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ByDr. Mojgan Afshari
Normal Distribution
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Properties Of Normal Curve
Normal curves are symmetrical.
Normal curves are unimodal.
Normal curves have a bell-shaped form. Mean, median, and mode all have the same
value.
Contains an infinite number of cases
X
f(X)
X
f(X)
Normal distributions differby mean & standard
deviation.
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Empirical RuleFor any normal curve, approximately 68% of the values fall within 1 standarddeviation of the mean in either direction
95% of the values fall within 2 standarddeviations of the mean in either direction
99.7% of the values fall within 3 standarddeviations of the mean in either direction
A measurement would be an extreme outlierif it fell more than 3 SD above or below the mean.
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The 68-95-99.7 Rule
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Heights of Adult Women
68% of adult women are between 62.5 and 67.5 inches,95% of adult women are between 60 and 70 inches,
99.7% of adult women are between 57.5 and 72.5 inches.
Since adult women
in U.S. have a meanheight of 65 incheswith a SD of 2.5inches and heightsare bell-shaped,approximately
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Standard Scores
One use of the normal curve is to exploreStandard Scores. Standard Scores are
expressed in standard deviation units, makingit much easier to compare variables measuredon different scales.
There are many kinds of Standard Scores.The most common standard score is the z
scores.
A z score states the number of standarddeviations by which the original score liesabove or below the mean of a normal curve.
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The Standard Normal Curve A normal curve with a mean of 0 and a
standard deviation of 1 is called a standard
normal curve. It is the curve that results when
any normal curve is converted to standardizedscores and is written as:
Z ~ N (0, 1)
X
X
= 0
= 1
Z= 0
= 1
Z
NormalDistributionNormalDistribution
Standardized NormalDistribution
Standardized NormalDistribution
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Direction of a Z-score
The sign of any Z-score indicates the direction
of a score: whether that observation fell above
the mean (the positive direction) or below the
mean (the negative direction)
If a raw score is below the mean, the z-
score will be negative, and vice versa
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Computing Z-Score
where:
Zx= standardized score for a value of X = number ofstandard deviations a raw score (X-score) deviatesfrom the mean
X= an interval/ratio variable
X= the mean of X
sx= the standard deviation of X
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Standardized Scores
Standardized Score (standard score or z-score):observed value mean
standard deviation
IQ scores have a normal distribution witha mean of 100 and a standard deviation of 16.
Suppose your IQ score was 116.
Standardized score = (116 100)/16 = +1 Your IQ is 1 standard deviation above the mean.
Suppose your IQ score was 84.
Standardized score = (84 100)/16 = 1 Your IQ is 1 standard deviation below the mean.
A normal curve with mean = 0 and standard deviation = 1
is called a standard normal curve.
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Example
X= 5
= 10
6.2 X= 5
= 10
6.2
NormalDistributionNormalDistribution
ZX
====
====
====
6 2 5
10
12.
.ZX
====
====
====
6 2 5
10
12.
.
Z= 0
= 1
.12 Z= 0
= 1
.12
Standardized NormalDistribution
Standardized NormalDistribution
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13
Your score is 3SD below the mean
X=40, 45, 50
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Comparing Scores from Different
Distributions
Interpreting a raw score requires additional
information about the entire distribution. Inmost situations, we need some idea about themean score and an indication of how muchthe scores vary.
For example, assume that an individual tooktwo tests in reading and mathematics. Thereading score was 32 and mathematics was
48. Is it correct to say that performance inmathematics was better than in reading?
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15
Z Scores Help in Comparisons One method to interpret the raw score is to
transform it to a z score.
The advantage of the z score transformation
is that it takes into account both the meanvalue and the variability in a set of raw scores.
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Did Sara improve?Did Sara improve?Did Sara improve?Did Sara improve?
Score in pretest was 18 and post test was 42
Saras score did increase. From 18 to 42.
But her relative position in the Class decreased.
Pretest Post test
Observation 18 42
Mean 17 49
Standard deviation 3 49
Z score 0.33 -0.14
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Normal Distribution Probability
)()( dxxfdxcPd
c=Probability is
area undercurve!
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ExampleExampleExampleExample
P(P(P(P(XXXX 8)8)8)8)
X = 5
= 10
8 X = 5
= 10
8
NormalDistributionNormalDistribution
Standardized NormalDistribution
Standardized NormalDistribution
ZX
====
====
====
8 5
10
30.ZX
====
====
====
8 5
10
30.
Z = 0
= 1
.30 Z = 0
= 1
.30
.3821
.3821
P( Z .30)
=.3821
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P(2.9 X 7.1)
5
= 10
2.9 7.1 X5
= 10
2.9 7.1 X
Normal
Distribution
Normal
Distribution
ZX
ZX
====
====
====
====
====
====
2 9 5
10
21
71 5
1021
..
..
ZX
ZX
====
====
====
====
====
====
2 9 5
10
21
71 5
1021
..
..
0
= 1
-.21 Z.210
= 1
-.21 Z.21
.1664.1664
.0832.0832
Standardized Normal
Distribution
Standardized Normal
Distribution
calculate the probability of scores between 2.9 and 7.1
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P(3.8P(3.8P(3.8P(3.8 XXXX 5)5)5)5)
X = 5
= 10
3.8 X = 5
= 10
3.8
Normal
Distribution
Normal
Distribution
ZX
====
====
====
3 8 5
10
12.
.ZX
====
====
====
3 8 5
10
12.
.
Z = 0
= 1
-.12 Z = 0
= 1
-.12
.0478.0478
Standardized Normal
Distribution
Standardized Normal
Distribution
calculate the probability of scores between 3.8 and 5
010
55=
=
=
XZ 0
10
55=
=
=
XZ
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P(7.1P(7.1P(7.1P(7.1 XXXX 8)8)8)8)
= 5
= 10
87.1 X = 5
= 10
87.1 X
Normal
Distribution
Normal
Distribution
Z
X
ZX
====
====
====
====
====
====
71 5
10 21
8 5
1030
.
.
.
Z
X
ZX
====
====
====
====
====
====
71 5
10 21
8 5
1030
.
.
.
= 0
= 1
.30 Z.21 = 0
= 1
.30 Z.21
.0347
.0347
Standardized Normal
Distribution
Standardized Normal
Distribution
calculate the probability of scores between 7.1 and 8
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P(2000P(2000P(2000P(2000 XXXX 2400)2400)2400)2400)
X = 2000
= 200
2400 X = 2000
= 200
2400
Normal
Distribution
Normal
Distribution
0200
20002000=
=
=
XZ 0
200
20002000=
=
=
XZ
Z = 0
= 1
2.0 Z = 0
= 1
2.0
.4772
.4772
Standardized NormalDistribution
Standardized NormalDistribution
0.2200
20002400=
=
=
XZ 0.2
200
20002400=
=
=
XZ
calculate the probability of scores between 2000 and 24000
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24
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Conclusions
Z-score is defined as the number of standard
deviations from the mean.
Z-score is useful in comparing variables with
very different observed units of measure.
Z-score allows for precise predictions to bemade of how many of a populations scores
fall within a score range in a normal
distribution.
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26
Exercise
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What is the z score for your test:
raw score = 80; mean = 75, S= 5?
S
XXz
= 1
5
7580=
=z
What is the z score of your friends test:
raw score = 80; mean = 75, S= 10?
S
XX
z
=5.
10
7580=
=z
Who do you think did better on their test? Why do you think this?
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28
4.15
6572
=
>z
7257.0
)6.0(
6.05
6568
=