StatisticsChapter 9

Day 1

Unusual Episode

MS133 Final Exam Scores

79 86 79 65 78

91 78 94 88 75

71 53 95 96 79

62 79 67 64 77

69 58 74 69 78

78 91 89 49 68

63 77 86 84 77

Line Plot or Dot Plot

Stem and Leaf

Stem and Leaf

9 1 1 4 5 6

8 6 8 9 6 4

7 9 1 8 8 9 7 9 8 5 9 7 4 8 7

6 2 9 3 5 7 4 9 8

5 3 8

4 9

Ordered Stem and Leaf

9 1 1 4 5 6

8 4 6 6 8 9

7 1 4 5 7 7 7 8 8 8 8 9 9 9 9

6 2 3 4 5 7 8 9 9

5 3 8

4 9

Frequency Table

Grade Score Tally Frequency

Frequency Table

Grade Score Tally Frequency

A 90-100 IIII 5

B 80-89 IIII 5

C 70-79 IIII IIII IIII 14

D 60-69 IIII III 8

F 0-59 III 3

Bar Graph

FREQUENCY

GRADES

MS133 Final Exam Grades

F'sD'sC'sB'sA's

14

12

10

8

6

4

2

Make a Pie Chart

• 5 A’s out of how many grades total?

• 5 A’s out of how many total grades? 35

• What percent of the class made an A?

• 5 A’s out of how many total grades? 35

• What percent of the class made an A?

5/35 ≈ 0.14 ≈ 14%

• What percent of the pie should represent the A’s?

• 5 A’s out of how many total grades? 35• What percent of the class made an A?

5/35 ≈ 0.14 ≈ 14%

• What percent of the pie should represent the A’s? 14%

• How many degrees in the whole pie?

• 5 A’s out of how many total grades? 35

• What percent of the class made an A?

5/35 ≈ 0.14 ≈ 14%

• What percent of the pie should represent the A’s? 14%

• How many degrees in the whole pie? 360°

• 5 A’s out of how many total grades? 35

• What percent of the class made an A?

5/35 ≈ 0.14 ≈ 14%

• What percent of the pie should represent the A’s? 14%

• How many degrees in the whole pie? 360°

• 14% of 360° is how many degrees?

• 5 A’s out of how many total grades? 35• What percent of the class made an A?

5/35 ≈ 0.14 ≈ 14%• What percent of the pie should represent

the A’s? 14%• How many degrees in the whole pie? 360°• 14% of 360° is how many degrees? .14 x 360° ≈ 51°

A's14%

• 5 B’s out of 35 grades total ≈ 14% ≈ 51°

A's14%

B's14%

A's14%

• 14 C’s out of 35 grades

• 14 C’s out of 35 grades

• 14/35 = .4 = 40%

• .4 x 360° = 144°

B's14%

A's14%

C's40%

B's14%

A's14%

• 8 D’s out of 35 grades total

• 8 D’s out of 35 grades

• 8/35 ≈ .23 ≈ 23% (to the nearest percent)

(keep the entire quotient in the calculator)

• x 360° ≈ 82°

C's40%

B's14%

A's14%

D's23%

C's40%

B's14%

A's14%

• 3 F’s out of 35 total

• 3 F’s out of 35 grades total

• 3/35 ≈ .09 ≈ 9% (to the nearest percent)(keep the entire quotient in the calculator)

• x 360° ≈ 31°

• Check the remaining angle to make sure it is 31°

D's23%

C's40%

B's14%

A's14%

MS133 Final Exam Grades

F's 9%

D's23%

C's40%

B's14%

A's14%

Make a Pie Chart

• Gross income: $10,895,000

• Labor: $5,120,650

• Materials: $4,031,150

• New Equipment: $326,850

• Plant Maintenance: $544,750

• Profit: $871,600

• Labor: $5,120,650 = 47% 169°

10,895,000 • Materials: $4,031,150 = 37% 133°

10,895,000• New Equipment: $326,850 = 3% 11°

10,895,000• Plant Maintenance: $544,750 = 5% 18°

10,895,000• Profit : $871,600 = 8% 29°

10,895,000

Profit 8%

5% Maintenance3%

Equipment

Materials 37%

Labor47%

Histogram

• Table 9.2 Page 527

Eisenhower High School Boys Heights

7371696765 747270686664

HEIGHTS (inches)

