2. Warm Up Solve for x 2) 2x + 80 The product of a number
and 9 increased by 4 is 58.
Find the number.
5(x + 2) 3x
5(x + 2) + 2x x = 180
9x + 4 = 58
9x = 54
5x x x = 180
x = 6
10x = 180
10x + 90 = 180
3) 9 – 4(2p – 1) = 45
10x = 90
4) 9x = 3(x – 2)
9 – 8p + 4 = 45
x = 9
9x = 3x - 6
13 – 8p = 45
6x = -6
-8p = 32
x = -1
p = -4

3. Chapter 6

4. Measures of Center Mean, Median, Mode and Range
Chapter 6.6
Measures of Center
Mean, Median, Mode and Range

5. Vocabulary Mean or Average Median Mode Range
The sum of all the numbers divided by the total number of numbers
Median
The middle number when the numbers are written in order
If there are two middle numbers you find the average of the two numbers
Mode
The number that occurs most often
You can have more then one mode
Range
The largest number subtracted by the smallest number

6. Find the mean, median, mode and range
4, 2, 10, 6, 10, 7, 10
First write the numbers in order from least to greatest
2, 4, 6, 7, 10, 10, 10
Mean:
2, 4, 6, 7, 10, 10, 10
Median: 10
Mode:
10 – 2 = 8
Range:

7. Find the mean, median, mode and range
5, 3, 10, 13, 8, 18, 5, 17, 2, 7, 9, 10, 4, 1
First write the numbers in order from least to greatest
1, 2, 3, 4, 5, 5, 7, 8, 9, 10, 10, 13, 17, 18
Mean:
Median:
1, 2, 3, 4, 5, 5, 7, 8, 9, 10, 10, 13, 17, 18
5 and 10
Mode:
18 – 1 = 17
Range:

8. What do you know using mean, median, mode and range?
Which is greater?
Which is smaller?
Are any equal?
8, 5, 6, 5, 6, 6
Mean = 6
Mean, Median and Mode
are all equal
Median = 6
Mode = 6
Range is the smallest
Range = 3

9. What do you know using mean, median, mode and range?
Which is greater?
Which is smaller?
Are any equal?
161, 146, 158, 150, 156, 150,
146, 150, 150, 156, 158, 161
Mean = 153.5
Mean, is the greatest
Median = 153
Range is the smallest
Mean > Median
Mode = 150
Median > Mode
Range = 15
Mode < Mean

10. Calculating and Interpreting
How does an outlier affect the measures of center and range?
Test grades:
50, 70, 62, 80, 70, 76
Test grades:
50, 70, 62, 80, 70, 76, 100
Mean: 68
Mean:
72.6
Median:
Median: 70 70
Mode:
Mode: 70 70
Range: 30
Range: 50

11. Calculating and Interpreting
How does an outlier affect the measures of center and range?
8, 5, 6, 5, 6, 6
8, 5, 6, 5, 6, 6, 15
Mean: 6
Mean:
7.3
Median:
Median: 6 6
Mode:
Mode: 6 6
Range: 3
Range: 10

12. Calculating and Interpreting
How does an outlier affect the measures of center and range?
161, 146, 158, 150, 156, 150
161, 146, 158, 150, 156, 150, 200
Mean:
153.5
Mean:
160.1
Median:
Median:
153
156
Mode:
Mode:
150
150
Range: 15
Range: 54

13. Homework: Page 371 problems 18-24 even
Find the mean, median, mode and range
18) 1, 2, 1, 2, 1, 3, 3, 4, 3,
20) 4, 4, 4, 4, 4, 4
22) 12, 5, 6, 15, 12, 9, 13, 1, 4, 6, 8, 14, 12
24) 161, 146, 158, 150, 156, 150

15. Warm Up Add up all the numbers then divide by how many
numbers there are
How do you find the mean?
How do you find the median?
How do you find the mode?
How do you find the range?
Find the mean, median, mode and range.
4, 2, 10, 6, 10, 7, 10
Put the numbers in order and find the middle number
Find the number that occurs most often
Subtract the largest number and the smallest number
Mean = 7
Median = 7
Mode = 10
Range = 8

16. Homework: Page 371 problems 18-24 even
Find the mean, median, mode and range
18) 1, 2, 1, 2, 1, 3, 3, 4, 3,
20) 4, 4, 4, 4, 4, 4
22) 12, 5, 6, 15, 12, 9, 13, 1, 4, 6, 8, 14, 12
24) 161, 146, 158, 150, 156, 150
Mean = 20/9
Median = 2
Mode = 1 and 3
Range = 3
Mean = 4
Median = 4
Mode = 4
Range = 0
Mean = 9
Median = 9
Mode = 12
Range = 14
Mean = 153.5
Median = 153
Mode = 150
Range = 15

