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Chapter 3
Z Scores & the Normal Distribution
Part 1
Z Scores
 Number of standard deviations a score
is above or below the mean
 Formula to change a raw score to a Z
score:
(X  M )
Z
SD
Sign of z indicates whether score is above or
below mean.
Magnitude of z – how many SDs above/below
Example in class:
Z Scores
 Formula to change a Z score to a raw
score:
X  ( Z )( SD)  M
 Example?
Z scores
• Distribution of Z scores
– Mean = 0
– Standard deviation = 1
• It’s a standard scale, so z scores can be
used to compare 2 scores from different
distributions or original scales
– Z= +3 represents score that is very deviant
from the mean, no matter what the distribution
Z as common unit of
comparison
• Comparing SAT and ACT scores
• Which is a higher score? Getting a 620 on
the SAT-verbal or a 27 on the ACT-verbal?
– Information about ACT and SAT M and SD:
– Did this person do better on the SAT or ACT?
The Normal Distribution
• In a normal curve (normal distribution), there
will always be a standard % of scores
between the mean and 1 and 2 standard
deviations from the mean:
–
–
–
–
50% all scores fall above the mean, 50% below
34% betw M and +1 SD, 34% betw M and –1SD
14% betw +1 and +2 SD, 14% betw –1 and –2SD
2% above +2 SD and below –2 SD
The Normal Distribution
• Can remember this as the ‘50-34-14 rule’
• Appendix A is the normal curve table with Z
scores
– Gives the precise % of scores between the mean
(which has a Z score of 0) and any other Z score
• What % of scores are betw M and z = .75?
– Table lists only positive Z scores, but since ND is
symmetrical, same % for neg z scores (just look up its
corresponding pos z)
Using the Z table: From z to %
• Steps for figuring the % of scores above or
below a particular raw or Z score:
1. Table uses z scores, so convert raw score to Z score
(if necessary)
2. Draw normal curve, indicate where the Z score falls
on it, shade in the area for which you are finding the
%
3. Make rough estimate of shaded area’s percentage
(using 50%-34%-14% rule):
– what % does it look like the shaded area covers?
(use your judgment – we’ll check on this later…)
(cont.)
4. Find exact % using normal curve table
– that is, look up the z score you calculated in step
1 and find the % associated with it.
5. If needed, add or subtract 50% from
this percentage
– since table only gives % between M and Z; if
you’re interested in % above Z, subtract from 50%
(see example)
6. Check to determine if your answer makes
sense given the graph you drew in Step 3