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Of Quantities MODULE 5 DESCRIBING VARIABILITY OF QUANTITIES The lessons in this module build on the data displays that you have used in elementary school, namely line plots, bar graphs, and circle graphs. You will be introduced to the fi eld of statistics, the study of data, and the statistical problem-solving process. You will calculate numerical summaries to describe a data set. You will also learn what separates mathematical and statistical reasoning—the presence of variability. Topic 1 The Statistical Process M5-3 Topic 2 Numerical Summaries of Data M5-67 CC01_SE_M05_INTRO.indd01_SE_M05_INTRO.indd 1 11/12/19/12/19 88:17:17 PMPM CC01_SE_M05_INTRO.indd01_SE_M05_INTRO.indd 2 11/12/19/12/19 88:17:17 PMPM TOPIC 1 The Statistical Process On average, one out of every 25 sheep has black wool. A quick way to estimate the size of a fl ock of sheep is to count the black sheep and multiply by 25. Lesson 1 What's Your Question? Understanding the Statistical Process . M5-7 Lesson 2 Get in Shape Analyzing Numerical Data Displays . M5-25 Lesson 3 Skyscrapers Using Histograms to Display Data . M5-47 CC01_SE_M05_T01_INTRO.indd01_SE_M05_T01_INTRO.indd 3 11/12/19/12/19 88:17:17 PMPM CC01_SE_M05_T01_INTRO.indd01_SE_M05_T01_INTRO.indd 4 11/12/19/12/19 88:17:17 PMPM Carnegie Learning Family Guide Course 1 Module 5: Describing Variability of Quantities TOPIC 1: THE STATISTICAL Where have we been? PROCESS In grade 1, students were expected to In this topic, students are introduced to organize, represent, and interpret data with the statistical problem-solving process: up to three categories. In grades 2 and 3, formulate questions, collect data, students created picture graphs and bar analyze data, and interpret the results. graphs of categorical data. In grades 4 and Students will use this process throughout 5, students made line plots to display data their studies of statistics, increasing the with fractions. And in grade 4, students complexity of each step of the process developed a conceptual understanding of as they develop their statistical literacy. angles and angle measurement, allowing Students use bar graphs and pie charts them to create pie charts. to analyze and interpret survey data, the fi nal steps of the statistical process. Where are we going? As students learn about and analyze In grade 7, students will use the data dot plots, stem-and-leaf plots, and displays learned in this topic to compare histograms, they practice using the four data distributions. They will use statistical steps of the statistical process. problem solving to investigate and draw inferences about populations. In grade 8, students will move into comparing two variables with data. Histograms A histogram is a graphical way to display Number of Floors in the Tallest Buildings in the Twin Cities quantitative or numerical data using vertical 30 25 bars. The width of a bar in a histogram 20 15 represents an interval of data and is often 10 5 referred to as a bin. The height of the bar Number of Buildings 0 1020 30 40 50 60 indicates the frequency, or the number of Number of Floors data values included in any given bin. Bins are represented by intervals of data instead of showing individual data values. TOPIC 1: Family Guide • M5-5 CC01_SE_FG_M05_T01.indd01_SE_FG_M05_T01.indd 5 11/12/19/12/19 88:13:13 PMPM Myth: Faster 5 smarter. In most cases, speed has nothing to do with how smart you are. Why is that? Because it largely depends on how familiar you are with a topic. For example, a bike mechanic can look at a bike for about 8 seconds and tell you details about the bike that you probably didn’t even notice (e.g., the front tire is on backwards). Is that person smart? Sure! Suppose, instead, you show the same bike mechanic a car. Will they be able to report the same amount of detail as they did for the bike? No! It’s easy to confuse speed with understanding. Speed is associated with the memorization of facts. Understanding, on the other hand, is a methodical, time-consuming process. Understanding is the result of asking lots of questions and seeing connections between different ideas. Many mathematicians who won the Fields Medal (i.e., the Nobel prize for mathematics) describe themselves as extremely slow thinkers. That’s because mathematical thinking requires understanding over memorization. #mathmythbusted Talking Points Key Terms You can support your student’s learning by categorical data approaching problems slowly. Students may Categorical data are data for which each observe a classmate learning things very piece of data fi ts into exactly one of several quickly, and they can easily come to believe different groups or categories. Categorical that mathematics is about getting the right data are also called qualitative data. answer as quickly as possible. When this quantitative data doesn’t happen for them, future encounters Quantitative