Review: Standard Deviation and the Standard Normal Distribution

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Review Standard Deviation and theStandard
Normal Distribution

• Sociology 315, Winter 2002
• Week 1 Jan. 7-11

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In this review we will be considering

• standard scores
• the standard normal distribution
• characteristics of normal distributions
• transforming raw scores to z-scores

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Standard Scores (Z-Scores) An ordinary score
converted so that it better describes the scores
place in its distribution. A z-score is the
number of standard deviations the score is above
the mean (for positive z-scores) or below the
mean (negative z-scores).
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X ( years served) (Xi X) (Xi X)2
7 1 1 8 2 4 8 2 4
7 1 1 3 -3 9 1 -5 25
6 0 0 9 3 9 3 -3 9
8 2 4 60 0 66
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s2 ? (Xi X ) 2 N
66 10 6.6 s ? 6.6 2.57
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What is the z-score for Mayor Hernandez who
served 9 years?
s
s
s
s
6.00
3.44
.86
8.57
11.14
Mean of years for 10 mayors
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Mayor Hernandez served 9 years s 2.57 X
6.0 z X - X s 9 - 6
2.57 1.17 z-score
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What is the z-score for a mayor who has served 6
years?
s
s
s
s
6.00
3.44
.86
8.57
11.14
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Z-Score
0
1
2
-2
-1
3.44
.86
8.57
11.14
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Raw Score
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What is the of years in office for someone with
a score of -0.7?
X z s X (-0.7) (2.57) 6 4.2
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Notation Sample z X - X and X z s
X s Population z X ? and X
z? ? ?
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(Standard) Normal Distributions are mathematical
abstractions that have a population mean (?)
of 0, and a population standard deviation (?) of
1 with a total area under the curve equal to
1.00 in other words, areas under the curve
are proportions of the total are symmetrical
have curves that approach, but never touch, zero
meaning that, in theory, there could always
be lower or higher values
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Transforming Raw Scores to z Scores
25
35
40
45
50
55
30
40
5
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What is the z-score of a score of 35? z
X ? ? 35 -
40 5 -1 What is the
score equivalent to a z-score of 2? X
z? ? (2) (5) 40
50
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Midterm Test Scores (Score and Percentage)
N Min. Max. Mean Std. Dev.
Midterm One x/70 44 30.0 66.0
55.4659 9.5079 OR
N Min. Max. Mean Std. Dev.
Midterm 44 42.86 94.29 79.2370 13.5827
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z X X s 52 55.4659
9.5079 -.3645
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(A) (B) (C) Area
between Area mean and
beyond z z z 0.36
.1406 .3594
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-.36
.3594
.1406
0
1 s
-1 s
-2 s
-3 s
3 s
2 s
Where does a z-score of -.36 lie? Area between
the mean and z is .1406 the area beyond z is
.3594
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s 5.5
s 9.5
40 45 50 55 60 65 70
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A person scores 81 on a test of verbal ability
and 6.4 on a test of quantitative ability. For
the verbal ability test, the mean for people in
general is 50, and the standard deviation is
20. For the quantitative ability test, the mean
for people in general is 0 and the standard
deviation is 5. Which is this persons stronger
ability, verbal or quantitative? Explain how
you arrived at your answer.
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