Chapter 6 The Normal Distribution. 2 Chapter 6 The Normal Distribution Major Points Distributions...

Chapter 6The Normal Distribution

2Chapter 6 The Normal Distribution

Major Points

Distributions and area The normal distribution The standard normal distribution Setting probable limits on an observation Measures related to z


Distributions and Area We can apply the concepts of

Area Percentages Probability Addition of areas Addition of probabilities

When using the normal distribution


12

Convert Histogram toFrequency Polygon of Distribution: connect the dots……..

0

1

2

3

4

5

6

7

8

9

10

1 to 10 11 to 20 21 to 30 31 to 40 41 to 50 51 to 60 61 to 70 71 to 80 81 to 90 91 to 100

N=50N=50


The Normal Distribution “Normal” refers to the general shape of

the distribution See next slide Remember our descriptors for distributions

The formula for the normal distribution allows us to Predict a proportion of cases that fall under

the normal curve between 2 given x values

Cont.


Normal Distribution--cont.

X is the value on the abscissa Y is the resulting height of the curve at X There are constants also in the formula Do you need to memorize or use the

formula? No! It’s just there to show you how they arrive at the percentages discussed in a minute.


The Standard Normal Distribution

We simply transform all X values to have a mean = 0 and a standard deviation = 1

Call these new values z Define the area under the curve to be

1.0, can look at the proportion of cases falling above mean and below mean


Ex. of a transformed Distribution

X

3.503.00

2.502.00

1.501.00

.500.00

-.50-1.00

-1.50-2.00

-2.50-3.00

-3.50-4.00

1200

1000

800

600

400

200

0

Std. Dev = 1.00

Mean = -.01

N = 10000.00


z Scores

Calculation of z

where is the mean of the population and is its

standard deviation This is a simple linear transformation of X.

σμX

z


Tables of z We use tables to find areas under the

distribution A sample table is on the next slide The following slide illustrates areas under the

distribution


zMean to

zLargerPortion

SmallerPortion

0.000 .0000 .5000 .5000

0.100 .0398 .5398 .4602

0.200 .0793 .5793 .4207

1.000 .3413 .8413 .1587

1.500 .4332 .9332 .0668

1.645 .4500 .9500 .0500

1.960 .4750 .9750 .0250

z Table


Using the Tables

Define “larger” versus “smaller” portion Distribution is symmetrical, so we don’t need

negative values of z Areas between z = +1.5 and z = -1.0

See next slide


Calculating areas

Area between mean and +1.5 = 0.4332 Area between mean and -1.0 = 0.3413 Sum equals 0.7745 Therefore about 77% of the observations

would be expected to fall between z = -1.0 and z = +1.5


Converting Back to X

Assume = 30 and = 5

5.3755.130

2550.130

Therefore

X

X

zX

Xz

So 77% of the distribution is expected to lie between 25 and 37.5


Probable Limits

X = + z Our last example has = 30 and = 5 We want to cut off 2.5% in each tail, so

z = + 1.96 (found from a z table, mean to z of 1.96 = .475, but we usually round to z = 2).

2.20596.1308.39596.130

XX

zX

Cont.


Probable Limits--cont.

We have just shown that 95% of the normal distribution lies between 20.2 and 39.8, or +/- 1.96 SD’s

Therefore the probability is .95 that an observation drawn at random will lie between those two values


20

Statistical Methods: The Normal Distributions

Walsh & Betz (1995), p29Walsh & Betz (1995), p29


Measures Related to z

Standard score Another name for a z score, there are other standard

scores, such as ….. T scores: Scores with a mean of 50 and and

standard deviation of 10, no negative values Others include IQ, SAT’s, etc

Percentile score/rank Percent of cases which fall at or below a given score

in the norming sample- THIS is how we typically find a percentile rank, not calculating using that somewhat confusing formula

Cont.


Related Measures--cont.

Stanines Scores which are integers between 1 and 9, with

mean = 5 and standard deviation = 2 Mainly used in education


Review Questions

Why do you suppose we call it the “normal” distribution?

What do we gain by knowing that something is normally distributed?

How is a “standard” normal distribution different?

Cont.


Review Questions--cont.

How do we convert X to z? How do we use the tables of z? If we know your test score, how do we

calculate your percentile? What is a T score and why do we care?

Chapter 6 The Normal Distribution. 2 Chapter 6 The Normal Distribution Major Points Distributions...

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