data are data for which each with math can raise anxiety, making problem piece of data can be placed on a numerical solving more diffi cult, and reinforcing a scale. Quantitative data are also called student’s view of himself or herself as numerical data. “not good at math.” Slowing down is not frequency the ultimate cure for math diffi culties. But it’s a good fi rst step for children who are Frequency is the number of times an item struggling. You can reinforce the view that or number occurs in a data set. learning with understanding takes time, and mode that slow, deliberate work is the rule, not the The mode is the value or values that exception. occur most frequently in a data set. M5-6 • TOPIC 1: The Statistical Process CC01_SE_FG_M05_T01.indd01_SE_FG_M05_T01.indd 6 11/12/19/12/19 88:13:13 PMPM What’s Your 1 Question? Understanding the Statistical Process WARM UP LEARNING GOALS Ms. White asked the 25 sixth • Recognize and design statistical questions and graders in her class, “How many anticipate variability in data related to the question. pets do typical 6th graders in our • Differentiate between surveys, observational class have?” Ms. White summarized studies, and experiments. the responses in the table. • Describe the four stages of the statistical process. • Discuss the different types of data that can be Number of Number of collected, displayed, and analyzed. Students Pets • Analyze and interpret bar graphs and circle graphs. 80 10 1 KEY TERMS 42 • variability • survey 23 • data • observational study 138 • statistical question • experiment • statistical process • bar graph 1. About how many pets does • categorical data • circle graph each sixth grader in Ms. White’s • quantitative data • frequency class own? How did you make • population • mode your decision? • sample You have been solving mathematical problems throughout this course. Now, you are going to study statistical problems. How are mathematics and statistics similar and different? LESSON 1: What’s Your Question? • M5-7 CC01_SE_M05_T01_L01.indd01_SE_M05_T01_L01.indd 7 11/12/19/12/19 88:17:17 PMPM Getting Started In statistics, Statistical or Not, That Is the Question variability means Have you ever wondered, “How much money do professional athletes that the value of make?” Or, “How long are the books assigned to sixth graders?” If the attribute being so, you have asked a statistical question. If you have sought out the studied can change answer to your question, you have engaged in the statistical process. from one person or thing to another. Cut out the questions provided at the end of the lesson. Read each question and sort them into as many groups as you would like. Data are categories, There must be more than one group and there must be at least two numbers, or questions per group. observations 1. Record your groups and the questions in each group. gathered in response to a statistical question. In this module, you will begin your formal study of statistics and the statistical process. Statistics is a problem-solving process, also called an investigative process, because the heart of statistics is about determining a possible answer to a question that has variability. Keep your Statistical problem solving begins with a statistical question. questions. A statistical question is a question that anticipates an answer based You will need on data that vary. them later in the lesson. 2. Which questions from your sort are statistical questions? Explain how you would expect the answers to those questions to vary. M5-8 • TOPIC 1: The Statistical Process CC01_SE_M05_T01_L01.indd01_SE_M05_T01_L01.indd 8 11/12/19/12/19 88:17:17 PMPM ACTIVITY Formulating Statistical NOTES 1.1 Questions The statistical process has four components: • Formulating a statistical question. • Collecting appropriate data. • Analyzing the data graphically and numerically. • Interpreting the results of the analysis. This lesson provides an overview of the statistical process, but you will continue to use the process throughout your study of statistics. Statistics is about posing interesting questions that you want to answer about varying attributes. Analyze the questions posed by Bianca and Rajan. Bianca Rajan “What clubs am I in?” “What clubs do my “How many students classmates belong to?” are in the Chess “How many members Club?” do the clubs at my school have?” 1. Explain why Bianca’s questions are not statistical questions but Rajan’s are. 2. What kinds of answers do you expect from Rajan’s questions? LESSON 1: What’s Your Question? • M5-9 CC01_SE_M05_T01_L01.indd01_SE_M05_T01_L01.indd 9 11/12/19/12/19 88:17:17 PMPM Answering a statistical question requires collecting variable data. Categorical data, or You will learn about two types of data: categorical data and qualitative data, are quantitative data. data for which each piece of data fits into 3.
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