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Standard Scores
Dr. Richard Jackson [email protected]
© Mercer University 2005© Mercer University 2005
All Rights ReservedAll 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
X ± 1 SD = 68% of subjects
You scored at the 84th centile (50% + 34%)
f
600400
34%34%
500
34%34%
50%50%
+1 SD- 1 SD
X = 500
SD = 100
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
X - Xs
Z = Formula:
50 60 704030
0 +1 +2-1-2
Raw
Z
X X - X Z
7070
6060
5050
4040
3030
20
10
0
-10
-20
+2.00
+1.00
0
-1.00
-2.00
x
X = 50SD = 10
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
500
f
600 700 800400300200
0 +1 +2 +3-1-2-3
68%68%
99%99%95%95%
X = 500SD = 100
Transforming Raw Scores into SS
Formula:
SS = what you want your X to be + (Z) what you want
your SD to be( )( )
Transforming Raw Scores into SS
Example
X = 70
80 - 7010Z =
X = 500
SS = 500 + 1 (100) = 600
Z = +1.00
SD = 100
SD = 10
70
f
8060RAW
0 +1-1Z
500 600400SS
Converting raw score of 80 to SS with a X of 500 and SD of 100
StepsCalculate Z Score
Choose what you want your mean and SD to be
Plug into the SS equation
Other Example of SS IQ Scores
X = 100SD = 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
100 115 1308570SS
95%95%
2.5%2.5% 2.5%2.5%
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
f
0x
y
Relationship of AUC to Proportion of Subjects in Study
f
0
x
y
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
0.00.10.20.30.4
Z
.0000
.0398
.0793
.1179
.1554
.00
1.31.41.5
1.61.71.81.92.0
.4032
.4192
.4332
.4452
.4554
.4641
.4713
.4772
.0040
.0438
.0832
.1217
.1591
.01
.4049
.4207
.4345
.4463
.4564
.4649
.4719
.4778
Examples Z = +1.50
0.00.10.20.3
Z
.0000
.0398
.0793
.1179
.00
1.31.41.5
.4032
.4192
.4332
.0040
.0438
.0832
.1217
.01
.4049
.4207
.4345
0 +1.50
0.4332 (43.32%)
Table IV Normal Curve Areas
50%50%
1.50from Table
IV
0.4332
What % of subjects fall below Z score of 1.5?
50% + 43.32% = 93.32%
C93.32
Examples Z = +2.00
0.00.10.2
Z
.0000
.0398
.0793
.00
1.71.81.92.0
.4554
.4641
.4713
.4772
.0040
.0438
.0832
.01
.4564
.4649
.4719
.4778
0 +2.00
0.4772 (47.72%)
Table IV Normal Curve Areas
0.5000.500
2.00 from Table
IV
0.4772
+2.0
0.0440 (4.4%)
0.5000.500
Z +1.5
Examples Find the AUC between Z=1.50 and
Z=2.00
2.00from Table
IV
0.4772
1.50from Table
IV
0.4332
(4.4%)0.4772 - 0.4332 = 0.0440+2.0
0.0440 (4.4%)
0.5000.500
Z +1.5
ExampleAssume 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 = 105SD = 9
90 - 1059
Z90 = = -1.67
125 - 1059
Z125 = = +2.22
2.22from
Table IV 0.4868
1.67from
Table IV 0.4525
(93.93%)0.4525 + 0.4868 = 0.9393
X=105 12590
0.48680.4525
Active Learning Exercise:SS and the Normal Curve
2. What level cuts off the lower 10 percent (10th centile) of diabetics?
X = 105
0.1000.100
0.4000.400
X - 1059
-1.28 =
X = 93.5
X - X
sZ =
Z1.28 from Table IV 0.4000
X = 105SD = 9
?
Active Learning Exercise:SS and the Normal Curve
3. What levels equidistant from the mean encompass 95 percent of diabetics?
X = 105
0.02500.0250
0.47500.4750
0.02500.0250
0.47500.4750
Z = +1.96Z = -1.96
X = 105SD = 9
X - 1059
-1.96 =
X = 87.4
X - X
sZ =
X - 1059
+1.96 =
X = 122.6
X - X
sZ =
Z +1.96from
Table IV 0.4750
Z -1.96from
Table IV 0.4750