot to find these values. b) What is the range of the rounded mean home attendance? c) As you look at your answers in part (a), are there any ways in which these values may be misleading? d) Based on this data, the dotplot, and the values of the mean, the median, and the mode, what predictions would you make about the future of Major League Baseball, in general or by specific team? Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 4

5 Major League Baseball Home Attendance September 1, 2002 Team Games Total Att. Mean Att. Anaheim 71 1,938,242 27,299 Arizona 68 2,720,175 40,002 Atlanta 66 2,210,247 33,485 Baltimore 64 2,158,432 33,725 Boston 67 2,208,708 32,965 Chicago Cubs 66 2,343,994 35,515 Chicago Sox 68 1,491,362 21,931 Cincinnati 70 1,610,776 23,011 Cleveland 71 2,326,839 32,772 Colorado 68 2,382,706 35,039 Detroit 68 1,294,332 19,034 Florida ,085 10,015 Houston 71 2,221,392 31,287 Kansas City 64 1,165,195 18,206 Los Angeles 68 2,633,777 38,732 Milwaukee 69 1,706,490 24,731 Minnesota 69 1,656,169 24,002 Montreal ,737 10,264 New York Mets 66 2,422,934 36,711 New York Yankees 63 2,830,794 44,933 Oakland 69 1,758,431 25,484 Philadelphia 67 1,423,322 21, 243 Pittsburgh 67 1,566,771 23,384 St. Louis 65 2,449,482 37,684 San Diego 69 1,955,701 28,343 San Francisco 65 2,619,341 40,297 Seattle 69 3,035,868 43,998 Tampa Bay ,678 13,271 Texas 69 2,072,956 30,042 Toronto 69 1,421,224 20,597 Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 5

6 Analyzing Mean, Median, Mode, and Range Answers: As analysis is a new activity for most students, these exercises effectiveness increases in proportion to class time spent discussing them. 1. One way for the median to be a favorable score would be for the top four scores to be close to the median and the other two much lower. One way for the median to be an unfavorable score would be for the bottom four scores to be close to the median and the other two scores to be much higher. 2. One way to be favorable to the skater would be for 9 scores to be close to the high score. The 10th score would be considerably lower than the other 9. One way to be unfavorable would be for 9 scores to be clustered near the bottom score and the tenth score to be very high in relationship to the others. 3. A mode of $100,000 would be favorable to new employees if the $100,000 was the bottom salary. It would be unfavorable if the mode came from experienced employees, and it was the top salary. 4. a) Median resale price means that there were as many homes that sold for less than $130,800 (Dallas) as there were that sold for more. b) Per capita income is the average income for each person based on the total income for the city divided by the population. c) For this question, the answers that students make should be justified by the data given. Accept any answer which fits this criteria. Example: Dallas is selected because the median resale of a home is the lowest of the three cities. Even though the per capita income is not the highest for the three cities, it is greater than San Antonio. d) Accept any generalization that students make and justify. Example: Higher home prices may be associated with higher incomes but not in all cases. In San Antonio there are several military bases that might affect the income/home price relationship. Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 6

7 5. a) Answers will vary. Since these data are categorical mean and median are not meaningful. Students should realize that the mode is the only meaningful value in this situation. b) Answers will vary widely. Some examples are medical research, car sales, fashion, etc. c) Sample answers: Unintentional injuries rise in because this age group is driving. Homicide rises in year olds because of life style choices. Suicide rises in year olds because this age is may be more stressful than early teens.. 6. a) X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 Mean (10,000 x , ,000 x ,000 x ,000 x ,000 x ,000 x ,000 x 2) 30. This gives 29,000. The median is 30,000, and can be found by counting 15 data points from the beginning of the line plot. The mode is 35,000 because this is the highest point on this line plot.. b) What is the range of mean attendance? The range is 45,000 10,000 = 35,000 c) The mode is a little misleading since it s so far from the mean and median. d) Answers will vary widely. Students may know that there are teams in difficulty and comment on the teams with the lowest mean attendance. Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 7