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z - SCORES
• standard score: allows comparison of
scores from different distributions
• z-score: standard score measuring in units
of standard deviations
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?
• 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!
• 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!
Calculating z-scores
• Your z-score in Dr. Difficult’s class was two
standard deviations above the mean. That
means z = +2.00.
• Your z-score in Dr. Easy’s class was half a
standard deviation below the mean. That
means z = -.50.
z - score formula
z
x
x
70  60
z
  2.00
5
85  90
z
 - 0.50
10
Cool Things About z-scores
• Any distribution, when converted to zscores, has
• a mean of zero
• a standard deviation of one
• the same shape as the raw score distribution
Finding Percentile Ranks with z-Scores
•
•
•
•
This only works for a normal distribution!
You have to know the  and x.
All it takes is a little calculus....
But the answer is in the back of the book.
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.
Another Example
Sam got a score of 515 on a normally
distributed aptitude test. The  of the test
is 500, with a  of 30. What is Sam’s
percentile rank?

500
515
STEP 1: Convert to a z-score.
z = (515-500)/30 = .50
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
STEP 3: Add the area below the mean.
total area below = .1915 + .5000 = .6915
STEP 4: Convert the proportion to a
percentage.
percentile rank = 69%
A Tricky Example
Sam got a score of 470 on a normally
distributed aptitude test. The  of the test
is 500, with a  of 30. What is Sam’s
percentile rank?
470

500
STEP 1: Convert to a z-score.
z = (470-500)/30 = -1.00
STEP 2: Look up the z-score in the Unit
Normal Table. Find the area beyond z.
area beyond z = .1587
STEP 3: Convert to a percentage.
.1587 = 16%
Working Backwards
The  of the test is 500, with a  of 30.
What score is at the 90th percentile?
90% or .9000

500
X=?
STEP 1: Look up the z-score.
proportion beyond z = .1000
z = +1.28
STEP 2: Convert the z-score into raw
score units, using x =  + z
x = 500 + (1.28)(30) = 500 + 38.40 = 538.40
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 = .0987 + .5000 = .5987
What proportion is between a z of -.25 and a
z of +.25?
area between mean and z = .0987
proportion between = .0987 + .0987 = .1974