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Mean Absolute Deviation Glossary What Is Statistics? Teacher Notes 6th Grade Statistical Variability 2017­02­16 www.njctl.org Table of Contents Click on a topic to go to that section What is Statistics? Measures of Center Mean Median Mode Central Tendency Application Problems Measures of Variation Minimum/Maximum Teacher Notes Range Quartiles Outliers Mean Absolute Deviation Glossary What is Statistics? Teacher Notes Return to Table of Contents Data Everyday we encounter numbers. We use numbers throughout the day without even realizing it. What time did each of us wake up? How many minutes did we have to get ready? How far from school do we live? How many students will be in class today? How long is each class? And so on... Data This information comes at us in the form of numbers, and this information is called data. If we start trying to put together all of this data and make sense of it, we can get overwhelmed. Statistics is what helps us along. Statistics is the study of data. Data Conclusions MP7 ­ Look for and make use Lets put statistics to work! of structure. Two 6th grade math classes Ask: took the same end of unit test. • What do you notice about every data point within the Here are their results. same class? Class 1: • How do the two class' data Math Practice 89, 88, 92, 78, 85, 89, 95, 71, 100, 88, points relate to each other? 97, 82, 77, 98, 86, 82, 95, 88 • What patterns do you notice between the data points? Class 2: 100, 53, 92, 91, 97, 93, 92 What conclusions can you draw by looking at the numbers? Statistics When we just look at a list of numbers, it can take a lot of time to make sense of what we are dealing with. Let's use statistics to analyze the scores. A good way to compare them is to start with their averages. Statistics Example Class 1: 89, 88, 92, 78, 85, 89, 95, 71, 100, 88, 97, 82, 77, 98, 86, 82, 95, 88 Class 1 average: 88Click Class 2: 100, 53, 92, 91, 97, 93, 92 Class 2 average: 88 Click So if we spread out the total points scored in each class, to give each student the same score, each student in both classes will have an 88. So can we assume that both classes had similar scores on the test? Graphing Statistics Look at the graphs below to see the results in another way. MP3 ­ Construct viable arguments and critique the reasoning of others. For #3 Ask: • What mathematical evidence would support your answer of fair or not fair? • What were you considering when giving your answer? • What is the same and different Math Practice The red line on the graph marks the class average. about your answer vs another students? Discuss the following questions with your group. • Is there a "right" answer to this 1. How does the number of students in each class affect the scores? question or can both answers be 2. How do the student scores compare to the average in each class? correct? How can you tell? 3. Is the average a fair way to compare the two classes scores? Why or why not? Statistics The average is one way to analyze data, but it is not always the best way. Sometimes there are other factors to consider. Such as: • The number of values in a data set. • The difference between the highest and lowest value. (Range) • If there are any values that are far apart from the rest. (Outliers) • How the values compare to the average or mean. (Mean Deviation) Statistics Example Lets use these statistical tools to compare the results. • # of values in each data set • Class 1 had 18 students, while class 2 only had 7. So the one poor score from class 2 had a significant impact on the class average. Statistics Range • Range • Class 2 has a much larger range than class 1. This shows that the scores were more spread out / farther apart, than class 1. Range: 29 Range: 47 Statistics Outliers • Outliers • The scores have been ordered to make it easier to see any outliers. As you can see, in class 1 the scores flow nicely from least to greatest. However, in class 2, the score 53 is much lower than the rest of the scores. The large range in class 2 was due to the score of 53. If we eliminate that score, the range is only 9. Statistics Mean Deviation • Mean Deviation • Every score in class 1 was within 17 points of the mean. • In class 2, the score of 53 was 35 points from the mean. If that outlier was eliminated, the mean would be 94 and would more accurately reflect the majority of the students' scores. Statistics Example Discuss the following question with your group. MP7 ­ Look for and make use After looking more closely at the data, of structure. what conclusions can we draw? Ask: • What do you notice about every data point within the same class? Math Practice • How do the two class' data points relate to each other? • What patterns do you notice between the data points? 1 Statistics is the study of data. True False True Answer 2 Data is a collection of facts, values or measurements. True False True Answer 3 The purpose of statistics is to: (Select all that apply) A Collect numerical information B Organize numerical data C Analyze numerical information A, B, C, D Answer D Interpret numerical data Statistical Questions A statistical question is a question that creates a variety of answers. Depending on the question, the data gathered can be numerical (hours spent studying) or categorical (favorite food). Our focus is on questions that create numerical data. Statistical Question Non­Statistical How many cupcakes of How many cupcakes each type were made by were made by the bakery the bakery last week? last week? How many cupcakes did How many cupcakes did I each person in my class eat last week? eat last week? 4 Which statistical question best fits the data graphed below? A How many glasses of milk does each member of our class drink a day? A B How many letters are in the last name of each class Answer member? C How many text messages did each class member send last week? D How many minutes did each class member spend doing homework last night? 5 Which statistical question best fits the data graphed below? C Answer A How many movies did each member of the class watch last night? B How many books did each member of the class bring home last night? C How many pencils does each member of the class have in his or her desk? D How many feet high is each member of the class? 6 Which question is a statistical question? A How tall is the oak tree? B How much did the tree grow in one year? C What are the heights of the oak trees in the schoolyard? C D What is the difference in height between the oak Answer tree and the pine tree? From PARCC EOY sample test non­calculator #12 Statistical Toolbox This was just a small look at statistics in action. Throughout this unit, you will learn many statistical tools that you can use to analyze and make sense of data. Think of it as a statistical toolbox. It is not just important to know how to use the tool, but what jobs to use the tool for. For example, you wouldn't use a hammer to staple your papers together. Measures of Center Return to Table of Contents Activity Each of your group members will draw a color card. Each person will take all the tiles of their color from the bag. Discussion Questions MP2 ­ Reason abstractly and quantitatively. • How many tiles does your group have in total? Ask: • • How can you equally share all the tiles? How many would each What are the different cards Teacher Notes & Math Practice representing? member receive? (Ignore the color) • • What are the different colors Each member has a different number of tiles according to color. representing? Write out a list of how many tiles each person has from least to • How can you tell which information is greatest. Look at the two middle numbers. What number is in relevant to decide what the values are? between these two numbers? Follow­Up Discussion What is the significance of the number you found when you shared the tiles equally? This number is called the mean (or average). It tells us that if you evenly distributed the tiles, each person would receive that number. What is the significance of the number you found that shows two members with more tiles and two with less? This number is called the median. It is in the middle of the all the numbers. This number shows that no matter what each person received, half the group had more than that number and the other half had less. Vocabulary Measures of Center: • Mean ­ The sum of the data values divided by the number of items; average • Median ­ The middle data value when the values are written in numerical order • Mode ­ The data value that occurs the most often Finding the Mean To find the mean of the ages for the Apollo pilots given below, add their ages. Then divide by 7, the number of pilots. Answer Mean Practice Find the mean 10, 8, 9, 8, 5 Answer 7 Find the mean 20, 25, 25, 20, 25 23 Answer 8 Find the mean 14, 17, 9, 2, 4,10, 5, 3 Answer Median Practice Given the following set of data, what is the median? 10, 7, 9, 3, 5 7 What do we do when finding the median of an even set of numbers? MP6 ­ Attend to precision.
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