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The Normal Distribution The Normal Distribution

The Normal Distribution - PowerPoint Presentation

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Uploaded On 2016-06-15

The Normal Distribution - PPT Presentation

History Abraham de Moivre 1733 consultant to gamblers Pronunciation Pierre Simon Laplace mathematician astronomer philosopher determinist Carl Friedrich Gauss mathematician and astronomer ID: 363200

scores normal fall curve normal scores curve fall 115 score 100 percentage table distribution probability distributions answer middle mathematician

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Slide1

The Normal DistributionSlide2

History

Abraham de

Moivre

(1733) – consultant to gamblers.

Pronunciation

.

Pierre Simon

Laplace – mathematician, astronomer, philosopher, determinist.

Carl Friedrich

Gauss – mathematician and astronomer.

Adolphe

Quetelet

--

mathematician, astronomer

, “social physics.”

Pronunciation

.Slide3

Importance

Many variables are distributed approximately as the

bell-shaped

normal

curve

The mathematics of the normal curve are well known and relatively simple

.

Many statistical procedures assume that the scores came from a normally distributed population.Slide4

Distributions of sums and means approach normality as sample size increases.Many other probability distributions are closely related to the normal curve.Slide5

Using the Normal Curve

From its PDF (probability density function) we use integral calculus to find the probability that a randomly selected score would fall between value

a

and value

b

.

This is equivalent to finding what proportion of the total area under the curve falls between

a

and

b

.Slide6

The PDF

F(Y) is the probability density, aka the height of the curve at value

Y.

There are only two parameters, the mean and the variance.

Normal distributions differ from one another only with respect to their mean and variance.Slide7

Avoiding the Calculus

Use the normal curve table in our text.

Use SPSS or another stats package.

Use an Internet resource.Slide8

IQ = 85, PR = ?z = (85 - 100)/15 = -1.

What percentage of scores in a normal distribution are less than minus 1?

Half of the scores are less than 0, so you know right off that the answer is less than 50%.

Go to the normal curve table.Slide9

Normal Curve Table

For each

z

score, there are three values

Proportion from score to mean

Proportion from score to closer tail

Proportion from score to more distant tailSlide10

Locate the |z| in the Table

34.13% of the scores fall between the mean and minus one.

84.13% are greater than minus one.

15.87% are less than minus oneSlide11

IQ =115, PR = ?

z

= (115 – 100)/15 = 1.

We are above the mean so the answer must be greater than 50%.

The answer is 84.13% .Slide12

85 < IQ < 115

What percentage of IQ scores fall between 85 (

z

= -1) and 115 (

z

= 1)?

34.13% are between mean and -1.

34.13% are between mean and

1.

68.26% are between -1 and 1.Slide13

115 < IQ < 130

What percentage of IQ scores fall between 115 (

z

= 1) and 130 (

z

= 2)?

84.13% fall below 1.

97.72% fall below 2.

97.72 – 84.13 = 13.59%Slide14

The Lowest 10%What score marks off the

lowest

10% of IQ scores

?

z

= 1.28

IQ = 100 – 1.28(15) = 80.8Slide15

The Middle 50%What scores mark off the middle 50% of IQ scores

?

-.67 <

z

< .67;

100 - .67(15) = 90

100 +

.67(15)

= 110Slide16

Online Calculator

This one is easy to use. Check it out.

http://

davidmlane.com/hyperstat/z_table.html

Slide17

Memorize These Benchmarks

This Middle Percentage of Scores

Fall Between Plus and Minus z =

50

.67

68

1.

90

1.645

95

1.96

98

2.33

99

2.58

100

3.Slide18

The Normal DistributionSlide19

The Bivariate Normal DistributionSlide20
Slide21
Slide22