6th Grade Math. Statistical Variability

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2 Slide 2 / 125 6th Grade Math Statistical Variability

3 Slide 3 / 125 Table of Contents What is Statistics? Measures of Center Mean Median Mode Central Tendency Application Problems Measures of Variation Minimum/Maximum Range Quartiles Outliers Mean Absolute Deviation Glossary Common Core: 6.SP.1,2,3,5(cd) Click on a topic to go to that section.

4 Slide 4 / 125 Vocabulary words are identified with a dotted underline. Sometimes when you subtract the fractions, you find that you can't because the first numerator is smaller than the second! When this happens, you need to regroup from the whole number. (Click on the dotted underline.) How many thirds are in 1 whole? How many fifths are in 1 whole? How many ninths are in 1 whole? The underline is linked to the glossary at the end of the Notebook. It can also be printed for a word wall.

5 Slide 5 / 125 The charts have 4 parts. 1 Factor Vocab Word 2 Its meaning A whole number that can divide into another number with no remainder Examples/ Counterexamples (As it is used in the lesson.) 5 R is a factor of 15 3 A whole number that multiplies with another number to make a third number. 3 x 5 = 15 3 is not a factor of 16 Back to 3 and 5 are factors of 15 4 Instruction Link to return to the instructional page.

6 What is Statistics? Teacher Note Slide 6 / 125 Return to Table of Contents

7 Slide 7 / 125 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...

8 Slide 8 / 125 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.

9 Slide 9 / 125 Lets put statistics to work! Two 6th grade math classes took the same end of unit test. Here are their results. Class 1: 89, 88, 92, 78, 85, 89, 95, 71, 100, 88, 97, 82, 77, 98, 86, 82, 95, 88 Class 2: 100, 53, 92, 91, 97, 93, 92 What conclusions can you draw by looking at the numbers?

10 Slide 10 / 125 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 two classes scores. A good way to compare them is to start with their averages.

11 Slide 11 / 125 Class 1: 89, 88, 92, 78, 85, 89, 95, 71, 100, 88, 97, 82, 77, 98, 86, 82, 95, 88 Class 2: 100, 53, 92, 91, 97, 93, 92 Class 1 average: 88 Class 2 average: 88 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?

12 Slide 12 / 125 Lets graph the results and have another look. The red line on the graph marks the class average. Discuss the following questions with your group. 1. How does the number of students in each class affect the scores? 2. How do the student scores compare to the average in each class? 3. Is the average a fair way to compare the two classes scores? Why or why not?

13 Slide 13 / 125 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)

14 Slide 14 / 125 Lets use these statistical tools to compare the classes 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.

15 Slide 15 / 125 Range: 47 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

16 Slide 16 / 125 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.

17 Slide 17 / 125 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.

18 Slide 18 / 125 Discuss the following question with your group. After looking more closely at the data, what conclusions can we draw?

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22 Slide 22 / 125 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 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.

23 Slide 23 / 125 Measures of Center Return to Table of Contents

24 Slide 24 / 125 Activity Check Here Before You Start 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 How many tiles does your group have in total? How can you equally share all the tiles? How many would each member receive? (Ignore the color) Each member has a different number of tiles according to color. Write out a list of how many tiles each person has from least to greatest. Look at the two middle numbers. What number is in between these two numbers?

25 Slide 25 / 125 Follow-Up 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.

26 Slide 26 / 125 Measures of Center Vocabulary: 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

27 Slide 27 / 125 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. Apollo Mission Pilot's age Click to reveal answer Mean = = 266 = 38 The mean of the Apollo pilots' ages is 38 years.

28 Slide 28 / 125 Find the mean 10, 8, 9, 8, 5 8

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31 Slide 31 / 125 Given the following set of data, what is the median? 10, 8, 9, 8, 5 8 What do we do when finding the median of an even set of numbers?

32 Slide 32 / 125 When finding the median of an even set of numbers, you must take the mean of the two middle numbers. Find the median 12, 14, 8, 4, 9, 3 8.5

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39 Slide 39 / 125 Find the mode 10, 8, 9, 8, 5 8 Find the mode 1, 2, 3, 4, 5 No mode What can be added to the set of data above, so that there are two modes? Three modes?