18

14

10

6

2

FREQUENCY

EHS Boys’ Heights

Height Frequency Relative

Frequency

64 1

65 1 70 14

66 3 71 10

67 7 72 6

68 15 73 2

69 19 74 2

EHS Boys’ Heights

Height Frequency Relative

Frequency

64 1 .0125

65 1 .0125 70 14 .175

66 3 .0375 71 10 .125

67 7 .0875 72 6 .075

68 15 .1875 73 2 .025

69 19 .2375 74 2 .025

Eisenhower High School Boys Heights

.25

.20

.15

.10

.05

FREQUENCY

7371696765 747270686664

HEIGHTS (inches)

RELATI VE

EHS Boys’ Heights

HEIGHTS (inches)

FREQUENCY

7473727170696867666564

18

14

10

6

2

Day 2

Measures of Central Tendency Lab

Print your first name below.

Getting Mean with Tiles

• Use your colored tiles to build a column 9 tiles high and another column 15 tiles high. Use a different color for each column.

Getting Mean with Tiles

• Use your colored tiles to build a column 9 tiles high and another column 15 tiles high. Use a different color for each column.

• Move the tiles one at a time from one column to another “evening out” to create 2 columns the same height.

• What is the new (average) height?

Getting Mean with Tiles

• Move the tiles back so that you have a column 9 tiles high and another 15 tiles high.

• Find another method to “even off” the columns?

Getting Mean with Tiles

• Use your colored tiles to build a column 19 tiles high and another column 11 tiles high. Use a different color for each column.

• “Even-off” the two columns using the most efficient method.

• What is the new (average) height?

Getting Mean with Tiles

• If we start with a column x tiles high and another y tiles high, describe how you could find the new (average) height?

• Let’s assume x is the larger number

• x – y (extra)

• x – y (extra) x – y 2

• x – y (extra) x – y 2

• y + x – y

2

• x – y (extra) x – y 2

• y + x – y

2

2y + x – y

2 2

• x – y (extra) x – y 2

• y + x – y

2

2y + x – y

2 2

2y + x - y

2

• x – y (extra) x – y 2

• y + x – y

2

2y + x – y

2 2

2y + x - y

2

x + y 2

Homework QuestionsPage 538

Measures of Central Tendency

• Mean – “Evening-off”

• Median – “Middle”

• Most – “Most”

Class R

71 71 7679 77 76 7072 92 74 8679 46 79 7281 67 77 7277 63 77 6176

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Mean = Sum of all grades

Number of grades

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Mean = Sum of all grades

Number of grades

Mean = 1771

24

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Mean = Sum of all grades

Number of grades

Mean = 1771

24

8.7324

1771x

Class S

72 77 75 7567 76 69 7671 68 77 7982 73 69 7668 69 71 7872 79 74 7373

67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Mean =

67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Mean = 25

1839

67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Mean = 25

1839

6.7325

1839x

Class T

74 79 86 8440 82 40 6140 49 70 8549 40 45 9174 96 81 8586 75 89 85

40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Mean =

40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Mean = 24

1686

40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Mean = 24

1686

3.7024

1686x

Median –”Middle”

Class R:46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Class S:67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Class T:40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Median

• Class R: 76

• Class S: 73

• Class T: 77

Mode – “Most”

Class R:46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Class S:67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Class T:40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Mode

• Class R: 77

• Class S: 69, 73, 76

• Class T: 40

Range - A measure of dispersion Greatest - Least

Class R:46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Class S:67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Class T:40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Range

