Z-Scores
Histogram
• A bar chart of a frequency distribution.
0—2
1—3
2—0
3—2
4—4
5—3
6—3
7—5
8—3
9—2
10—2
Normal Distribution
12
10
8
6
4
2
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
With some frequency distributions the histograms have
lots of examples in the middle and fewer that are higher
and lower.
Normal Distribution
12
10
8
6
4
2
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Measurements based on an underlying random
phenomena distribute normally if you have enough
data points.
Normal Distribution
12
10
8
6
4
2
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Normal Distribution
12
10
8
6
4
2
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
A Normal Curve is the theoretical line that represents all of the responses in a
normal distribution. The area under the curve encloses the frequency
distribution of the normally distributed phenomena.
Characteristics of Normal Curves
A line connecting the tops of the bars of a histogram.
Mean, median and mode appear at the same point.
Lots of responses in the middle fewer in the extremes.
One More Interesting Characteristic of Normal Curves
Mean
1 Standard Deviation
One More Interesting Characteristic of Normal Curves
1 Standard Deviation
Mean
50.0
One More Interesting Characteristic of Normal Curves
1 Standard Deviation
Mean
50.0
34.13
One More Interesting Characteristic of Normal Curves
2 Standard Deviations
Mean
50.0
34.13
13.59
Cognitive abilities are normally distributed.
A test score
Then, if the tests are designed carefully the assessment of
cognitive ability should be normally distributed as well.
A test score
Results from a standardized test are, by definition,
normal distributed.
A test score
z Scores
Mean
-3
-2
-1
1
2
3
SD
A test score
If you know the percentage of scores that are lower than the target
score you will know the Percentile Rank of the target score.
Normal Curves
2 Standard Deviations
Mean
50.0
34.13
13.59
1. The relationship between SD and percentage of the area
under the curve is constant regardless of the distribution.
2. If the mean and SD of the distribution are known then the
percentage of scores lower than every possible score can be
computed.
3. In other words, percentile rank of every score can be computed.
z Score Computation
• Subtract the mean of the distribution from
the target score.
• Divide by the standard deviation.
• Look up the z score on the z table.
Subtract the mean of the distribution from the target score.
Divide by the standard deviation.
Look up the z score on the z table.
Target Score
(28)
Mean
(25.69)
1 SD
(2.72)
-3
-2
-1
Percentage of scores lower
than the target score
1
2
3
SD
Number of SD away
from the mean
z Score
•
•
•
•
•
•
Target score (28)
Score - Mean (28 - 25.69 = 2.31)
Result / SD (2.31 / 2.72 = .85)
Look up on z table (0.85)
z = 0.85; area = .80
Score is in the 80th percentile
z Scores
28
25.69
-3
-2
-1
1
.85 SD
80 % of scores
2
3
SD
z Scores
Target Score
Mean
-3
-2
-1
1
2
3
SD
z Scores
23
25.69
-3
-2
-1
-.99 SD
16 % of scores
1
2
3
SD
z Score
•
•
•
•
•
•
Target score (23)
Score - Mean (23 - 25.69 = -2.69)
Result / SD (-2.69 / 2.72 = -.99)
Look up on z table (-.99)
Z = -0.99; area = .16
Score is in the 16th percentile
z Scores
23
25.69
-3
-2
-1
-.99 SD
16 % of scores
1
2
3
SD
Percentile Rankings
• z Scores
• Assume a normal distribution
• Based on knowing everyone in the
population
• Allows comparison of individual
to the whole
What is Molly’s percentile ranking
on the test? What is Karl’s
•
•
•
•
•
•
•
34
56
22
31
44
47
37
Karl
•
•
•
•
•
•
•
42
51
37
40
29
36
39
•
•
•
•
•
•
•
52
39
29
51
45
47
42
Molly
Bobby and Sally take a standardized test
that has 160 questions
• Bobby gets a raw score of 140 and has a percentile
rank of 52. Sally gets a raw score of 142 and has a
percentile rank of 67.
• This doesn’t make sense to Bobby’s mother. First,
how can 140 out of 160 be 52nd percentile. And
second, why should just a couple of points on the
test make such a huge percentile rank difference?
What are you going to tell her?