chapter 6 the normal distribution. 2 chapter 6 the normal distribution major points distributions...
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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
3Chapter 6 The Normal Distribution
Distributions and Area We can apply the concepts of
Area Percentages Probability Addition of areas Addition of probabilities
When using the normal distribution
4Chapter 6 The Normal Distribution
12
Convert Histogram toFrequency Polygon of Distribution: connect the dots……..
0
1
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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
5Chapter 6 The Normal Distribution
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.
6Chapter 6 The Normal Distribution
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.
7Chapter 6 The Normal Distribution
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
8Chapter 6 The Normal Distribution
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
9Chapter 6 The Normal Distribution
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
10Chapter 6 The Normal Distribution
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
11Chapter 6 The Normal 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
12Chapter 6 The Normal Distribution
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
13Chapter 6 The Normal Distribution
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
14Chapter 6 The Normal Distribution
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
15Chapter 6 The Normal Distribution
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.
16Chapter 6 The Normal Distribution
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
17Chapter 6 The Normal Distribution
20
Statistical Methods: The Normal Distributions
Walsh & Betz (1995), p29Walsh & Betz (1995), p29
18Chapter 6 The Normal Distribution
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.
19Chapter 6 The Normal Distribution
Related Measures--cont.
Stanines Scores which are integers between 1 and 9, with
mean = 5 and standard deviation = 2 Mainly used in education
20Chapter 6 The Normal Distribution
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.
21Chapter 6 The Normal Distribution
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?
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