Name that tune.
Song title? Performer(s)?
R.G. Bias | rbias@ischool.utexas.edu |
1
Scientific Method (continued)
“Finding New Information”
4/11/2011
R.G. Bias | rbias@ischool.utexas.edu |
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Objectives
 I want you to understand z scores. “Standard scores.”
 (Note – not standard deviation!)
 Pencils down. I don’t want you to take notes – just get
the concept.
-
Randolph – remember to take roll.
R.G. Bias | rbias@ischool.utexas.edu |
Which when?
Mode
Range
Median
SIQR
Mean
(µ)
SD
(σ)
-“Most common score.”
-Easy to calculate.
-Maybe be misleading.
-Capture the center.
-Not influenced by
extreme scores.
-Take every score into
account.
-Allow later
manipulations.
R.G. Bias | rbias@ischool.utexas.edu |
4
So far . . .
 . . . we’ve talked of summarizing ONE
distribution of scores.
– By ordering the scores.
– By organizing them in graphs/tables/charts.
– By calculating a measure of central tendency
and a measure of dispersion.
 What happens when we want to compare
TWO distributions of scores?
5
R.G. Bias | rbias@ischool.utexas.edu |
“Now, why would I want to do that”?
 Is your child taller or heavier?
 Is this month’s SAT test any easier or harder than last
month’s?
 Is my 91 in my History class better than my 95 in my
Math class?
 Is the new library lay-out better than the old one?
 Can more employees sign up, more quickly, for benefits
with our new intranet site than with our old one?
 Did my class perform better on the TAKS test than they
did on the TAAS test?
6
R.G. Bias | rbias@ischool.utexas.edu |
Well?
 COULD it be the case that your 91 in your
History class is better than your 95 in your
Math class?
 How?
7
R.G. Bias | rbias@ischool.utexas.edu |
What if . . .
 The mean in History
was 50, and the
mean in Digital
Libraries Math was
99?
 (What, besides the
fact that everyone
else is trying to drop
the History class!)
 So:
You
Mean
History
91
50
Math
95
99
R.G. Bias | rbias@ischool.utexas.edu |
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The Point
 As I said last week, you need to know
BOTH a measure of central tendency AND
a measure of spread to understand a
distribution.
 BUT STILL, this can be convoluted . . .
 “Well, daughter, how are you doing in
school this semester”?
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R.G. Bias | rbias@ischool.utexas.edu |
“Well, Mom . . .
 “. . . I had a 91 on my History test but the
mean was 50 and the standard deviation
was 12. But I got a 95 on Math test,
whereas the mean on that test was 99 with
a standard deviation of 1.”
 Of course, your mom’s reaction will be,
“Just call home more often, dear.”
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R.G. Bias | rbias@ischool.utexas.edu |
Wouldn’t it be nice . . .
 . . . if there could be one score we could
use for BOTH classes, for BOTH the TAKS
test and the TAAS test, for BOTH your
child’s height and weight?
 There is – and it’s called the “standard
score,” or “z score.”
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R.G. Bias | rbias@ischool.utexas.edu |
Standard Score
 z = (X - µ)/σ
 “Hunh”?
 Each score can be expressed as the number of
standard deviations it is from the mean of its
own distribution.
 “Hunh”?
 (X - µ) – This is how far the score is from the
mean. (Note: Could be negative! No squaring,
this time.)
 Then divide by the SD to figure out how many
SDs you are from the mean.
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R.G. Bias | rbias@ischool.utexas.edu |
Z scores (cont’d.)
 z = (X - µ)/σ
 Notice, if your score (X) equals the mean,
then z is, what?
 If your score equals the mean PLUS one
standard deviation, then z is, what?
 If your score equals the mean MINUS one
standard deviation, then z is, what?
13
R.G. Bias | rbias@ischool.utexas.edu |
An example
Test 1
Test 2
Kris
76
76
Robin
52
86
Marty
58
80
Terry
58
90
ΣX
244
332
µ
Mode, median?
R.G. Bias | rbias@ischool.utexas.edu |
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Let’s calculate σ – Test 1 – Mean = 61
X
X-µ
(X-µ)2
Kris
76
15
225
Robin
52
-9
81
Marty
58
-3
9
Terry
58
-3
9
Σ
244
0
324
/N
61
σ
81
9
R.G. Bias | rbias@ischool.utexas.edu |
15
Let’s calculate σ – Test 2= Mean = 83
X
X-µ
(X-µ)2
Kris
76
-7
49
Robin
86
3
9
Marty
80
-3
9
Terry
90
7
49
Σ
332
0
116
/N
83
σ
29
5.4
R.G. Bias | rbias@ischool.utexas.edu |
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So . . . z = (X - µ)/σ
 Kris had a 76 on both tests.
 Test 1 - µ = 61, σ = 9
– So her z score was (76-61)/9 or 15/9 or 1.67. So we say that
Kris’s score was 1.67 standard deviations above the mean. Her
z score was 1.67.
 Test 2 - µ = 83, σ = 5.4
– So her z score was (76-83)/5.4 or -7/5.4 or –1.3. So we say that
Kris’s score was 1.3 standard deviations BELOW the mean. Her
z score was -1.3.
 Given what I said earlier about two-thirds of the scores
being within one standard deviation of the mean . . . .
 Wouldn’t it be nice if we knew exactly how many . . . ?
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R.G. Bias | rbias@ischool.utexas.edu |
z = (X - µ)/σ
 If I tell you that the average IQ score is
100, and that the SD of IQ scores is 16,
and that Bob’s IQ score is 2 SD above the
mean, what’s Bob’s IQ?
 If I tell you that your 75 was 1.5 standard
deviations below the mean of a test that
had a mean score of 90, what was the SD
of that test?
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R.G. Bias | rbias@ischool.utexas.edu |
z scores – table values
 z = (X - µ)/σ
 It is often the case that we want to know
“What percentage of the scores are above
(or below) a certain other score”?
 Asked another way, “What is the area
under the curve, beyond a certain point”?
 THIS is why we calculate a z score, and
the way we do it is with the z table.
19
R.G. Bias | rbias@ischool.utexas.edu |
Z distribution
20
R.G. Bias | rbias@ischool.utexas.edu |
Study Guide . . .
 Later tonight, online.
R.G. Bias | rbias@ischool.utexas.edu | 21