History Abraham de Moivre 1733 consultant to gamblers Pronunciation Pierre Simon Laplace mathematician astronomer philosopher determinist Carl Friedrich Gauss mathematician and astronomer ID: 363200
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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 DistributionSlide20Slide21Slide22