13 What number can be added to the data set below so that the median is 16.5? 17, 9, 4, 16, 29, Answer 39
Mean, Median, and Data What do the mean and median tell us about the data ? Mr. Smith organized a scavenger hunt for his students. They had to find all the buried "treasure". The following data shows how many The mean is 6 and the median is 7. coins each student found. Teacher Notes The mean tells us that if the data 10, 7, 3, 8, 2 were evenly distributed, each student would have 6 coins. Find the mean and median of the data. What does the mean and median tell us about the data? The median tells us that half the class has more than 7 coins and the other half has less than 7. 40
Mode Practice Find the mode 10, 8, 9, 8, 5 Math Practice Answer & MP8 Look for and express regularity in repeated reasoning Find the mode Ask: What do we now about the mode that we can 1, 2, 3, 4, 5 apply to this new situation? How is this new situation similar to finding the mode in the first place? Different? What can be added to the set of data above, so that there are two modes? Three modes? 41
14 What number(s) can be added to the data set so that there are 2 modes: 3, 5, 7, 9, 11, 13, 15 ? 3 A B 6 Answer C 8 A & D D 9 10 E 42
15 What value(s) must be eliminated so the data set has 1 mode: 2, 2, 3, 3, 5, 6 ? Answer 2 or 3 43
16 Find the mode(s): 3, 4, 4, 5, 5, 6, 7, 8, 9 4 A 5 B 9 C Answer A & B No mode D 44
17 What number can be added to the data set below so that the mode is 7? 5, 3, 4, 4, 6, 9, 7, 7 Answer 45
Central Tendency Application Problems Return to Table of Contents 46
Teachers: Use this Mathematical Practice Pull Tab to compliment slides 4851. & Math Practice Teacher Notes 47
Which Measure of Center to Use? Sherman and his friends had a paper 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. 48
Which Measure of Center to Use? 13 ft, 2 ft, 19 ft, 18 ft and 16 ft 13.6 ft Mean: Click for answer 16 ft Median: Click for answer Mode: no mode Click for answer Which measure of center best describes the data? The median is closest to most of the Click for answer values, so it best describes the data. The mean is less than 3 out of the 5 values, and there was no mode. 49
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 50
Measures of Center Data 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. 51
18 Which measure of center best describes the data set? 2, 2, 2, 4, 4, 7, 7, 7 A mean B median Answer B C mode 52
19 Sarah records the number of texts she receives each day. During one week, she receives 7, 3, 10, 5, 5, 6, and 6 texts. Which measure of center best describes this data? A mean Answer A B median C mode 53
20 Thomas Middle School held a track meet. The times for the 200meter dash, in seconds, were 22.3, 22.4, 23.3, 24.5 and 22.5. Does the mean, median or mode best describe the runners' times? A mean Answer B median A C mode 54
21 Julie is comparing prices for a new pair of shoes. The prices at seven different stores are $18.99, $17.99, $19.99, $17.99, $17.99, $17.00 and $10.99. Which measure of center best describes the set of prices? A mean Answer C B median C mode 55
Teachers: Use this Mathematical Practice Pull Tab for the next 3 slides (5759). MP1 Make sense of problems and persevere in solving them. Ask: Math Practice How would you describe what the problem is asking? What information is given in the problem? Does the answer you get make sense for the problem's context? Which method would be most efficient for answer the problem? 56
Method 1 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 Answer 57
Method 2 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? Answer Method 2: Work Backwards 58
Method 3 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 Let x = Jae's cost for the last gift. she spend for the last gift? 24 + 26 + 20 + 18 + x = 24 5 Method 3: Write an Equation 88 + x = 24 Answer 5 88 + x = 120 (multiplied both sides by 5) x = 32 (subtracted 88 from both sides) 59
