1. Analyzing Statistics Core Focus on Ratios, Rates & Statistics Lesson 4.6

2. Warm-Up For each data set find the five-number summary and create a box-and-whisker plot. 1.1, 4, 4, 7, 9, 10, 11, 17, 22 2.49, 54, 56, 42, 51, 58, 53 42 ~ 49 ~ 53 ~ 56 ~ 58 1 ~ 4 ~ 9 ~ 14 ~ 22

3. Analyzing Statistics Analyze how characteristics of a data set affect the measures of center. Lesson 4.6

4. Mr. Hinton was curious about the average number of letters in each of his students’ names. The names of eight of his students are listed below. Paul, Rob, Ana, Javon, Savannah, Ali, Juan, Alexandria Step 1Find the number of letters in each of the eight names. Step 2For each student, make a stack of cubes with the height matching the number of letters in their name. For example, Paul’s stack should be 4 cubes tall. Step 3Look at the stacks. What is the mode? Step 4Put the stacks in order from shortest to tallest. Find the median. Step 5Without adding any more cubes, “level” the stacks by redistributing blocks so that all eight stacks have the same height. How many cubes are in each stack? Step 6If each of Mr. Hinton’s eight students had the same number of letters in their name, how many letters would each person have? Which measure of center does this value represent? Step 7Which measure of center best represents the data? Explain your reasoning.

5. Step 8Copy the dot plot below onto a sheet of paper. The numbers (instead of dots) on the top of the dot plot represent how far each value is away from the mean. These numbers are called absolute deviations from the mean. For example, Paul is represented by a “1” since his name has 4 letters – 1 away from the mean of 5. Rob and Ana are each represented by a “2” since they are 2 away from the mean. Which student does the “3” represent? Step 9The dot plot is missing numbers representing Ali, Juan and Alexandria. Find where they should be located on the dot plot and put the correct absolute deviation number to represent each of them. Step 10Add the numbers to the left of the 0 on the dot plot. Then add the numbers to the right of the 0. What do you notice? What does this tell you about the mean as a “balancing” value for the data set? Each absolute deviation from the mean is written as a positive number.

6. Vocabulary Outlier An extreme value that varies greatly from the other values in a data set. Good to Know! When analyzing statistics, it is important to take the following into account: where the data came from, how much data was collected, how spread out the data is and any clusters that are present. Outliers can have a large impact on the mean, but little impact on the median or mode. Statistics from a larger data set are usually more reliable than statistics from a smaller data set.

7. Example 1 Find the mean, median and mode of each data set. Determine which measure of center best represents each data set. a. 4, 2, 2, 8, 6, 2, 8, 5, 8 Mean = = 5Median = 5Modes = 2 and 8 Since there is no mode, that would not be the best measure of center. The mean or median represents the data the best.

8. Example 1 Continued… Find the mean, median and mode of each data set. Determine which measure of center best represents each data set. b. 18, 19, 12, 17, 1, 19, 19 Mean = = 15Median = 18Mode = 19 The median best represents this data set. The mean is affected by the outlier (1) and the mode does not really represent the middle of the data set.

9. Find the mean, median and mode of each data set. Determine which measure of center best represents each data set. c. 10, 10, 7, 10, 10, 8, 10, 10, 10, 10 Mean = = 9.5Median = 10Mode = 10 All three measures of center represent this data set well. However, when there are many values that are the same, the mode is the best choice. The mode of 10 best represents this data set. Example 1 Continued…

10. Example 2 Jessie and Samantha compared how many movies they watched per month for a year. The two dot plots show each set of data: a. Compare the dot plots and describe the differences you see. Jessie’s dot plot is more spread out than Samantha’s. He has more low values, but also has some high numbers. Samantha’s dot plot is symmetrical.

11. Example 2 Continued… Jessie and Samantha compared how many movies they watched per month for a year. The two dot plots show each set of data: b. Find the measures of center for both Jessie and Samantha. How do they compare? Jessie: Mean = 3.5; Median = 2; Mode = 1 Samantha: Mean = 4.5; Median = 4.5; Mode = 4, 5 Jessie’s measures of center are lower than Samantha’s.

12. Example 2 Continued… Jessie and Samantha compared how many movies they watched per month for a year. The two dot plots show each set of data: c.Find the range for each data set. How do they compare? Jessie’s Range = 11 – 0 = 11 Samantha’s Range = 7 – 2 = 5 Jessie’s values are much more spread out.

13. Example 2 Continued… Jessie and Samantha compared how many movies they watched per month for a year. The two dot plots show each set of data: d.Which person is likely to watch more than 4 movies in a month? Which is more likely to watch more than 7 movies in a month? Samantha is more likely to watch more than 4 movies in a month. Samantha did this six out of the twelve months compared to Jessie, who watched more than 4 movies only three times. Jessie is more likely to watch more than 7 movies in a month. Samantha never watched more than 7 movies in a month.

14. Communication Prompt Which measure of center is typically used by teachers to calculate your grade in a class? If you got to choose which measure of center was used for calculating grades, which would you choose? Why?

15. Exit Problems 1. Find the mean, median and mode of the following data set. 17, 18, 13, 15, 3, 11, 20, 18, 14 2. Which measure of center best represents the data in #1? Explain. Since the outlier (3) affects the mean, the median probably best represents the data.