# Students summarize a data set using box plots, the median, and the interquartile range. Students use box plots to compare two data distributions.

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1 Student Outcomes Students summarize a data set using box plots, the median, and the interquartile range. Students use box plots to compare two data distributions. Lesson Notes The activities in this lesson engage students in thinking about everything they learned in the last several lessons about summarizing and describing distributions of data using counts and position relative to the other data values. They consider displays of numerical data in plots on the number line box plots and dot plots. They use quantitative measures of center (median) and variability (interquartile range) and describe overall patterns in the data with reference to the context. If possible, students should have technology that allows them to construct the box plots so they can focus on what can be learned from analyzing the plots as a way to summarize the data, rather than on figuring out the scale and plotting points. If students can use technology, it would be important to transfer the data sets to the students in order to reduce the time spent entering the data and also the time spent tracking down entry errors. Classwork Exercise 1 (7 10 minutes): Supreme Court Chief Justices This example should take students a short time to do if they understood the concepts from the prior lessons. One error might be forgetting to order the data before finding the 5 number summary. If time permits, it is also an opportunity to make connections to social studies. Ask students if they know what cases are before the current Supreme Court, whether they think any data would be involved in those cases, and whether any of the analysis techniques might involve what they have been learning about statistics. Note that this discussion might extend the time necessary for the activity. Ask the following questions to students as they discuss the answers to the example questions in small groups: Why is it important to order the data before you find a median? The middle value, or median, is based on the order of the data. What other mistakes do you think most people make when thinking about a box plot? Allow students to indicate their own problems when they work with a box plot (for example, not counting correctly to locate the median, or not figuring out the median correctly based on whether there is an even number or odd number of data). Date: 4/3/14 177

2 Exercise 1: Supreme Court Chief Justices The Supreme Court is the highest court of law in the United States, and it makes decisions that affect the whole country. The Chief Justice is appointed to the Court and will be a justice the rest of his or her life unless he or she resigns or becomes ill. Some people think that this gives the Chief Justice a very long time to be on the Supreme Court. The first Chief Justice was appointed in The table shows the years in office for each of the Chief Justices of the Supreme Court as of 2013: Name Years Appointed in John Jay John Rutledge Oliver Ellsworth John Marshall Roger Brooke Taney Salmon P. Chase Morrison R. Waite Melville W. Fuller Edward D. White William Howard Taft Charles Evens Hughes Harlan Fiske Stone Fred M. Vinson Earl Warren Warren E. Burger William H. Rehnquist John G. Roberts Data Source: 1. Use the table to answer the following: a. Which Chief Justice served the longest term and which served the shortest term? How many years did each of these Chief Justices serve? John Marshall had the longest term, which was years. He served from to. John Rutledge served the shortest term, which was one year in. MP.1 b. What is the median number of years these Chief Justices have served on the Supreme Court? Explain how you found the median and what it means in terms of the data. First, you have to put the data in order. There are justices so the median would fall at the value ( years) counting from the top or from the bottom. The median is. Half of the justices served less than or equal to years as chief, and half served greater than or equal to years. MP.3 c. Make a box plot of the years the justices served. Describe the shape of the distribution and how the median and IQR relate to the box plot. The distribution seems to have more justices serving a small number of years (on the lower end). The range (max min) is years, from year to years. The IQR is:.., so about half of the Chief Justices had terms in the. year interval from. to. d. Is the median half way between the least and the most number of years served? Why or why not? The halfway point on the number line between the lowest number of years served,, and the highest number of years served,, is., but because the data are clustered in the lower end of the distribution, the median,, is to the left of (smaller than).. The middle of the interval from the smallest to the largest data value has no connection to the median. The median depends on how the data are spread out over the interval. Date: 4/3/14 178

5 Exercises 4 5: Rainfall 4. Data on average rainfall for each of the twelve months of the year were used to construct the two dot plots below. a. How many data points are in each dot plot? What does each data point represent? There are data points in St. Louis. There are also data points in San Francisco. Each data point represents the average monthly precipitation in inches. b. Make a conjecture about which city has the most variability in the average monthly amount of precipitation and how this would be reflected in the IQRs for the data from both cities. San Francisco has the most variability in the average monthly amount of precipitation. It has the largest IQR of the two cities. c. Based on the dot plots, what are the approximate values of the interquartile ranges (IQR) of the amount of average monthly precipitation in inches for each city? Use each IQR to compare the cities. For St. Louis, the IQR is... ; for San Francisco, the IQR is.... About the middle half of the precipitation amounts in St. Louis are within inch of each other. In San Francisco, the middle half of the precipitation amounts is within about inches of each other. d. In an earlier lesson, the average monthly temperatures were rounded to the nearest degree Fahrenheit. Would it make sense to round the amount of precipitation to the nearest inch? Why or why not? Answers will vary: It would not make sense because the numbers are pretty close together, or yes, it would make sense because you would still get a good idea of how the precipitation varied. If you rounded to the nearest inch, the IQR for San Francisco would be because three of the values round to and three round to. St. Louis would be because most of the values round to or. In both cases, that is pretty close to the IQR using the numbers to the hundredths place. 5. Use the data from Exercise 4 to answer the following. a. Make a box plot of the amount of precipitation for each city. Date: 4/3/14 181

7 Name Date Exit Ticket The number of pets per family for students in a sixth grade class is below: 1. Can you tell how many families have two pets? Explain why or why not. 2. Given the plot above, which of the following statements are true? If the statement is false, modify it to make the statement true. a. Every family had at least one pet. b. About one fourth of the families had six or more pets. c. Most of the families had three pets. d. Half of the families had five or fewer pets. e. Three fourths of the families had two or more pets. Date: 4/3/14 183

8 Exit Ticket Sample Solutions The number of pets per family for students in a sixth grade class is below: Number of Pets Can you tell how many families have two pets? Explain why or why not. You cannot tell from the box plot. You only know that the lower quartile (Q1) is pets. You do not know how many families are included in the data set. 2. Given the plot above, which of the following statements are true? If the statement is false, modify it to make the statement true. a. Every family had at least one pet. True. b. About one fourth of the families had six or more pets. True. c. Most of the families had three pets. False because you cannot determine the number of any specific data value. Revise to You cannot determine the number of pets most families had. d. Half of the families had five or fewer pets. False. Revise to More than half of the families had five or fewer pets. e. Three fourths of the families had two or more pets. True. Date: 4/3/14 184

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COMMON CORE Locker LESSON Reporting with Precision and Accuracy Common Core Standards The student is expected to: COMMON CORE N-QA3 Choose a level of accuracy appropriate to limitations on measurement