Mean Problem Your test scores are 87, 86, 89, and 88. You have one more test in the marking period. You want your average to be a 90. What score must you get on your last test? Answer 60
22 Your test grades are 72, 83, 78, 85, and 90. You have one more test and want an average of an 82. What must you earn on your next test? Answer 61
23 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 Yes my friend is correct because average. Is your friend correct? Why or why not? I would need a 102 on the next test. Answer 72 + 83 + 78 + 85 + 90 + x = 85 Yes 6 408 + x = 85 No 6 408 + x = 510 x = 102 62
Data Problems Consider the data set: 50, 60, 65, 70, 80, 80, 85 The mean is: The median is: Answer The mode is: What happens to the mean, median and mode if 60 is added to the set of data? Mean: Median: Mode: 63
Data Problem Practice Consider the data set: 55, 55, 57, 58, 60, 63 • The mean is: • the median is: • and the mode is: Answer What would happen if a value x was added to the set? How would the mean change: • if x was less than the mean? • if x equals the mean? • if x was greater than the mean? 64
Data Problem Practice 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 Answer 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? 65
Data Problem Practice Consider the data set: 10, 15, 17, 18, 18, 20, 23 • The mean is 17.3 • the median is 18 • and the mode is 18 Answer What would happen if the value of 20 was added to the data set? How would the mean change? How would the median change? How would the mode change? 66
Data Problem Practice Consider the data set: 55, 55, 57, 58, 60, 63 If x was 55, the mode would stay • The mean is 58 the same at 55. • the median is 57.5 • and the mode is 55 If x was another number on the list Answer other than 55, there What would happen if a value, "x", was added to the set? would be another mode. How would the mode change: If x was a number not on the list, if x was 55? the mode would stay if x was another number in the list other than 55? the same at 55. if x was a number not in the list? 67
24 Consider the data set: 78, 82, 85, 88, 90. Identify the data values that remain the same if "79" is added to the set. mean A median B Answer mode C D range E minimum 68
Measures of Variation Return to Table of Contents 69
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. 70
Minimum and Maximum 14, 17, 9, 2, 4, 10, 5 What is the minimum in this set of data? Answer What is the maximum in this set of data? 71
Maximum and Minimum Practice Given a maximum of 17 and a minimum of 2, what is the range? Answer 15 72
25 Find the range: 4, 2, 6, 5, 10, 9 5 A 8 B Answer B C 9 D 10 73
26 Find the range, given a data set with a maximum value of 100 and a minimum value of 1 Answer 99 74
27 Find the range for the given set of data: 13, 17, 12, 28, 35 Answer 23 75
28 Find the range: 32, 21, 25, 67, 82 Answer 61 76
Quartiles There are three quartiles for every set of data. Lower Upper Half Half 10, 14, 17, 18, 21, 25, 27, 28 Q1 Q3 Q2 The lower quartile (Q1) is the median of the lower half of the data which is 15.5. 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 19.5. The interquartile range is Q3 Q1 which is equal to 10.5. 77
Analyzing Data 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). First Quartile: 2 Answer 1, 2, 2, 3 , 5, 7, 8 Median: 3 Third Quartile: 7 What is the lower quartile, upper quartile, and interquartile range? Interquartile Range: 7 2 = 5 First Quartile: Median: Third Quartile: Interquartile Range: 78
29 The median (Q2) of the following data set is 5. 3, 4, 4, 5, 6, 8, 8 True Answer False 79
30 What are the lower and upper quartiles of the data set 3, 4, 4, 5, 6, 8, 8? A Q1: 3 and Q3: 8 Answer Q1: 3.5 and Q3: 7 B Q1: 4 and Q3: 7 C D Q1: 4 and Q3: 8 80