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44 Slide 44 / 125 Central Tendency Application Problems Return to Table of Contents

45 Slide 45 / 125 Which Measure of Center to Use? Sherman and his friends had a paper airplane competition. The distances each plane traveled were 13 ft, 2 ft, 19 ft, 18 ft and 16 ft. Should Sherman use the mean, median or mode to describe their results? Find the mean, median and mode and compare them.

46 Slide 46 / ft, 2 ft, 19 ft, 18 ft and 16 ft Click for answer ft Mean: 13.6 answer Median: Click 16forft Click for answer Mode: no mode Which measure of center best describes the data? for answer The mean isclick closest to most of the values, so it best describes the data. The mean is less than 4 out of the 5 values, and there was no mode.

47 Slide 47 / 125 Using Measures of Center to Describe Data Foodie grocery store sells several juice brands in 12 oz bottles. Which measure of center best describes the cost for a 12 oz bottle of juice? Brand A $1.25 Brand D $0.99 Brand B $0.95 Brand E $1.99 Brand C $1.09 Brand F $0.99

48 Slide 48 / 125 In order to see how the measures of center compare to the data, the data needs to be in order from least to greatest. The data has been graphed to help you see the comparisons. Mean: $1.21 The Mean is greater than most of the data. Median: $1.04 Half of the data is greater than the median, and half of the data is less than the median. Mode: $0.99 The mode reflects the lower 4 values very well, but is much lower than the top two values.

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53 Slide 53 / 125 Jae bought gifts that cost $24, $26, $20 and $18. She has one more gift to buy and wants her mean cost to be $24. What should she spend for the last gift? 3 Methods : Method 1: Guess & Check Try $ = Try a greater price, such as $ = 24 5 The answer is $32.

54 Slide 54 / 125 Jae bought gifts that cost $24, $26, $20 and $18. She has one more gift to buy and wants her mean cost to be $24. What should she spend for the last gift? Method 2: Work Backwards In order to have a mean of $24 on 5 gifts, the sum of all 5 gifts must be $24 x 5 = $120. The sum of the first four gifts is $88. So the last gift should cost $120 - $88 = $ x 5 = = 32

55 Slide 55 / 125 Jae bought gifts that cost $24, $26, $20 and $18. She has one more gift to buy and wants her mean cost to be $24. What should she spend for the last gift? Method 3: Write an Equation Let x = Jae's cost for the last gift x = x = x = 120 (multiplied both sides by 5) x = 32 (subtracted 88 from both sides)

56 Slide 56 / 125 Your test scores are 87, 86, 89, and 88. You have one more test in the marking period. Pull You want your average to be a 90. What score must you get on your last test?

57 Slide 57 / Your test grades are 72, 83, 78, 85, and 90. You have one Pull more test and want an average of an 82. What must you earn on your next test?

58 Slide 58 / Yes No Pull Your test grades are 72, 83, 78, 85, and 90. You have one more test and want an average of an 85. Your friend figures out what you need on your next test and tells you that there is "NO way for you to wind up with an 85 average. Is your friend correct? Why or why not?

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60 Slide 60 / 125 Consider the data set: 55, 55, 57, 58, 60, 63 The mean is 58 the median is 57.5 and the mode is 55 How would the mean change: if x was less than the mean? if x equals the mean? if x was greater than the mean? Pull What would happen if a value x was added to the set?