• Class R: 92 - 46 = 46

• Class S: 82 – 67 = 15

• Class T: 96 – 40 = 56

Class R Class S Class T

Mean = 73.8 73.6 70.3

Median = 76 73 77

Mode = 77 69,73,76 40

Range = 46 15 56

Weighted Mean Example 9.7

Owner/Manager earned $850,000

Assistant Manager earned $48,000

16 employees $27,000 each

3 secretaries $18,000 each

Find the MEAN, MEDIAN, MODE

MEAN

Salary

$18,000

$27,000

$48,000

$850,000

MEAN

Salary Frequency

$18,000 3

$27,000 16

$48,000 1

$850,000 1

MEAN

Mean = 3(18,000)+16(27,000)+48,000+850,000

21

= 1384000 21

≈ $65,905

MEDIAN

Salary Frequency

$18,000 3

$27,000 16

$48,000 1

$850,000 1

MEDIAN

Salary Frequency Cumulative Frequency

$18,000 3 1 – 3

$27,000 16 4 - 19

$48,000 1 20

$850,000 1 21

MODE

Salary Frequency Cumulative Frequency

$18,000 3 1 – 3

$27,000 16 4 - 19

$48,000 1 20

$850,000 1 21

RANGE

Salary Frequency Cumulative Frequency

$18,000 3 1 – 3

$27,000 16 4 - 19

$48,000 1 20

$850,000 1 21

• Mean = $65,905

• Median = $27,000

• Mode = $27,000

• Range = $832,000

Grade Point AverageA weighted mean

quality points earned

hours attempted

Quality PointsEvery A gets 4 quality points per hour. For

example, an A in a 3 hour class gets 4 quality points for each of the 3 hours, 4x3=12. An A in a 4 hour class gets 4 quality points for each of the 4 hours, 4X4=16 quality points.

Every B gets 3 quality points per hour.

Every C gets 2 quality points per hour.

Every D gets 1 quality points per hour.

No quality points for an F.

Sally Ann’s First Semester Grades

Hours Grade

3 D

4 F

2 B

3 C

2 C

1 A

Sally Ann’s First semester GPAto the nearest hundredth

53.115

23

Sally Ann’s Second Semester

Hours Grade

3 C

3 C

3 B

3 B

Sally Ann’s Second Semester GPA

5.212

30

Sally Ann’s Cumulative GPA

Total quality points earned

Total hours attempted

Sally Ann’s New GPAto the nearest hundredth

96.127

53

Day 3

Class X

60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90

Find the mean, median, mode, and range.

Mean

60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90

15

)90(2)85(2)82(4)78(2)72(3)60(2

Mean

60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90

15

)90(2)85(2)82(4)78(2)72(3)60(2

7815

1170

Median – Mode – Range

60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90

• Mean = 78

• Median = 82

• Mode = 82

• Range = 30

Standard Deviation

The standard deviation is a measure of dispersion. You can think of the standard deviation as the “average” amount each data is away from the mean. Some data are close, some are farther. The standard deviation gives you an average.

Find the standard deviation of class x.

Standard Deviation

60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90

Mean = 78

Standard Deviation of Class X

15

)9078(2)8578(2)8278(4)7878(2)7278(3)6078(2 222222

15

)12(2)7(2)4(4)0(2)6(3)18(2 222222

15

)12(2)7(2)4(4)0(2)6(3)18(2 222222

15

)144(2)49(2)16(4)0(2)36(3)324(2

15

)12(2)7(2)4(4)0(2)6(3)18(2 222222

15

)144(2)49(2)16(4)0(2)36(3)324(2

15

28898640108648

15

)12(2)7(2)4(4)0(2)6(3)18(2 222222

15

)144(2)49(2)16(4)0(2)36(3)324(2

15

28898640108648

97.84.8015

1206

Page 558Example 9.11

Find the mean (to the nearest tenth):

35, 42, 61, 29, 39

Page 558Example 9.11

Find the mean (to the nearest tenth): ≈ 41.2

Standard deviation (to the nearest tenth):

35, 42, 61, 29, 39

Page 558Example 9.11

Find the mean (to the nearest tenth): ≈ 41.2

Standard deviation (to the nearest tenth): ≈ 10.8

Box and Whisker Graph

• Graph of dispersion

• Data is divided into fourths

• The middle half of the data is in the box

• Outliers are not connected to the rest of the data but are indicted by an asterisk.

Box and Whisker Graph• Class R:46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =

Upper Quartile = Lower Quartile =

100908070605040

Outliers

• Any data more than 1 ½ boxes away from the box (middle half) is considered an outlier and will not be connected to the rest of the data.

• The size of the box is called the Inner Quartile Range (IQR) and is determined by finding the range of the middle half of the data.