31 What is the interquartile range of the data set 3, 4, 4, 5, 6, 8, 8? Answer 81
32 What is the median of the data set 1, 3, 3, 4, 5, 6, 6, 7, 8, 8? A 5 Answer B 5.5 6 C D No median 82
33 What are the lower and upper quartiles of the data set 1, 3, 3, 4, 5, 6, 6, 7, 8, 8? (Pick two answers) Answer D Q3: 6 Q1: 1 A Q3: 7 E B Q1: 3 F Q3: 8 Q1: 4 C 83
34 What is the interquartile range of the data set 1, 3, 3, 4, 5, 6, 6, 7, 8, 8? Answer 84
Outliers Practice Outliers Numbers that are relatively much larger or much smaller than the data. Which of the following data sets have outlier(s)? Answer A & B A. 1, 13, 18, 22, 25 B. 17, 52, 63, 74, 79, 83, 120 C. 13, 15, 17, 21, 26, 29, 31 D. 25, 32, 35, 39, 40, 41 85
Outliers Practice 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 Q1: 3 above Q3. Q2: 6 Consider the set 1, 5, 6, 9, 17. Q3: 13 IQR: 10 Q1: 3 Answer 1.5 x IQR = 1.5 x 10 = 15 Q2: 6 Q3: 13 Q1 15 = 3 15 = 12 IQR: 10 Q3 + 15 = 13 + 15 = 28 In order to be an outlier, a number 1.5 x IQR = 1.5 x 10 = 15 should be smaller than 12 or larger than 28. Q1 15 = 3 15 = 12 Q3 + 15 = 13 + 15 = 28 In order to be an outlier, a number should be smaller than 12 or larger than 28. 86
35 Which of the following data sets have outlier(s)? 13, 18, 22, 25, 100 A 17, 52, 63, 74, 79, 83 B Answer A, B, C, D C 13, 15, 17, 21, 26, 29, 31, 75 D 1, 25, 32, 35, 39, 40, 41 87
36 The data set: 1, 20, 30, 40, 50, 60, 70 has an outlier which is ________ than the rest of the data. A higher B B lower C neither Answer Even though 1 does not follow the general rule, it is obvious that it does not belong. 88
37 In the following data what number is the outlier? { 1, 2, 2, 4, 5, 5, 5, 13} Answer 89
38 In the following data what number is the outlier? { 27, 27.6, 27.8 , 27.8, 27.9, 32} Answer 32 90
39 In the following data what number is the outlier? { 47, 48, 51, 52, 52, 56, 79} Answer 79 91
40 The data value that occurs most often is called the mode A range B C median Answer A D mean 92
41 The middle value of a set of data, when ordered from lowest to highest is the _________ A mode range B Answer median C D mean 93
42 Find the maximum value: 15, 10, 32, 13, 2 2 A B 15 13 C Answer D D 32 94
43 Identify the outlier(s): 78, 81, 85, 92, 96, 145 Answer 95
44 If you take a set of data and subtract the minimum value from the maximum value, you will have found the ______ outlier A B median Answer D mean C D range 96
Analyzing Data Practice Find the mean, median, range, quartiles, interquartile range and outliers for the data below. High Temperatures for Halloween Year Temperature 2003 91 2002 92 2001 92 2000 89 1999 96 1998 88 1997 97 1996 95 97
High Temperatures for Halloween 88 89 90 91 92 93 94 95 96 97 High Temperatures for Halloween & Math Practice Teacher Notes YearTemperature 740/8 = 92.5 Mean 2003 91 2002 92 92 2001 92 Median 2000 89 1999 96 Range 9788 = 9 1998 88 1997 97 1996 95 90 Lower Quartile 95.5 Upper Quartile Interquartile Range 5.5 None Outliers 98
Analyzing Data Practice Find the mean, median, range, quartiles, interquartile range and outliers for the data. Candy Calories Butterscotch Discs 60 Candy Corn 160 Caramels 160 Gum 10 Dark Chocolate Bar 200 Gummy Bears 130 Jelly Beans 160 Licorice Twists 140 Lollipop 60 Milk Chocolate Almond 210 Milk Chocolate 210 99
Calories from Candy 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 Candy Calories Butterscotch Discs 60 1500/11 = Candy Corn 160 136.36 Caramels Mean 160 Gum 10 Dark Chocolate Bar 200 160 Median Gummy Bears 130 Jelly Beans 160 Licorice Twists 21010 = 200 Range 140 Lollipop 60 Milk Chocolate Almond 60 Milk Chocolate 210 Lower Quartile 210 200 Upper Quartile 140 Interquartile Range Outliers 10 100
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