1. z - SCORES
standard score: allows comparison of scores from different distributions
z-score: standard score measuring in units of standard deviations

2. Comparing Scores from Different Distributions
Suppose you got a score of 70 in Dr. Difficult’s class, and you got an 85 in Dr. Easy’s class.
In relative terms, which score was better?

3. Suppose the M in Dr. Difficult’s class was 60 and the SD was 5.
So your score of 70 was two standard deviations above the mean.
That’s good!

4. In Dr. Easy’s class, the M was 90, with a SD of 10.
So your score of 85 was half of a standard deviation below the mean.
Not as good!

5. Calculating z-scores
Your z-score in Dr. Difficult’s class was two standard deviations above the mean. That means z =
Your z-score in Dr. Easy’s class was half a standard deviation below the mean. That means z = -.50.

6. z - score formula

7. Cool Things About z-scores
Any distribution, when converted to z-scores, has
a mean of zero
a standard deviation of one
the same shape as the raw score distribution

8. Finding Percentile Ranks with z-Scores
This only works for a normal distribution!
You have to know the m and sx.
All it takes is a little calculus....
But the answer is in the back of the book.

9. A Really Easy Example Suppose your score is at the mean of a distribution, and the distribution is normal. What is your percentile rank?
Answer: 50th percentile rank
The mean = the median
50% of the scores are below the median.

10. Another Example Sam got a score of 515 on a normally distributed aptitude test. The m of the test is 500, with a s of 30. What is Sam’s percentile rank?

11. m
515
500

12. STEP 1: Convert to a z-score.
STEP 2: Look up the z-score in the Normal
Curve Table. Find the area between mean
and z.
area between mean and z = .1915

13. STEP 3: Add the area below the mean.
total area below = = .6915
STEP 4: Convert the proportion to a
percentage.
percentile rank = 69%

14. A Tricky Example Sam got a score of 470 on a normally distributed aptitude test. The m of the test is 500, with a s of 30. What is Sam’s percentile rank?

15. m
470
500

16. STEP 1: Convert to a z-score.
STEP 2: Look up the z-score in the Unit
Normal Table. Find the area beyond z.
area beyond z = .1587

17. STEP 3: Convert to a percentage.
.1587 = 16%

18. Working Backwards The m of the test is 500, with a s of 30
Working Backwards The m of the test is 500, with a s of 30. What score is at the 90th percentile?

19. 90% or .9000 m
500
X=?

20. STEP 1: Look up the z-score.
proportion beyond z = z = +1.28
STEP 2: Convert the z-score into raw
score units, using x = m + zs
x = (1.28)(30) = =

21. Finding Other Proportions
What proportion is above a z of .25?
area beyond z = .4013
What proportion is above a z of -.25?
area between mean and z = .0987
proportion above = = .5987

22. What proportion is between a z of -.25 and a z of +.25?
area between mean and z = .0987
proportion between = = .1974