Box and Whisker Graph• Class R:46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =76

Upper Quartile = 78 Lower Quartile = 71

Inner Quartile Range =

Box and Whisker Graph• Class R:46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =76

Upper Quartile = 78 Lower Quartile = 71

Inner Quartile Range = 7 IQR x 1.5 =

Box and Whisker Graph• Class R:46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =76

Upper Quartile = 78 Lower Quartile = 71

Inner Quartile Range = 7 IQR x 1.5 = 10.5

Checkpoints for Outliers:

Box and Whisker Graph• Class R:46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =76

Upper Quartile = 78 Lower Quartile = 71

Inner Quartile Range = 7 IQR x 1.5 = 10.5

Checkpoints for Outliers: 60.5, 88.5

Outliers =

**

100908070605040

Box and Whisker Graph• Class R:46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =76

Upper Quartile = 78 Lower Quartile = 71

Inner Quartile Range = 7 IQR x 1.5 = 10.5

Checkpoints for Outliers: 60.5, 88.5

Outliers = 46, 92 Whisker Ends =

Box and Whisker Graph• Class R:46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =76

Upper Quartile = 78 Lower Quartile = 71

Inner Quartile Range = 7 IQR x 1.5 = 10.5

Checkpoints for Outliers: 60.5, 88.5

Outliers = 46, 92 Whisker Ends = 61, 86

**

100908070605040

Box and Whisker Graph• Class S:67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Median =

UQ = LQ =

IQR = IQR x 1.5 =

Checkpoints for outliers:

Outliers = Whisker Ends =

Box and Whisker Graph• Class S:67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Median = 73

UQ = 76.5 LQ = 70

IQR = 6.5 IQR x 1.5 = 9.75

Checkpoints for outliers: 60.25, 86.25

Outliers = none Whisker Ends = 67, 82

**

100908070605040

Box and Whisker Graph• Class T:40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Median =

UQ = LQ = IQR =

IQR x 1.5 =

Checkpoints for Outliers:

Outliers= Whisker Ends=

Box and Whisker Graph• Class T:40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Median = 77

UQ = 85 LQ = 49 IQR = 36

IQR x 1.5 = 54

Checkpoints for Outliers: -5, 139

Outliers = none Whisker Ends = 40, 96

**

100908070605040

Day 4

Homework QuestionsPage 561

Statistical Inference

• Population

• Sampling

• Random Sampling

• Page 576 #2, 4, 5, 17, 18, 19, 21, 22

Example 9.15, Page 569

Getting a random sampling

5,5,2,9,1,0,4,5,3,1,2,4,1,9,4,6,6,9,1,7

5,5,2,9,1,0,4,5,3,1,2,4,1,9,4,6,6,9,1,7

55 29 10 45 3124 19 46 69 17

5,5,2,9,1,0,4,5,3,1,2,4,1,9,4,6,6,9,1,7

55 29 10 45 3124 19 46 69 17

Sample

65 64 68 65 63

63 64 62 64 67

Find the mean of the sample

65 64 68 65 63

63 64 62 64 67

Mean = 62 + 2(63) + 3(64) + 2(65) + 67 + 68

10

Sample Mean

Mean = 62 + 2(63) + 3(64) + 2(65) + 67 + 68

10

Mean = 645

10

Mean = 64.5

Standard Deviation of the Sample

62 63 63 64 64 64 65 65 67 68

Standard Deviation of the Sample

62 63 63 64 64 64 65 65 67 68

10

)685.64()675.64()655.64(2)645.64(3)635.64(2)625.64( 222222

Standard Deviation

10

)685.64()675.64()655.64(2)645.64(3)635.64(2)625.64( 222222

10

)5.3()5.2()5.(2)5(.3)5.1(2)5.2( 222222

Standard Deviation

10

)685.64()675.64()655.64(2)645.64(3)635.64(2)625.64( 222222

10

)5.3()5.2()5.(2)5(.3)5.1(2)5.2( 222222

10

25.1225.6)25(.2)25(.3)25.2(225.6

Standard Deviation

10

)685.64()675.64()655.64(2)645.64(3)635.64(2)625.64( 222222

10

)5.3()5.2()5.(2)5(.3)5.1(2)5.2( 222222

10

25.1225.6)25(.2)25(.3)25.2(225.6

75.105.310

5.30

Random Sample

• Mean = 64.5

• Standard deviation = 1.75

• Compare the sample to the mean and standard deviation of the entire population. (example 9.14)

• Compare our sample to the author’s sample. (example 9.14)

Beans or Fish

Normal Distribution

• The distribution of many populations form the shape of a “bell-shaped” curve and are said to be normally distributed.

