c hapter 6. c hapter 6.6 measures of center mean, median, mode and range

CHAPTER 6

CHAPTER 6.6Measures of CenterMean, Median, Mode and Range

VOCABULARY Mean or Average

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

FIND THE MEAN, MEDIAN, MODE AND RANGE

4, 2, 10, 6, 10, 7, 10First write the numbers in order from least to greatest

2, 4, 6, 7, 10, 10, 10Mean:

Median:

Mode:

Range:

7

1010107642

7

497

2, 4, 6, 7, 10, 10, 1010

10 – 2 =8

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

Mean:

Median:

Mode:

Range:

14

1128

5 and 10

18 – 1 =17

1, 2, 3, 4, 5, 5, 7, 8, 9, 10, 10, 13, 17, 18

1, 2, 3, 4, 5, 5, 7, 8, 9, 10, 10, 13, 17, 18

2

87

2

155.7

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

Median = 6

Mode = 6Range = 3

Mean, Median and Mode are all equal

Range is the smallest

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, 161Mean = 153.5

Median = 153

Mode = 150Range = 15

Mean, is the greatest

Range is the smallest

Mean > Median

Median > Mode

Mode < Mean

Calculating and Interpreting

How does an outlier affect the measures of center and range?

Test grades:

50, 70, 62, 80, 70, 76

Mean:

Median:

Mode:Range:

68

70

30

70

• Test grades:50, 70, 62, 80, 70, 76,

100Mean:Median:

Mode:

Range:

72.6

70

70

50



8, 5, 6, 5, 6, 6

Mean:

Median:

Mode:Range:

6

6

3

6

• 8, 5, 6, 5, 6, 6, 15

Mean:Median:

Mode:

Range:

6

6

10

7.3



161, 146, 158, 150, 156, 150

Mean:

Median:

Mode:Range:

153.5

153

15

150

• 161, 146, 158, 150, 156, 150, 200

Mean:Median:

Mode:

Range:

160.1

156

150

54

CHAPTER 6.6Stem-and-Leaf Plot

STEM-AND-LEAF PLOT

Arrangement of digits that is used to display and order numerical data

MAKING A STEM-AND-LEAF PLOT

60, 74, 75, 63, 78, 70, 50, 74, 52, 74, 65, 78, 54

567

0 2 4

0

0

3

4

5

4 4 5 8 8

PRACTICE ON YOUR OWN

4, 31, 22, 37, 39, 24, 2, 28, 1, 26, 28, 30, 28, 3, 20, 20, 5

0123

1 2 3 4 5

0 0 2 4 6 8 8 8

0 1 7 9

CHAPTER 6.7Box-and-Whisker Plots

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

12, 5, 3, 8, 10, 7, 6, 5

Find the first, second and third quartiles

3, 5, 5, 6, 7, 8, 10, 12Second =

2

765.6

3, 5, 5, 6 7, 8, 10, 12

First = Third =2

555

2

1089

2 3 4 5 6 7 8 9 10 11 12 13

1, 12, 6, 5, 4, 7, 5, 10, 3, 4


1, 3, 4, 4, 5, 5, 6, 7, 10, 12Second =

2

555

1, 3, 4, 4, 5 5, 6, 7, 10, 12

First = Third =4 7

1 2 3 4 5 6 7 8 9 10 11 12

6, 7, 10, 6, 2, 8, 7, 7, 8


2, 6, 6, 7, 7, 7, 8, 8, 10Second =7

2, 6, 6, 7 7, 8, 8, 10

First = Third =6 8

1 2 3 4 5 6 7 8 9 10 11 12

2

66

2

88

STATISTICSBell Curve, Standard Deviation, Z-curve, etc.

What’s Normal?

Descriptive Statistics

What is standard deviation?

- What is the 68-95-99.7 rule in a normal distribution?

- measures the spread of the data from the mean

mean :

s. deviation :

μ

σ

Use your copy to shade in

the regions shown.

The 68-95-99.7 RuleFor 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.

Notation

x

.x vs μSigma notation: sum of all the elements Average of a set

of data values

Read as x bar

sample

Read as mu

population

μ

8

i 1

i = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8

5

ii 1

x = x1 + x2 + x3 + x4 + x5

Mean Absolute Deviation

n

ii

x

n

1

Average of the DISTANCES between each data value and the mean

Variance

( )n

ii

x

n

2

1

Average of the squares of the differences between each data value and the mean

Standard Deviation

( )n

ii

x

n

2

1

Square root of the variance

Warm Up

1) 26 – 9p = -12) 6m – 3 = 10 - 6(2 – m)

-9p = -27

p = 3

6m – 3 = 10 – 12 + 6m

6m – 3 = -2 + 6m

-3 = -2

No solution3) S = 2πrh, solve for h

hr

S

2

4) Name the property(5 + x)6 = 6(5 + x)

Commutative

5) Name the property9 + 0 = 9 Identity

Measures of Dispersion

• describes the average distance from the mean

• describes the spread of the data

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


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

x

-170

-46

-43

-39

-35

10

57

62

89

115


• What is the sum of the differences from the mean?

