1. Z Scores & the Normal Distribution
Chapter 3
Z Scores & the Normal Distribution
Part 1
Thurs. Aug. 29, 2013

2. Z Scores
Number of standard deviations a score is above or below the mean
Formula to change a raw score to a Z score:
Sign of z indicates whether score is above or below mean.
Magnitude of z – how many SDs above/below
Example in class:

3. Z Scores
Formula to change a Z score to a raw score:
Example?

4. Z scores Distribution of Z scores It’s a standardized scale Mean = 0
Standard deviation = 1
It’s a standardized scale
z scores can be used to compare 2 scores from same or different distributions
Z= +3 represents score that is very deviant from the mean, no matter the distribution

5. 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?

6. 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

7. The Normal Distribution
Can remember this as the ‘ 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)

8. 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…)

9. (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
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