17. Chapter 6.6
Stem-and-Leaf Plot

18. Stem-and-Leaf Plot
Arrangement of digits that is used to display and order numerical data

19. Making a Stem-and-Leaf Plot
60, 74, 75, 63, 78, 70,
50, 74, 52, 74, 65, 78, 54 5 6 7 2 4 3 5 4 4 4 5 8 8

20. Practice on your own
4, 31, 22, 37, 39, 24, 2, 28,
1, 26, 28, 30, 28, 3, 20, 20, 5 1 2 3

21. Box-and-Whisker Plots
Chapter 6.7
Box-and-Whisker Plots

22. Box-and-Whisker Plots
Divides a set of data into four parts
Median or Second Quartile
Separates the set into two halves
Numbers below the median
Numbers above the median
First Quartile
Median of the lower half
Third Quartile
Median of the upper half

23. 12, 5, 3, 8, 10, 7, 6, 5
Find the first, second and third quartiles
3, 5, 5, 6, 7, 8, 10, 12
Second =
7, 8, 10, 12
3, 5, 5, 6
First =
Third = 2 3 4 5 6 7 8 9 10 11 12 13

24. 1, 12, 6, 5, 4, 7, 5, 10, 3, 4
Find the first, second and third quartiles
1, 3, 4, 4, 5, 5, 6, 7, 10, 12
Second =
5, 6, 7, 10, 12
1, 3, 4, 4, 5
First =
Third = 1 2 3 4 5 6 7 8 9 10 11 12

25. 6, 7, 10, 6, 2, 8, 7, 7, 8
Find the first, second and third quartiles
2, 6, 6, 7, 7, 7, 8, 8, 10
Second =
7, 8, 8, 10
2, 6, 6, 7
First =
Third = 1 2 3 4 5 6 7 8 9 10 11 12

26. Practice Answer the questions.
Draw a Box-and-Whisker plot for each question.

27. Homework: page 371 problems 12 and 14 page 378 problems 16 and 18
Make a Stem-and-Leaf plot
12) 24, 29, 17, 50, 39, 51, 19, 22, 40, 45, 20, 18, 23, 30
14) 15, 39, 13, 31, 46, 9, 38, 17, 32, 10, 12, 45, 30, 1, 32, 23, 32, 41
Make a Box-and-Whisker plot
16) 10, 5, 9, 50, 10, 3, 4, 15, 20, 6
18) 8, 8, 10, 10, 1, 12, 8, 6, 5, 1, 9, 10

29. Warm Up Draw a Stem-and-Leaf plot
Find the mean, median, mode and range
Draw a Box-and-Whisker plot
1130, 695, 900, 220, 350, 500, 630, 180, 170, 145, 185, 140
Second Quartile = 285
Mean =
No mode
First Quartile = 175
Median = 285
Range = 990
Third Quartile = 662.5

30. Homework: page 371 problems 12 and 14 page 378 problems 16 and 18
Make a Stem-and-Leaf plot
12) 24, 29, 17, 50, 39, 51, 19, 22, 40, 45, 20, 18, 23, 30
14) 15, 39, 13, 31, 46, 9, 38, 17, 32, 10, 12, 45, 30, 1, 32, 23, 32, 41
Make a Box-and-Whisker plot
16) 10, 5, 9, 50, 10, 3, 4, 15, 20, 6
18) 8, 8, 10, 10, 1, 12, 8, 6, 5, 1, 9, 10

31. Chapter 6 Quiz Review

32. Find the mean, median, mode and range
4, 8, 10, 6, 12, 16, 10, 22
Mean = 11
Median = 10
Mode = 10
Range = 18
What can you tell from the data?

33. Draw a Stem-and-Leaf Plot
60, 74, 75, 63, 78, 70, 50, 74, 52, 74, 65, 78, 54
Draw a Box-and-Whisker Plot
12, 5, 3, 8, 10, 7, 6, 5
Second Quartile = 6.5
First Quartile = 5
Third Quartile = 9

34. Quiz on Chapter 6

36. Bell Curve, Standard Deviation, Z-curve, etc.
Statistics
Bell Curve, Standard Deviation, Z-curve, etc.

37. Descriptive Statistics
What’s Normal?
Descriptive Statistics

38. What is standard deviation?
- measures the spread of the data from the mean
- What is the rule in a normal distribution?
Use your copy to shade in the regions shown.

39. The 68-95-99.7 Rule For a normal distribution,
68% of the data generally falls within 1 standard deviation of the mean.
95% of the data generally falls within 2 standard deviations of the mean.
99.7% of the data generally falls within 3 standard deviations of the mean.