-170 – 46 – 43 – 39 – 35 + 10 + 57 + 62 + 89 + 115 =0

• Will this always happen?

• What can be done to getting around the problem of always getting zero?

• Test grades: 50, 70, 62, 80, 70, 76

Is there a way to get rid of the negatives?


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

x

115

89

62

57

10

35

39

43

46

170


• What is the sum of the absolute value distances? 666

• The Mean Absolute Deviation = 66.6

The 68-95-99.7 Rule

SAT problem

956 1036 1116 1196 1276 1356 1436

How many SAT scores fall within one standard deviation from the mean?

What % of the data does this represent? 70%

7

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.

10 90 30 2050 30 60 4070 40 30 6080 20

What is the mean for the data?

10 90 30 2050 30 60 4070 40 30 6080 20

45

Distance from the mean

Mean = 45

2030

50

6070

8090

30

10

30

40 40

60

20

What if we find the average of the difference between each data value and the mean?

Mean = 45

2030

50

6070

8090

30

10

30

40 40

60

20

-35 -15

5

15 25

-15

3545

-25

-15-5 -5

15

-25

-35-15+5+15+25-15+35+45-25-15-5-5+15-25 = 0

What if we find the average of the DISTANCES from each data value to the mean?

Mean = 45

2030

50

6070

8090

30

10

30

40 40

60

20

3515

5

15 25

3545

1525

15 5 5

15

25

35+15+5+15+25+15+35+45+25+15+5+5+15+25= 14

280 14 =20

One Standard Deviation from the mean

Mean = 45

2030

50

6070

8090

30

10

30

40 40

60

20

=68.222

=21.778

Calculating Standard Deviation

How much time does it take for a dead cell phone battery to completely recharge?

Calculating Standard DeviationMr. 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.

3.75 3.25 4 4.5 4.75

3.75 4 3.5 4.25 5

4.25 4 3.75 4.5 2.5

4 4.5 3.5 4 4.25

Calculating Standard Deviation

What are the mean, mode, and median of the data?

mean: 4, mode: 4, median: 4

Calculating Standard DeviationCalculate 1-Var Stats

σx = 0.548

What does the standard deviation represent in this data?

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?

A) 3.4 B) 4.3 C) 4.5 D) 5

What is the interpretation of the mean absolute value deviation of 4.3?

Sample Question for A.9Student 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?

A) 49, 5 B) 50, 5.5 C) 49.5, 5.5 D) 50, 4.5

Z-Scores for Algebra I

Descriptive Statistics

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.

ix x

s

z-score = describes the location of a data value within a distribution

referred to as a standardized value

μ

σ

ixSample Population

Z-score

In order to calculate a z-score you must know:

• a data value

• the mean

• the standard deviation

μ

σ

ix

Z-scores

What is the mean score?

What is the standard deviation?

Here are 23 test scores from Ms. Bienvenue’s stat class.

79 81 80 77 73 83 74 93 78 80 75 67 73

77 83 86 90 79 85 83 89 84 82

80.5

5.9

Z-scores

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?

Here are 23 test scores from Ms. Bienvenue’s stat class.

79 81 80 77 73 83 74 93 78 80 75 67 73

77 83 86 90 79 85 83 89 84 82

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

5.9

= 0.93

Therefore, Michele’s standardized test score is 0.93. Nearly one standard deviation above the class mean.

μ

σ

ix

9.5

5.8086

80.5

x x σ86.6

2x σ92.7

3x σ98.8

x σ74.4

2x σ68.3

3x σ

62.2

0 1 2 3-1-2-3

Michele’s Score = 86

Michele’s z-score = .93

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.

Calculate a z-score

Teacher : 63,000 Grocery Bagger: 30,000 or

Who has the better salary relative to the mean? A Big State teacher making 63,000 or a grocery bagger making 30,000?

63,000 45,0003.6

5000

30,000 21,0004.5

2000

What is the interpretation of the two z-scores?

Who has a better salary relative to the mean?


Height 46 51 50 42 56 48 57

Which students’ heights have a z-score greater than 1?

A) All of themB) Bill, Carrie, Ed and Gus

C) Ed and Gus

D) None of them

Mean = 50

Standard Deviation = 5.3

501

5.3

x

55.3x


Height 46 51 50 42 56 48 57

Which students have a z-score less than -2?

A) All of them

B) Dan and Andy

C) Only Dan

D) None of them

Mean = 50


502

5.3

x

39.4x


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


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?

A) Less than -5

B) Between -5 and -1

C) Between -1 and 0

D) Greater than 0

Mean = 125

Standard Deviation = 20

120 125

20

z

1

4z


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?

A) 36

B) 35

C) 34.5

D) 32.5

Mean = 30


302

2.5

x

35x


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



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.



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

c hapter 6. c hapter 6.6 measures of center mean, median, mode and range

Documents

middle numbers

measures of centermean

numbersmodethe number

standard deviation

moderangethe largest

leaf plotstem

leaf plot60

test grades