1 PDF 1 Normal Distribution

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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



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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


ZX

====

====

====

6 2 5

10

12.

.ZX

====

====

====

6 2 5

10

12.

.

Z= 0

= 1

.12 Z= 0

= 1

.12




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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




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



0.2200

20002400=

=

=

XZ 0.2

200

20002400=

=

=

XZ

calculate the probability of scores between 2000 and 24000


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23


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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

=

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