1. Standard Scores Dr. Richard Jackson jackson_r@mercer.edu © Mercer University 2005 All Rights Reserved

2. Standard Scores (SS) and the Unit Normal Curve Example: SAT and GRE

3. 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

4. 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% 500 34% 50% +1 SD- 1 SD X = 500 SD = 100

5. 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

6. Z Score X - X s Z = Formula: 5060704030 0+1+2-2 Raw Z XX - XZ 7060504030 20 10 0 -10 -20 +2.00 +1.00 0 -2.00 x X = 50 SD = 10

7. 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 600700800400300200 0+1+2+3-2 -3 68% 99% 95% X = 500 SD = 100

8. Transforming Raw Scores into SS Formula: SS = what you want your X to be + (Z) what you want your SD to be ()()

9. Transforming Raw Scores into SS Example X = 70 80 - 70 10 Z = X = 500 SS = 500 + 1 (100) = 600 Z = +1.00 SD = 100 SD = 10 70 f 8060RAW 0+1Z 500600400SS Converting raw score of 80 to SS with a X of 500 and SD of 100 Steps Calculate Z Score Choose what you want your mean and SD to be Plug into the SS equation

10. 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 1001151308570 SS 95% 2.5%2.5%

11. 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 0 x y

12. Relationship of AUC to Proportion of Subjects in Study f 0 x y

13. Table IV Normal curve area The numbers in body of table represent the AUC between the mean and a particular Z Score value 0.0 0.1 0.2 0.3 0.4 Z.0000.0398.0793.1179.1554.00 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0.4032.4192.4332.4452.4554.4641.4713.4772.0040.0438.0832.1217.1591.01.4049.4207.4345.4463.4564.4649.4719.4778

14. Examples Z = +1.50 0.0 0.1 0.2 0.3 Z.0000.0398.0793.1179.00 1.3 1.4 1.5.4032.4192.4332.0040.0438.0832.1217.01.4049.4207.4345 0+1.50 0.4332 (43.32%) Table IV Normal Curve Areas 50% 1.50 from Table IV 0.4332 What % of subjects fall below Z score of 1.5? 50% + 43.32% = 93.32% C 93.32

15. Examples Z = +2.00 0.0 0.1 0.2 Z.0000.0398.0793.00 1.7 1.8 1.9 2.0.4554.4641.4713.4772.0040.0438.0832.01.4564.4649.4719.4778 0+2.00 0.4772 (47.72%) Table IV Normal Curve Areas 0.500 2.00 from Table IV 0.4772

16. +2.0 0.0440 (4.4%) 0.500 Z+1.5 Examples Find the AUC between Z=1.50 and Z=2.00 2.00 from Table IV 0.4772 1.50 from Table IV 0.4332 (4.4%) 0.4772 - 0.4332 = 0.0440 +2.0 0.0440 (4.4%) 0.500 Z+1.5

17. 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 (10 th centile) of diabetics? 3. What levels equidistant from the mean encompass 95 percent of diabetics?

18. 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 9 Z 90 = = -1.67 125 - 105 9 Z 125 = = +2.22 2.22 from Table IV 0.4868 1.67 from Table IV 0.4525 (93.93%) 0.4525 + 0.4868 = 0.9393 X=105 12590 0.48680.4525

19. Active Learning Exercise: SS and the Normal Curve 2. What level cuts off the lower 10 percent (10 th centile) of diabetics? X = 105 0.100 0.400 X - 105 9 -1.28 = X = 93.5 X - X s Z = Z1.28 from Table IV 0.4000 X = 105 SD = 9 ?

20. Active Learning Exercise: SS and the Normal Curve 3. What levels equidistant from the mean encompass 95 percent of diabetics? X = 105 0.0250 0.4750 0.0250 0.4750 Z = +1.96Z = -1.96 X = 105 SD = 9 X - 105 9 -1.96 = X = 87.4 X - X s Z = X - 105 9 +1.96 = X = 122.6 X - X s Z = Z +1.96 from Table IV 0.4750 Z -1.96 from Table IV 0.4750