Measures of Central Tendency SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Interactive Word Wall, Marking the Text, Summarize/Paraphrase/Retell, Think/Pair/Share Matthew is a student reporter for the Culliver A.Tomas Middle School newspaper. His assignment for the next issue is to write an article about the fans who attend school basketball games. To begin, Matthew decides to investigate the ages of the people who attend the next game. On game day, he randomly selects 21 fans and asks their age. The numerical data he collects is listed in the chart below. Age of Fans 42 12 11 12 43 13 9 13 12 12 31 11 36 3 13 40 39 10 14 30 36 ACTIVITY 5.2 The mean, median, and mode of a numerical data set are called measures of central tendency. Since these measures are usually located in the middle of the data, they are chosen as representatives of the entire data set. 1. Recall that the mean is the sum of the data values divided by the number of data items. Determine the mean age of fans who attended the basketball game. MATH TERMS Measures of central tendency the mean, median, and mode describe what is near the middle of a distribution of numbers; for example, the average age of middle school students (mean). 2. The mode is the value or values that occur most often. What is the mode of the age of fans who attended the basketball game? 3. The median divides the data into two sets of data with the same number of elements in each set. Follow the steps below to find the median of the age of fans in the data set. a. Arrange all the data items in order from smallest to largest. b. Determine the number that divides the ordered list of data into two equal halves. If a data set has an odd number of data items, the median is the middle element of the ordered list. If the list has an even number of data items, the median is the average of the middle two data items in the ordered list. Unit 5 Probability and Statistics 265
ACTIVITY 5.2 Measures of Central Tendency ACADEMIC VOCABULARY Numerical data is quantitative. It is a collection of numbers. Categorical variables are qualitative. Gender and eye color are examples of categorical data. SUGGESTED LEARNING STRATEGIES: Debrief, Discussion Group, Quickwrite The ages of the cheerleaders on the middle school cheerleading squad are given below. Ages of Cheerleaders 11 11 12 12 12 13 13 13 13 14 14 4. Do these numbers represent numerical data or categorical variables? Explain. 5. Determine all three measures of central tendency for the ages of the cheerleaders on the squad. The cheerleading squad has decided to have a cheerleader mascot. She is the little sister of one of the cheerleaders on the squad, and she is 3 years old. 6. Determine the mean, median, and mode of the cheerleaders if the age of their mascot is included. MATH TERMS Outliers are individual data points that do not fit the overall pattern of the data set. 7. Describe how this outlier affects the mean, median, and mode of the data set. 8. Which single measure serves as the best representative of the data with the outlier? Explain. 9. Which of the measures of central tendency is the most affected by the inclusion of the outlier? Explain. 10. Which measure of central tendency: mean, median, or mode must be a value in the data set? Explain. 266 SpringBoard Mathematics with Meaning TM Level 3
Measures of Central Tendency ACTIVITY 5.2 SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Debrief, Discussion Group, Think/Pair/Share For Matthew s article, he has decided to include a box-and-whisker plot to represent the numerical data he collected regarding the age of the fans who attend the game. The data he collected is shown below. Age of Fans 42 12 11 12 43 13 9 13 12 12 31 11 36 3 13 40 39 10 14 30 36 MATH TERMS A box-and-whisker plot is a diagram that summarizes data by dividing it into four parts, each representing 25% of the data entries. Box-and-whisker plots have the following shape: 11. List the data in the table from least to greatest, and circle the median. The median of the lower half of the data is called the lower quartile, and the median of the upper half is called the upper quartile. 12. Find the lower and upper quartiles of your data list. 13. Graph the values of the lower quartile, median, and upper quartile as points above the number line below. 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 14. Use the number line in Question 13 for the following: a. Create a rectangle using the lower quartile and the median as the midpoint of the sides of the rectangle. A five-number summary consists of the five numbers needed to make a box-and-whisker plot: minimum, lower quartile, median, upper quartile, maximum. b. Now, create another rectangle using the upper quartile and the median as the midpoints of the sides of the rectangle. c. Finally, graph the values of the maximum and the minimum as points above the number line given above. Connect the lower quartile to the minimum with a line segment. Connect the upper quartile to the maximum with a line segment. Unit 5 Probability and Statistics 267
ACTIVITY 5.2 Measures of Central Tendency SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Debrief, Think/Pair/Share, Quickwrite 15. Use percentages to describe the distribution of the data represented by the box-and-whisker plot. 16. The interquartile range (IQR) is the difference between the lower and upper quartiles. a. Determine the interquartile range for the age of the fans. b. What percent of the data set is represented by the IQR? 17. Explain why you cannot determine the mean or the mode of the data set by looking at a box-and-whisker plot. 268 SpringBoard Mathematics with Meaning TM Level 3
Measures of Central Tendency ACTIVITY 5.2 SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Debrief, Think/Pair/Share, Quickwrite, Discussion Group The box-and-whisker plot below shows the average number of points scored per game by each player on the basketball team this season. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 18. What is the median number of points scored by a player? 19. What is the lower quartile? 20. What percent of the players have average scores that are below 10 points? 21. What is the range of the average number of points scored in a game as displayed in the box-and-whisker plot above? 22. Two box-and-whisker plots have the same median and equally long whiskers. If the box of one plot is longer than the box of the other, what can you say about the two data sets? ACADEMIC VOCABULARY As a measure of central tendency, the range of data is the difference between the minimum and maximum values of the data set. 9 10 11 12 13 14 15 16 17 18 19 20 21 Unit 5 Probability and Statistics 269
ACTIVITY 5.2 Measures of Central Tendency CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. 5. Two box-and-whisker plots have the same Show your work. median and lower and upper extremes. If 1. Kylie surveyed 9 of her friends about how the box of one plot is longer than the box many people lived in their households. of the other, what can you say about the Her results are shown in the table below. two data sets? Number of People Living in a Household 3 4 3 5 2 8 10 4 4 a. What is the mean of Kylie s data set? b. What is the median of Kylie s data set? c. What is the mode of Kylie s data set? d. Give the outlier for this data set. Describe how the outlier affects the mean, median and mode of the data. e. Which measure of central tendency is most representative of this data set? Explain. 2. The values in a data set are 10, 7, 9, 5, 13, 10, 7, 14, 8, and 11. Which measure of central tendency gives the answer 9.5 for this data set? 3. Which is not a measure of central tendency for the data set: 4, 6, 6, 7, 9, 10, 11, 11? a. 6 b. 8 c. 9 d. 11 4. Consider the data set: 14, 8, 13, 20, 15, 17, 1, 12, 18, 10 a. Make a box-and-whisker plot of the data set. b. What percent of the data is greater than 13.5? c. What is the lower quartile? d. What is the upper quartile? e. What is the interquartile range? 9 10 11 12 13 14 15 16 17 18 19 20 21 6. MATHEMATICAL REFLECTION Compare using measures of central tendency and the range to describe data to using a box-and-whisker plot to describe data. Discuss the strengths and weaknesses of both. 270 SpringBoard Mathematics with Meaning TM Level 3