Data Analysis and Probability Measure of Central Tendency (Notes) Name I can Date Essential Question(s): Mean Key Concepts Notes Definition When is it most appropriate to use? Median Definition When is it most appropriate to use? Mode Definition When is it most appropriate to use? Range Definition Outlier Definition Steps for identifying an outlier
Example #1 The following numbers represent Mr. Abraham s bowling scores: 167, 148, 180, 124, 254, 200, 175, 180, and 220. a) Find the mean of the data. b) Find the median of the data. c) Find the mode of the data. d) Find the range of the data. Example #2 During the summer, Mr. Rice is an avid golfer. The following numbers below represent his 18-hole scores: 77, 85, 78, 90, 72, 85, 79, 83, 74, and 86 a) Find the mean of the data. b) Find the median of the data. c) Find the mode of the data. d) Find the range of the data. Example #3 Weekly salaries of the employees who work at a local video store are shown in the table below: a) How would I write out this data? b) Find the mean of the data. c) Find the median of the data. d) Find the mode of the data. e) Find the range of the data.
Now that we know WHAT mean, median, and mode are, how do we know WHEN is the best time to use them? Example #4 a) The grades on a geometry quiz that were given to Mrs. Grant s were 20, 19, 18, 18, 17, 15, 13, 13, 12, 10, and 4. Which measure of central tendency would best describe the data? Why? b) The following numbers represent some of the weights (in pounds) of the players on the Maple Heights football team: 195, 210, 230, 184, 345, 238, 177, 200, and 213. Which measure of central tendency would best describe the data? Why? c) The following table contains math scores of two teams in a high school math contest. Calculate the mean and median for each team. Then, determine which team scored better in the contest by comparing these measures.
Assorted Measures of Central Tendency Examples Example #5 1. A set of data contains 10 negative and 4 positive numbers. Which one of these statements must be true? Explain why each statement is either false (by giving a counterexample) or true. a) The mean is a negative number. b) The median is a negative number. c) The mode is a negative number. d) The range is a negative number. Example #6 Deven wants to get at least a 70% in his English class. His scores on his first five exams are 74%, 68%, 65%, 71%, and 77%. What is the lowest score that Deven can get on his sixth test to attain his goal of 70%? Example #7 The data table to the right represents the number of points that Kevin Durant had scored in the first ten games of the season. However, the team statistician noticed that there is a typo in the data. Kevin Durant actually only scored 12 points in the 6 th game. How much does this affect the mean?
Example #8 The average salary for all department store workers in a certain area is 4255. The weekly salaries of the 7 employees in the Acme department store are given in the table to the right. a) Determine the measures of central tendency of the 7 salaries. b) Specify which of these measures of central tendency the management could use to represent the salaries in an argument against pay increases. Explain your answer. c) Specify which of these measures of center the labor union could use to represent the salaries in an argument for pay increases. Summary, Reflection, & Analysis
Data Analysis and Probability Measures of Central Tendency (Exercise) Name Date 1. The following numbers represent the number of wins the Detroit Red Wings have over the past several seasons: 51, 47, 46, 40, 56, 50, 48, 46, 53, 57, and 51. a. Find the mean of the data. b. Find the median of the data. c. Find the mode of the data. d. Find the range of the data. 2. In 1993, the median age of brides in Ohio was 27.0 years and the median age of grooms was 28.9 years. Based on this information, which of the following statements must be true? Explain why each statement is either false (by giving a counter-example) or true. a. Most brides were 27 years old. b. The youngest groom was 28.9 years old. c. Half of the grooms were 28.9 years old or older. d. All the grooms were older than the brides were. 3. A video game company reports the sales of video games for the first 11 months. The company s goal is to sell an average of 5 million games per month. What is the minimum number of video games the company must sell in the 12 th month to meet this goal?
