z-Scores
Standardized Scores
Standardizing scores
• With non-equivalent assessments it is not
possible to develop additive summary statistics.
– e.g., averaging the scores from assessments with
different maximum scores.
• In classrooms we solve this problem by
assigning grades which represent a uniform
evaluation for every assessment.
– i.e., A, B, C, D, F
Standardizing scores
• Grades often have a subjective interpretation
which makes comparing students using average
grades problematic.
• Assuming that assessments themselves are
minimally subjective (i.e., well-designed) some
consistent measure of student achievement can
be used across assessments.
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
Summary Statistics
• The summary statistics used to describe
interval data are mean and standard deviation.
• Standard deviation represents the average
distance of observed scores from the mean.
• Any given score can be represented as the
number of standard deviations from the mean.
z-Scores
• Hence:
A z-score is the distance a given score is from the
mean divided by the standard deviation.
(score – mean)/standard deviation = z
• A z-score is a given score translated into units
of standard deviation.
Common standard score for any interval assessment
z-Score Computation
• Subtract the mean of the distribution from
the target score.
• Divide by the standard deviation.
• Positive z-scores are above the mean.
• Negative z-scores are below the mean.
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
Mean = 5.74
sd = 2.61
z-Score Computation
• For a score of 9:
z = (9-5.74)/2.61 = 1.25 sd above the mean
• For a score of 4:
z = (4-5.74)/2.61 = - 0.67 sd below the mean
• Positive z-scores are above the mean.
• Negative z-scores are below the mean.
What is Molly’s z-score 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
z-Scores
A special case with
Normal Distributions
Normal Distribution
12
10
Histogram
of a frequency
distribution
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
Lots of naturally occurring phenomena distribute
normally if you have enough data points.
Normal Distribution
12
10
Histogram
of a frequency
distribution
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
Things that distribute normally have lots of examples in the
middle of the range of possibilities and fewer examples that
are farther from the middle of the range.
Normal Distribution
12
10
Histogram
of a frequency
distribution
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
Things that are normally distributed have equal
numbers of examples that are above and below the
average example. Normal distributions are symmetrical.
Normal Distribution
12
Histogram
of a frequency
10
distribution
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.
4
2
0
Examples of Normally
Distributed Phenomena
12345678910
11 2
13
14
15
16
17
18
19
20
21
22 3
24
25
26
27
28
•
•
•
•
•
•
The height of 12 year old boys
The life of a 60 watt light bulb
The weight of Red Delicious apples
Recovery time of surgery patients
Spatial problem solving ability of 10 year olds
Scores on norm-referenced tests of anything
Normal Curves
A line connecting the tops of the bars of a histogram
Normal Curves
In a normal distribution the mean, median and mode appear at
the same point.
Normal Curves
Mean
1 Standard Deviation
Normal Curves
1 Standard Deviation
Mean
50.0
Normal Curves
1 Standard Deviation
Mean
50.0
34.13
Normal Curves
2 Standard Deviations
Mean
50.0
34.13
13.59
Normal Curves
0.13%
0.13%
2.14% 13.59%
Standard
Deviations
-3
-2
-1
34.13%
34.13%
0
+1
13.59% 2.14%
+2
+3
Cognitive abilities are normally distributed.
Then, if the tests are designed carefully the assessment of
cognitive ability should be normally distributed as well.
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.
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 = .8023
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 = .1611
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
Practice
• Open the Excel file from the course
webpage: OWM Data
• What is the percentile rank of a score of 10
on the variable pre?
• What is the percentile rank of a score of 10
on the variable delayed?
Normal Curves
0.13%
2.14% 13.59%
Standard Deviations (z-scores) -3
Cumulative Percentages
-2
0.01%
Percentile Equivalents
-1
2.3%
1
34.13%
5
15.9%
1
0
2
0
34.13%
0
50.0%
IQ Scores
55
70
85
4 6
3 5
7
0 0
0 0
0
100
SAT Scores (sd 209)
400
608
817
1026
0.13%
13.59% 2.14%
+1
+2
84.1%
97.7%
8
0
9
5
9
0
+3
99.9%
9
9
115
130
145
1235
1444
1600
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?
• Explain this to Bobby’s mother in a paragraph or two and
send me what you have written.
• For a challenge (if you have time), what is the mean and
standard deviation for this test? (solution)