40. Average of a set of data values
Notation
Sigma notation: sum of all the elements
Average of a set of data values =
Read as x bar
sample
= x1 + x2 + x3 + x4 + x5
Read as mu
population

41. Mean Absolute Deviation
Average of the DISTANCES between each data value and the mean

42. Variance
Average of the squares of the differences between each data value and the mean

43. Square root of the variance
Standard Deviation
Square root of the variance

44. Measures of Dispersion
describes the average distance from the mean
describes the spread of the data

45. Investigating Dispersions Based on the Mean
The SAT scores for ten students are given. The school wants to determine spread about the mean to fill out a report.
1026, 1150, 1153, 1157, 1161, 1206, 1253, 1258, 1285, 1311
Calculate the mean.
= 1196

46. Investigating Dispersions Based on the Mean
Create a chart of values for the SAT data set and determine the distance each data piece is from the mean. x
1026
1150
1153
1157
1161
1206
1253
1258
1285
1311
-170
-46
-43
-39
-35 10 57 62 89
115

47. Investigating Dispersions Based on the Mean
What is the sum of the differences from the mean?
-170 – 46 – 43 – 39 – =
Will this always happen?
Test grades: 50, 70, 62, 80, 70, 76
What can be done to getting around the problem of always getting zero?
Is there a way to get rid of the negatives?

48. Investigating Dispersions Based on the Mean
Mathematically, we can take the absolute value of a number to ensure that it is positive. x
1026
1150
1153
1157
1161
1206
1253
1258
1285
1311
170 46 43 39 35 10 57 62 89
115

49. Investigating Dispersions Based on the Mean
What is the sum of the absolute value distances?
666
The Mean Absolute Deviation =
66.6 80
The Standard Deviation =

50. The Rule
SAT problem
How many SAT scores fall within one standard deviation from the mean? 7
What % of the data does this represent?
70%

51. Homework
How many texts do you send today?

53. Warm Up 2) 6m – 3 = 10 - 6(2 – m) 1) 26 – 9p = -1
-3 = -2
No solution
3) S = 2πrh, solve for h
4) Name the property
(5 + x)6 = 6(5 + x)
Commutative
5) Name the property
9 + 0 = 9
Identity

54. In a park that has several basketball courts a student samples the number of players playing basketball over a two week period and has the following data.
80 20

55. What is the mean for the data?
80 20

56. Distance from the mean 90 80 70 60 60 50 Mean = 45 40 40 30 30 30 2010

57. 90
What if we find the average of the difference between each data value and the mean? 80 45 70 35 60 60 25 15 15 50 5
Mean = 45 -5 -5
-15
-15 40 40
-15
-35
-25
-25 = 30 30 30 20 20 10

58. 90
What if we find the average of the DISTANCES from each data value to the mean? 80 45 70 35 60 60 15 25 15 50 = 14 5
Mean = 45
280 14 5 5 15
=20 15 15 40 40 25 35 25 30 30 30 20 20 10

59. 90
One Standard Deviation
from the mean 80 70
=68.222 60 60 50
Mean = 45 40 40
=21.778 30 30 30 20 20 10

60. Calculating Mean Absolute Deviation
How many texts did you send yesterday? 4
Calculate the mean absolute deviation
for the data set.
Calculate the standard deviation
for the data set.

61. Calculating Standard Deviation
How much time does it take for a dead cell phone battery to completely recharge?

62. Calculating Standard Deviation
Mr. Bolling’s homework assignment for his students was to determine how much time it takes for their dead cell phone battery to completely recharge. The results for the amount of time (to the nearest quarter hour) for 20 students are shown below.

63. Calculating Standard Deviation
What are the mean, mode, and median of the data?
mean: 4, mode: 4, median: 4

64. Calculating Standard Deviation
Calculate 1-Var Stats
= 0.548
What does the standard deviation represent in this data?

65. Sample Question for A.9
Student
Andy
Bill
Carrie
Dan Ed
Frank
Gus
Height 46 51 50 42 56 48 57
Henry
Izzi
Jack
Ken
Louise
Manny
Ned
Owen 45 52 49 41 53 46 43 56
What is the approximate mean absolute deviation?
D) 5
A) 3.4
B) 4.3
C) 4.5
What is the interpretation of the mean absolute value deviation of 4.3?

