Standard Scores
Dr. Richard Jackson
jackson_r@mercer.edu
© Mercer University 2005
All Rights Reserved
Standard Scores (SS) and the
Unit Normal Curve
Example: SAT and GRE
Standard Scores (SS) and the
Unit Normal Curve
SS is any measurement (score) that has
been transformed from a raw score to a
more meaningful score
Example: SAT and GRE
SAT Scores
Example: You scored 600 on Math section
50%
X = 500
SD = 100
X ± 1 SD = 68% of subjects
You scored at the 84th centile
(50% + 34%)
f
34%
400
- 1 SD
34%
500
600
+1 SD
Z Score
Special type of standardized score
Represents measures that have been
transformed from raw scores /measures
Represents the number of standard
deviations a particular measure/score is
above or below the mean
Z Score
Formula:
Z=
X-X
s
x
X
X-X
Z
70
60
50
40
30
20
10
0
-10
-20
+2.00
+1.00
0
-1.00
-2.00
X = 50
SD = 10
Raw
30
40
50
60
70
Z
-2
-1
0
+1
+2
Z Scores
The mean of all Z
scores is 0
The SD of all Z
scores is 1
All GRE scores
are transformed
into scores with
mean of 500 and
SD of 100 to
make them more
meaningful
X = 500
SD = 100
f
68%
95%
99%
200
-3
300
-2
400
-1
500
0
600
+1
700
+2
800
+3
Transforming Raw Scores into SS
Formula:
SS =
(
) + (Z) (
what you want
your X to be
what you want
your SD to be
)
Transforming Raw Scores into SS
Example
Converting raw score
of 80 to SS with a X
of 500 and SD of 100
Steps
X = 70
SD = 10
80 - 70
Z=
10
Z = +1.00
f
Calculate Z Score
Choose what you want
your mean and SD to be
X = 500
SD = 100
Plug into the SS
equation
SS = 500 + 1 (100) = 600
RAW 60
Z
-1
70
80
0
+1
SS 400 500 600
Other Example of SS
IQ Scores
X = 100
SD = 15
IQ of 130 is 2 SD’s above
the mean and it places you at
the 97.5 centile
Only 2.5% of people scored
higher than you
2.5%
2.5%
95%
SS
70
85 100
115
130
Normal Curve
Bell Shaped
Has its max y value at its mean
Includes approximately 3 SD’s
on each side
Not skewed
Mesokurtic
Unit Normal Curve
Total Area Under a Curve (AUC) is
regarded as being equal to Unity
(or 1)
X=0
SD = 1
y
f
x
0
Relationship of AUC to Proportion
of Subjects in Study
y
f
x
0
Relationship of AUC to Proportion
of Subjects in Study
Table IV
Normal curve area
The numbers in body
of table represent the
AUC between the
mean and a particular
Z Score value
Z
.00
.01
0.0
0.1
0.2
0.3
0.4
.0000
.0398
.0793
.1179
.1554
.0040
.0438
.0832
.1217
.1591
1.3
1.4
1.5
.4032
.4192
.4332
.4049
.4207
.4345
1.6
1.7
1.8
1.9
2.0
.4452
.4554
.4641
.4713
.4772
.4463
.4564
.4649
.4719
.4778
Table IV Normal Curve Areas
Examples
Z = +1.50
1.50
from
Table IV
Z
.00
.01
0.0
0.1
0.2
0.3
.0000
.0398
.0793
.1179
.0040
.0438
.0832
.1217
1.3
1.4
1.5
.4032
.4192
.4332
.4049
.4207
.4345
0.4332
50%
0.4332
(43.32%)
What % of subjects fall below
Z score of 1.5?
50% + 43.32% = 93.32%
C93.32
0
+1.50
Table IV Normal Curve Areas
Examples
Z = +2.00
2.00
from
Table IV
Z
.00
.01
0.0
0.1
0.2
.0000
.0398
.0793
.0040
.0438
.0832
1.7
1.8
1.9
2.0
.4554
.4641
.4713
.4772
.4564
.4649
.4719
.4778
0.4772
0.500
0.4772
(47.72%)
0
+2.00
Examples
Find the AUC between
Z=1.50 and Z=2.00
0.500
1.50
from
Table IV
0.4332
2.00
from
Table IV
0.4772
0.4772 - 0.4332 = 0.0440 (4.4%)
Z
0.0440
(4.4%)
+1.5 +2.0
Example
Assume that among diabetics the fasting blood level of
glucose is approximately normally distributed with a mean of
105 mg per 100 ml and an SD of 9 mg per 100 ml.
1. What proportion of diabetics have levels between 90
and 125mg per 100ml?
2. What level cuts off the lower 10 percent (10th centile)
of diabetics?
3. What levels equidistant from the mean encompass 95
percent of diabetics?
Active Learning Exercise:
SS and the Normal Curve
1. What proportion of diabetics have levels
between 90 and 125mg per 100ml?
X = 105
SD = 9
90 - 105
= -1.67
Z90 =
9
125 - 105
= +2.22
Z125 =
9
1.67
from
Table IV
0.4525
2.22
from
Table IV
0.4868
0.4525 + 0.4868 = 0.9393 (93.93%)
0.4525
0.4868
90 X=105 125
Active Learning Exercise:
SS and the Normal Curve
2. What level cuts off the lower 10
percent (10th centile) of diabetics?
X = 105
SD = 9
?
Z1.28
from
Table IV
X-X
Z=
s
X - 105
-1.28 =
9
X = 93.5
0.4000
0.100
0.400
X = 105
Active Learning Exercise:
SS and the Normal Curve
3. What levels equidistant from the mean
encompass 95 percent of diabetics?
Z -1.96
from
Table IV
0.4750
Z +1.96
from
Table IV
0.4750
Z=
-1.96 =
X-X
s
X - 105
9
X = 87.4
Z=
+1.96 =
X-X
s
X - 105
9
X = 122.6
X = 105
SD = 9
0.0250
0.0250
0.4750 0.4750
Z = -1.96
X = 105
Z = +1.96