61 Let's further consider the data set: 55, 55, 57, 58, 60, 63 The mean is 58 the median is 57.5 and the mode is 55 What would happen if a value, "x", was added to the set? How would the median change: if x was less than 57? if x was between 57 and 58? if x was greater than 58? Pull Slide 61 / 125

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63 Slide 63 / 125 What would happen if a value, "x", was added to the set? How would the mode change: if x was 55? if x was another number in the list other than 55? if x was a number not in the list? Pull Consider the data set: 55, 55, 57, 58, 60, 63 The mean is 58 the median is 57.5 and the mode is 55

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65 Slide 65 / 125 Measures of Variation Return to Table of Contents

66 Slide 66 / 125 Measures of Variation Vocabulary: Minimum- The smallest value in a set of data Maximum- The largest value in a set of data Range - The difference between the greatest data value and the least data value Quartiles - are the values that divide the data in four equal parts. Lower (1st) Quartile (Q1) - The median of the lower half of the data Upper (3rd) Quartile (Q3) - The median of the upper half of the data. Interquartile Range - The difference of the upper quartile and the lower quartile. (Q3 - Q1) Outliers - Numbers that are significantly larger or much smaller than the rest of the data

67 Slide 67 / 125 Minimum and Maximum 14, 17, 9, 2, 4, 10, 5 What is the minimum in this set of data? 2 What is the maximum in this set of data? 17

68 Slide 68 / 125 Given a maximum of 17 and a minimum of 2, what is the range? 15

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73 Slide 73 / 125 Quartiles There are three quartiles for every set of data. Lower Half Upper Half 10, 14, 17, 18, 21, 25, 27, 28 Q1 Q2 Q3 The lower quartile (Q1) is the median of the lower half of the data which is The upper quartile (Q3) is the median of the upper half of the data which is 26. The second quartile (Q2) is the median of the entire data set which is The interquartile range is Q3 - Q1 which is equal to 10.5.

74 Slide 74 / 125 To find the first and third quartile of an odd set of data, ignore the median (Q2) when analyzing the lower and upper half of the data. 2, 5, 8, 7, 2, 1, 3 First order the numbers and find the median (Q2). 1, 2, 2, 3, 5, 7, 8 What is the lower quartile, upper quartile, and interquartile range? First Quartile: Median: Third Quartile: Interquartile Range: =5 Click to Reveal

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82 Slide 82 / 125 When the outlier is not obvious, a general rule of thumb is that the outlier falls more than 1.5 times the interquartile range below Q1 or above Q3. Consider the set 1, 5, 6, 9, 17. Q1: 3 Q2: 6 Q3: 13 IQR: x IQR = 1.5 x 10 = 15 Q1-15 = 3-15 = -12 Q = = 28 In order to be an outlier, a number should be smaller than -12 or larger than 28.

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93 Slide 93 / 125 Find the mean, median, range, quartiles, interquartile range and outliers for the data below. High Temperatures for Halloween Year Temperature

94 Slide 94 / 125 High Temperatures for Halloween Mean 740/8 = 92.5 Median 92 Range = 9 Lower Quartile 90 Upper Quartile 95.5 Interquartile Range 5.5 Outliers None Year Temperature Pull High Temperatures for Halloween

95 Slide 95 / 125 Find the mean, median, range, quartiles, interquartile range and outliers for the data. Candy Butterscotch Discs Candy Corn Caramels Gum Dark Chocolate Bar Gummy Bears Jelly Beans Licorice Twists Lollipop Milk Chocolate Almond Milk Chocolate Calories

96 Slide 96 / 125 Calories from Candy Mean 1500/11 = Median 160 Range = 200 Lower Quartile 60 Upper Quartile 200 Interquartile Range 140 Outliers 10 Candy Butterscotch Discs Candy Corn Caramels Gum Dark Chocolate Bar Gummy Bears Jelly Beans Licorice Twists Lollipop Milk Chocolate Almond Milk Chocolate Calories

97 Slide 97 / 125 Mean Absolute Deviation Return to Table of Contents

98 Slide 98 / 125 Activity Find the mean of the data. What is the difference between the data value 52 and the mean? Which value is farthest from the mean? Overall, are the data values close to the mean or far away from the mean? Explain. Phone Usage (Minutes) Pull The table below shows the number of minutes eight friends have talked on their cell phones in one day. In your groups, answer the following questions.

99 Slide 99 / 125 The mean absolute deviation of a set of data is the average distance between each data value and the mean. Steps 1. Find the mean. 2. Find the distance between each data value and the mean. That is, find the absolute value of the difference between each data value and the mean. 3. Find the average of those differences. *HINT: Use a table to help you organize your data.