66. Sample Question for A.9
Student
Andy
Bill
Carrie
Dan Ed
Frank
Gus
Height 46 51 50 42 56 48 57
Henry
Izzi
Jack
Ken
Louise
Manny
Ned
Owen 45 52 49 41 53 46 43 56
Use your calculator to find the mean and standard deviation of the data set to the nearest inch?
D) 50, 4.5
A) 49, 5
B) 50, 5.5
C) 49.5, 5.5

68. Warm Up 1) Find mean, median, mode and range:
56, 58, 63, 71, 78, 84, 85, 86, 82, 78, 65, 58
Mean = 72
Median = 74.5
Mode = 58 & 78
Range = 30
3) Rewrite so y is a function of x
x = y + 3 2)
y = x – 3

69. Descriptive Statistics
Z-Scores for Algebra I
Descriptive Statistics

70. How Close is Close? Activity

71. Z-score
Position of a data value relative to the mean.
Tells you how many standard deviations above or below the mean a particular data point is.
z-score = describes the location of a data value within a distribution referred to as a standardized value
Population
Sample

72. In order to calculate a z-score you must know:
a data value
the mean
the standard deviation

73. Z-scores Here are 23 test scores from Ms. Bienvenue’s stat class.
What is the mean score?
80.5
What is the standard deviation?
5.9

74. Z-scores Here are 23 test scores from Ms. Bienvenue’s stat class.
The bold score is Michele’s. How did she perform relative to her classmates?
Michele’s score is “above average”, but how much above average is it?

75. Z-scores
If we convert Michele’s score to a standardized value, then we can determine how many standard deviations her score is away from the mean.
What we need:
Michele’s score
mean of test scores
standard deviation 86
80.5
= 0.93
5.9
Therefore, Michele’s standardized test score is Nearly one standard deviation above the class mean.

76. Michele’s Score = 86
Michele’s z-score = .93
62.2
68.3
74.4
80.5
86.6
92.7
98.8 -3 -2 -1 1 2 3

77. Consider this problem:
Calculate a z-score
Consider this problem:
The mean salary for math teachers in Big State is $45,000 per year with a standard deviation of $5,000.
The mean salary of a Piggly-Wiggly bagger is $21,000 with a standard deviation of $2,000.

78. Calculate a z-score
Who has the better salary relative to the mean? A Big State teacher making 63,000 or a grocery bagger making 30,000?
Teacher : 63,000 or
Grocery Bagger: 30,000
What is the interpretation of the two z-scores?
Who has a better salary relative to the mean?

80. 4) Rewrite so y is a function of x
Warm Up 1)
2) 5x – 2 = 8
5x = 10
x = 2
4) Rewrite so y is a function of x
3(y – x) = 10 – 4x
3) Name the property
If a = b and b =c, then a = c
3y – 3x = 10 – 4x
3y = 10 – x
Transitive Property

81. Sample Question for A.9
Student
Andy
Bill
Carrie
Dan Ed
Frank
Gus
Height 46 51 50 42 56 48 57
Which students’ heights have a z-score greater than 1?
A) All of them
Mean = 50
Standard Deviation = 5.3
B) Bill, Carrie, Ed and Gus
C) Ed and Gus
D) None of them

82. Sample Question for A.9 Which students have a z-score less than -2?
Andy
Bill
Carrie
Dan Ed
Frank
Gus
Height 46 51 50 42 56 48 57
Which students have a z-score less than -2?
Mean = 50
Standard Deviation = 5.3
A) All of them
B) Dan and Andy
C) Only Dan
D) None of them

83. Sample Question for A.9 Which student’s height has a z-score of zero?
Andy
Bill
Carrie
Dan Ed
Frank
Gus
Height 46 51 50 42 56 48 57
Which student’s height has a z-score of zero?
A) Bill
B) Carrie
C) Frank
D) None of them

84. Sample Question for A.9
Given a data set with a mean of 125 and a standard deviation of 20, describe the z-score of a data value of 120?
Mean = 125
Standard Deviation = 20
A) Less than -5
B) Between -5 and -1
C) Between -1 and 0
D) Greater than 0

85. Sample Question for A.9
Given a data set with a mean of 30 and a standard deviation of 2.5, find the data value associated with a z-score of 2?
Mean = 30
Standard Deviation = 2.5
A) 36
B) 35
C) 34.5
D) 32.5

86. Sample Question for A.9
Suppose the test scores on the last exam in Algebra I are normally distributed. The z-scores for some of the students in the course were:
1.5, 0, -1.2, -2, 1.95, 0.5
1) List the z-scores of students that were above the mean.
1.5, 1.95, and 0.5

87. Sample Question for A.9
Suppose the test scores on the last exam in Algebra I are normally distributed. The z-scores for some of the students in the course were:
1.5, 0, -1.2, -2, 1.95, 0.5
2) If the mean of the exam is 80, did any of the students selected have an exam score of 80? Explain.
One student with a z-score of 0.

88. Sample Question for A.9
Suppose the test scores on the last exam in Algebra I are normally distributed. The z-scores for some of the students in the course were:
1.5, 0, -1.2, -2, 1.95, 0.5
3) If the standard deviation of the exam was 5 and the mean is 80, what was the actual test score for the student having a z-score of 1.95? 90

89. Quiz