Z-Scores for Algebra I

Z-Scores for Algebra I

Descriptive Statistics

Mean

• Mean = the average of the numbers. To find the mean of a set of numbers, you must add up the numbers then divide by the number of numbers.

• Example: 18 23 10 39 22 17 16 15

18+23+10+39+22+17+16+15 = 160

160

= 208

Mean Practice

• Find the mean of the following sets of numbers.

25 30 15 40 20 80 60 50

34 20 56 82 78 30

Mean Practice

20 + 30 + 15 + 40 + 20 + 80 + 60 + 50 = 320

320

34 + 20 + 56 +82 + 78 + 30 = 300

300

8 = 40

6 = 50

How Close is Close?

Z-Scores

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 Mr. Barnes 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

What is the mean score?

What is the standard deviation?

Here are 23 test scores from Mr. Barnes 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

6.1

Z-scores

The bold score is Michele’s. How did she perform relative to her classmates?

Here are 23 test scores from Mr. Barnes 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

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 Mr. Barnes 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

6.1

= 0.90

Therefore, Michele’s standardized test score is 0.90. Nearly one standard deviation above the class mean.

μ

σ

ix

86 – 80.5 6.1

Z=

Z-score Number Line

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 = .90

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 Food Lion 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?

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? Both scores are above the mean.

Who has a better salary relative to the mean? The grocery bagger.

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 themB) Bill, Carrie, Ed and Gus

C) Ed and Gus

D) None of them

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 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

Sample Question for A.9

Student Andy Bill Carrie Dan Ed Frank Gus

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

Sample Question for A.9

Student Andy Bill Carrie Dan Ed Frank Gus

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

Standard Deviation = 5.3

502

5.3

x

39.4x

Sample Question for A.9

Student 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

Sample Question for A.9

Student 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

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?

A) Less than -5

B) Between -5 and -1

C) Between -1 and 0

D) Greater than 0

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?

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

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?

A) 36

B) 35

C) 34.5

D) 32.5

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?

A) 36

B) 35

C) 34.5

D) 32.5

Mean = 30

Standard Deviation = 2.5

302

2.5

x

35x

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.

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

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.

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.

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?

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