100 Slide 100 / 125 Let's continue with the "Phone Usage" example. Step 1 - We already found the mean of the data is 56. Step 2 - Now create a table to find the differences. Data Value Absolute Value of the Difference Data Value - Mean

101 Slide 101 / 125 Step 3 - Find the average of those differences = The mean absolute deviation is The average distance between each data value and the mean is 3.75 minutes. This means that the number of minutes each friend talks on the phone varies 3.75 minutes from the mean of 56 minutes.

102 Slide 102 / 125 The table shows the maximum speeds of eight roller coasters at Eight Flags Super Adventure. Find the mean absolute deviation of the set of data. Describe what the mean absolute deviation represents. Maximum Speeds of Roller Coasters (mph) Pull Try This!

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107 Slide 107 / 125 Glossary Return to Table of Contents

108 Slide 108 / 125 Analyze To examine the detail or structure of something, in order to provide an explanation or interpretation of it. what why how when Back to Instruction

109 Slide 109 / 125 Data A collection of facts, such as values or measurements. Back to Instruction

110 Slide 110 / 125 Interquartile Range The difference between the upper and the lower quartile in a set of data. Q1 Q2 Q3 25% 25% 25% 25% = Q3 - Q1 Q1 Q2 Q3 Q1 Q2 Q3 1,3,3,4,5,6,6,7,8,8 = Q3 - Q Back to Instruction

111 Slide 111 / 125 Lower (1st) Quartile Range The median of the lower half of a set of data. Q1 Q1 Q2 Q3 25% 25% 25% 25% } Q1 Q2 Q3 1,3,3,4,5,6,6,7,8,8 } Median Q2 Q3 Median Back to Instruction

112 Slide 112 / 125 Maximum The highest or greatest amount or value. Maximum includes the highest value. It means or less. Back to Instruction

113 Slide 113 / 125 Mean The value/amount of each item when the total is distributed across each item equally =9 =9 3=3 Back to Instruction

114 Slide 114 / 125 Mean Absolute Deviation The average distance between each data value and the mean of a set of data. Find the 1. mean 2,2,3,4,4 15 5=3 2. Subtract the mean from each data point 3-2=1 3-2=1 3-3=0 4-3=1 4-3=1 Find the mean of the 3. differences =4 5=.8 Back to Instruction

115 Slide 115 / 125 Measures of Center Statistics used to describe the "center" of the distribution of data. (mean, median, mode) mode mean = 4 median Back to Instruction

116 Slide 116 / 125 Median The middle value in a set of ordered numbers. 1, 2, 3, 4, 5 Median 1, 2, 3, 4 Median is = 10 10/4 = 2.5 Back to Instruction

117 Slide 117 / 125 Minimum The lowest or least amount or value. You must drive at least 40 mph. Minimum includes the smallest possible value. You must be at least this tall to ride. It means or more. Back to Instruction

118 Slide 118 / 125 Mode The number that occurs most often in a set of numbers. 2, 4, 6, 3, 4 The mode is 4. Back to Instruction

119 Slide 119 / 125 Outlier A value in a set of data that is much lower or much higher than the other values. 1,3,5,5,6,12 outlier outlier Back to Instruction

120 Slide 120 / 125 Quartile One of three values that divide a set of data into four quarters. Q1 Q1 Q2 Q3 25% 25% 25% 25% Q2 Q3 Q1 Q2 Q3 1,3,3,4,5,6,6,7,8, Back to Instruction

121 Slide 121 / 125 Range The difference between the lowest and the highest value in a set of data. 2, 4, 7, = 10 The range is 10. Back to Instruction

122 Slide 122 / 125 Relatively To evaluate something based on how it compares to something else. relatively small relatively large brother mother cousin uncle Back to Instruction

123 Slide 123 / 125 Upper (3rd) Quartile Range The median of the upper half of a set of data. Q1 Q1 Q2 Q3 25% Q1 Q2 Q3 25% 25% 25% } 1,3,3,4,5,6,6,7,8,8 } Median Median Q2 Q Back to Instruction

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