4. The noon temperatures are recorded for 7 consecutive days in Cleveland, Ohio. The sum of the 7 temperatures is positive. Which statement about these 7 temperatures must be true? Explain why each statement is either false (by giving a counter-example) or true. a. The mean of the temperatures is positive. b. The mode of the temperatures is positive. c. The median of the temperatures is positive. d. The minimum of the temperatures is positive. 5. The following table represents the shoe sizes of a 4 th grade class. a. Write out all the data from the chart. b. Find the mean of the data. c. Find the median of the data. d. Find the mode of the data. e. Find the range of the data. 6. To test the effect of a new medication on reaction time, two groups of 5 adults were asked to step on a pedal as soon as they saw a flashing light on a video screen. One group received the medication, while the other group did not receive the medication. The reaction times, in fractions of a second, are show for each of the two groups. Groups receiving medication: 0.55, 0.65, 0.60, 0.50, 0.70 Groups not receiving medication: 0.20, 0.30, 0.25, 0.20, 0.30 How much longer is the mean reaction time of the group receiving the medication than the group not receiving the medication?
7. The bar graph shows the number of customers in Kendre s restaurant each day last week. What is the approximate mean number of customers per day in Kendre s restaurant last week? Explain your answer and show your work. 8. In Steve s Math Transitions II class, each test is worth 100 points. His scores on his first four tests are 46, 85, 82, and 90. Which of the measures of center would be best to describe this data? Explain why. 9. Tonya and her friends earned extra money by mowing lawns around their neighborhood. Tonya created a table to show their earnings. However, Pedro noticed that Tonya made a mistake. His earnings were $24, not $8. How does this change affect the mean of the group s earnings? Does it increase or decrease the group mean and by how much? 10. The varsity track coach recorded the height, in inches, of each senior on the team and listed the results: 62, 76, 57, 69, 60, 73, 56, 75, 71, 60, 58, and 51. a. Find the mean of the data. b. Find the median of the data. c. Find the mode of the data. d. Find the range of the data.
11. The line plot below shows the ages of swim team members for two local schools. Each team currently has 8 swimmers. a. Find the mean of the data. b. Find the median of the data. c. Find the mode of the data. d. Find the range of the data. e. Let s say that we take away two 17-year olds with a 16-year old and an 18-year old. Which changes more, the mean or the median, and by how much did it change? Explain your answer. 12. Robert s test scores are 80, 90, 92, and 88. He just found out that he needs to have an average of a 90 in order to make the honor roll. What is the lowest score that he can get on his next test in order to make the honor roll? 13. A data set contains 7 single-digit numbers and 2 double-digit numbers. Which statement must be true? Explain why each statement is either false (by giving a counter-example) or true. a. The mode is a single-digit number. b. The mean is a single-digit number. c. The range is a single-digit number. d. The median is a single-digit number.
14. Cetorré recorded the amount she spent each day for lunch this week. Which of the following statements is true? Explain why each statement is either false (by giving a counter-example) or true. a. The mean is less than the median. b. The mean and mode are equal in value. c. The median and mode are equal in value. d. There is no mode. 15. The line plot shows the heights of players on a basketball team. Which measure of central tendency (mean, median, mode, and range) does not change when a new player who is 72 inches tall joins the team? Explain your answer. 16. The attendance for Mr. Rice s 1 st block class last week was as follows: 28, 27, 29, 26, and 30. Which measure of center is the most appropriate to describe the data? What is its value?
17. Justin s test scores are 88, 90, 94, and 98. He just found out that he needs to have an average of a 95 in order to make the dean s list. What is the lowest score that he can get on his next test in order to achieve his goal? 18. The following are scores from the last six games of the Cleveland Cavaliers and the Detroit Pistons. Compare and Contrast each of the four types of measures of center and using the data, make a convincing agreement on which team would win if they played each other. Cavaliers: 98, 93, 118, 93, 91, and 99. Pistons: 87, 87, 103, 101, 96, and 109