What if. . . .
• You recently finished taking a test that you
received a score of 90
• It was out of 200 points
• The highest score was 110
• The average score was 95
• The lowest score was 90
Z-score
• A mathematical way to modify an
individual raw score so that the result
conveys the score’s relationship to the
mean and standard deviation of the other
scores
Z-score
• Ingredients:
X
Raw score
Mean of scores
S
The standard deviation of scores
Z-score
What it does
• xTells you how far from the mean
you are and if you are > or < the mean
• S Tells you the “size” of this difference
Example
• Sample 1:
X=8
=6
S =5
Example
• Sample 1:
X=8
=6
S =5
Z score = .4
Example
• Sample 1:
X=8
=6
S = 1.25
Example
• Sample 1:
X=8
=6
S = 1.25
Z-score = 1.6
6
=6
S=5
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
6
=6
S=5
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
6
=6
S = 1.25
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
6
=6
S=5
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
6
=6
S = 1.25
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
6
=6
S=5
X=8
5
4
3
2
1
0
Z = .4
1
2
3
4
5
6
7
8
9
10
11
X
6
=6
5
4
3
S = 1.25
2
1
X=8
0
1
2
3
4
5
6
7
8
Z=1.6
X
9
10
11
Bobby
Raw
Score
26
Z-score
Raw
Score
26
Cindy
4
66
Greg
8
62
Peter
8
70
Z-score
Bobby
Raw
Score
26
Cindy
4
66
Greg
8
62
Peter
8
70
= 11.5
S = 8.52
Z-score
Raw
Score
26
Z-score
Z-score
Bobby
Raw
Score
26
1.70
Raw
Score
26
Cindy
4
-.88
66
Greg
8
-.41
62
Peter
8
-.41
70
= 11.5
S = 8.52
Z-score
Z-score
Bobby
Raw
Score
26
1.70
Raw
Score
26
Cindy
4
-.88
66
Greg
8
-.41
62
Peter
8
-.41
70
= 11.5
S = 8.52
= 56
S =17.55
Z-score
Z-score
Bobby
Raw
Score
26
Z-score
1.70
Raw
Score
26
Cindy
4
-.88
66
.57
Greg
8
-.41
62
.34
Peter
8
-.41
70
.80
= 11.5
S = 8.52
= 56
S =17.55
-1.71
Practice
• SAT and GRE
•
•
•
•
•
•
Mean = 500
S = 100
Your scores: Verbal = 400
Analyt = 500
Quant = 700
How did you do on each section?
Practice
• SAT and GRE
•
•
•
•
•
•
Mean = 500
S = 100
Your scores: Verbal = 400 Z = -1
Analyt = 500 Z = 0
Quant = 700 Z = 2
How did you do on each section?
Practice
• The history teacher Mr. Hand announced
that the lowest test score for each student
would be dropped. Jeff scored a 85 on his
first test. The mean was 74 and the SD
was 4. On the second exam, he made
150. The class mean was 140 and the SD
was 15. On the third exam, the mean was
35 and the SD was 5. Jeff got 40. Which
test should be dropped?
Practice
• Test #1
Z = (85 - 74) / 4 = 2.75
• Test #2
Z = (150 - 140) / 15 = .67
• Test #3
Z = (40 - 35) / 5 = 1.00
Practice
Time
(sec)
30
Distance
(feet)
6
Joey
40
8
Ross
25
4
Monica
45
10
Chandler
33
9
Rachel
Did Ross do worse in the endurance challenge
then in the throwing challenge? Did Monica do
better in the throwing challenge than the
endurance?
Time
(sec)
30
Distance
(feet)
6
Joey
40
8
Ross
25
4
Monica
45
10
Chandler
33
9
Rachel
Practice
Time
(sec)
30
Distance
(feet)
6
Joey
40
8
Ross
25
4
Monica
45
10
Chandler
33
9
Rachel
= 34.6
S = 7.12
= 7.4
S = 2.15
Practice
Rachel
Time
(sec)
30
Distance
(feet)
-.65
6
Joey
40
.76
8
Ross
25
-1.35
4
Monica
45
1.46
10
Chandler
33
-.22
9
= 34.6
S = 7.12
= 7.4
S = 2.15
Practice
Rachel
Time
(sec)
30
Distance
(feet)
-.65
6
-.65
Joey
40
.76
8
.28
Ross
25
-1.35
4
-1.58
Monica
45
1.46
10
1.21
Chandler
33
-.22
9
.74
= 34.6
S = 7.12
= 7.4
S = 2.15
Ross did worse in the throwing challenge than
the endurance and Monica did better in the
endurance than the throwing challenge.
Rachel
Time
(sec)
30
Distance
(feet)
-.65
6
-.65
Joey
40
.76
8
.28
Ross
25
-1.35
4
-1.58
Monica
45
1.46
10
1.21
Chandler
33
-.22
9
.74
= 34.6
S = 7.12
= 7.4
S = 2.15
Z-Scores
• A distribution of scores has a standard
deviation = 10. Find the z-score
corresponding to each of the following
values.
•
•
•
•
A score that is 20 points above the mean
A score 10 points below the mean
A score 15 points above the mean
A score 30 points below the mean
Z-Scores
• A distribution of scores has a standard
deviation = 10. Find the z-score
corresponding to each of the following
values.
•
•
•
•
A score that is 20 points above the mean = 2.00
A score 10 points below the mean = -1.00
A score 15 points above the mean = 1.50
A score 30 points below the mean = -3.00
Z-Scores
• A score that is 12 points above the mean
corresponds to a Z-score of Z = 2.00.
What is the standard deviation for this
population?
Z-Scores
• A score that is 12 points above the mean
corresponds to a Z-score of Z = 2.00.
What is the standard deviation for this
population?
• 12 / y = 2
• y=6
One-Step Beyond
• For a population of exam scores, a score
of X = 58 corresponds to Z = .50 and a
score of X = 46 corresponds to Z = -1.00.
Find the mean and standard deviation for
the population.
Z-Scores
• 1. Sketch out the distribution to help
• 2. Notice that the difference between the
two raw scores (X = 58 and X = 46) is 12
raw units.
• 3. Notice the difference between the two
raw scores is 1.5 SD.
Z-Scores
• 4. Thus, 1.5 (SD) = 12
• 5. SD = 8
• 6. Plug in the SD into either Z-score
formula
Z-Scores
• 7. Z score = 46 - y / 8 = -1.00
• Z score = 46 - 54 / 8 = -1.00
• Mean = 54!
• SD = 8