• If a population is normally distributed, approximately 68% of the population lies within 1 standard deviation of the mean. About 95% within 2 standard deviations. About 99.7% within 3 standard deviations.

Normal Curve

x + 3sx - 3s x - 2s x + 2sx - s x + sx

68% of the data is within 1 standard deviation of the mean

x - s x + sx

< >68%

95% of the data is within 2 standard deviations of the mean

x - 2s x + 2sx

< >95%

99.7% of the data is within 3 standard deviations of the mean

x - 3s x + 3sx

< >99.7%

Normal Distribution

x + 3sx - 3s

99.7%

95%

68%

x - 2s x + 2sx - s x + sx

Normal Distribution Example

• Suppose the 200 grades of a certain professor are normally distributed. The mean score is 74. The standard deviation is 4.3.

• What whole number grade constitutes an A, B, C, D and F?

• Approximately how many students will make each grade?

74x 3.4.. ds students200

61.1 65.4 69.7 86.982.678.374

• A: 83 and above 200 students• B: 79 – 82• C: 70 – 78• D: 66 – 69• F: 65 and below

61.1 65.4 69.7 86.982.678.374

• A: 83 and above 5 people

• B: 79 – 82 27 people

• C: 70 – 78 136 people

• D: 66 – 69 27 people

• F: 65 and below 5 people

Normal Distribution

• The graph of a normal distribution is symmetric about a vertical line drawn through the mean. So the mean is also the median.

• The highest point of the graph is the mean, so the mean is also the mode.

• The area under the entire curve is one.

Normal Distribution

x + 3sx - 3s x - 2s x + 2sx - s x + sx

Standardized form of the normal distribution (z curve)

-3 -2 -1 3210

Z Curve

• The scale on the horizontal axis now shows a z – Score.

Any normal distribution in standard form will have mean 0 and standard deviation1.

• 68% of the data will lie between -1 and 1.

• 95% of the data will lie between -2 and 2.

• 99.7% of the data will lie between -3 and 3.

Z- Scores

• By using a z-Score, it is possible to tell if an observation is only fair, quite good, or rather poor.

• EXAMPLE: A z-Score of 2 on a national test would be considered quite good, since it is 2 standard deviations above the mean.

• This information is more useful than the raw score on the test.

Z- Scores

• z – Score of a data is determined by subtracting the mean from the data and dividing the result by the standard deviation.

• z = x - µ

σ

62,62,63,64,64,64,64,66,66,66

• Mean = 64.1

• Standard deviation ≈ 1.45

• Convert these data to a set of z-scores.

62,62,63,64,64,64,64,66,66,66

62, 63, 64, 66

z-scores: -1.45, -0.76, -0.07, 1.31

Percentiles

• The percentile tells us the percent of the data that is less than or equal to that data.

Percentile in a sample:62,62,63,64,64,64,64,66,66,66

• The percentile corresponding to 63 is the percent of the data less than or equal to 63.

• 3 data out of 10 data = .3 = 30% of the data is less than or equal to 63.

• For this sample, 63 is in the 30th percentile.

Percentile in a Population

• Remember that the area under the normal curve is one.

• The area above any interval under the curve is less than one which can be written as a decimal.

• Any decimal can be written as a percent by multiplying by 100 (which moves the decimal to the right 2 places).

• That number would tell us the percent of the population in that particular region.

Percentiles

• Working through this process, we can find the percent of the data less than or equal to a particular data – the percentile.

• The z-score tells us where we are on the horizontal scale.

• Table 9.4 on pages 585 and 586 convert the z-score to a decimal representation of the area to the left of that data.

• By converting that number to a percent, we will have the percentile of that data.

• If the z-score of a data in a normal distribution is -0.76,what is it’s percentile in the population?

• Table 9.4 page 585• Row marked -0.7• Column headed .06• Entry .2236• 22.36% of the population lies to the left of -0.76

Note the difference in finding the percentile in a sample and the

entire population.

Interval Example

• Show that 34% of a normally distributed population lies between the z scores of -0.44 and 0.44

Interval Example

• Show that 34% of a normally distributed population lies between the z scores of -0.44 and 0.44

• Table 9.4, page 585

• 33% to the left of -0.44

• 67% to the left of 0.44

• 67% - 33% = 34%

Day 5

Homework QuestionsPage 576

Normal Distribution Lab

Day 6

Lab Questions

Statistics Review

M&M Lab