Transcript Chapter 3

Psy B07
THE NORMAL DISTRIBUTION
Chapter 3
Slide 1
Psy B07
Outline
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A quick look back
The normal distribution
Relationship between bars and lines
Area under the curve
Standard Normal Distribution
z-scores
Chapter 3
Slide 2
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A quick look back
 In Chapter 2, we spent a lot of time
plotting distributions and calculating
numbers to represent the distributions.
 This raises the obvious question:
WHY BOTHER?
Chapter 3
Slide 3
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A quick look back
 Answer: because once we know (or
assume) the shape of the distribution
and have calculated the relevant
statistics, we are then able to make
certain inferences about values of the
variable.
 In the current chapter, this will be show
how this works using the Normal
Distribution
Chapter 3
Slide 4
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The Normal Distribution
 As shown by Galton (19th century guy),
just about anything you measure turns
out to be normally distributed, at least
approximately so.
 That is, usually most of the observations
cluster around the mean, with
progressively fewer observations out
towards the extremes
Chapter 3
Slide 5
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The Normal Distribution
 Example:
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 Thus, if we don’t know how some variable is
distributed, our best guess is normality
Chapter 3
Slide 6
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The Normal Distribution
 A note of caution
 Although most variables are normally
distributed, it is not the case that all
variables are normally distributed.
 Values of a dice roll.
 Flipping a coin.
 We will encounter some of these critters
(i.e. distributions) later in the course
Chapter 3
Slide 7
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Relationship between
bars and lines
 Any Histogram:
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 Can be shown
as a line graph:
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Chapter 3
Slide 8
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Relationship between
bars and lines
 Example: Pop Quiz #1
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Chapter 3
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2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
Slide 9
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Relationship between
bars and lines
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2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
Chapter 3
Slide 10
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Area under the curve
 Line graphs make it easier to talk of the
“area under the curve” between two
points where:
area=proportion (or percent)=probability
 That is, we could ask what proportion of
our class scored between 7 & 9 on the
quiz
Chapter 3
Slide 11
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Area under the curve
 If we assume that the total area under
the curve equals one. . . .
 then the area between 7 & 9 equals the
proportion of our class that scored
between 7 & 9 and also indicates our
best guess concerning the probability
that some new data point would fall
between 7 & 9.
Chapter 3
Slide 12
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Area under the curve
 The problem is that in order to calculate
the area under a curve, you must either:
1) use calculus
2) use a table that specifies the area
associated with given values of you
variable.
Chapter 3
Slide 13
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Area under the curve
 The good news is that a table does exist,
thereby allowing you to avoid calculus. The
bad news is that in order to use it you must:
1) assume that your variable is normally
distributed
2) use your mean and standard deviation to
convert your data into z-scores such that the
new distribution has a mean of 0 and a
standard deviation of 1 - standard normal
distribution or N(0,1).
Chapter 3
Slide 14
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Standard Normal Distribution
0.5
0.4
0.3
0.2
0.1
0
-3
Chapter 3
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-1
0
Z-SCORE
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z
Mean to
z
Larger
Portion
Smaller
Portion
.....
.98
.99
1.00
1.01
.....
........
.3365
.3389
.3413
.3438
........
........
.8365
.8389
.8413
.8438
........
........
.1635
.1611
.1587
.1562
........
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z-scores
 It would be too much work to provide a
table of area values for every possible
mean and standard deviation.
 Instead, a table was created for the
standard normal distribution, and the
data set of interest is converted to a
standard normal before using the table.
Chapter 3
Slide 16
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z-scores
 How do we get our mean equal to zero?
Simple, subtract the mean from each data
point.
 What about the standard deviation? Well, if
we divide all values by a constant, we divide
the standard deviation by a constant. Thus, to
make the standard deviation 1, we just divide
each new value by the standard deviation.
Chapter 3
Slide 17
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z-scores
 In computational form then,
z
X 

 where z is the z-score for the value of X we
enter into the above equation
Chapter 3
Slide 18
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z-scores
 Once we have calculated a z-score, we
can then look at the z table in Appendix
Z to find the area we are interested in
relevant to that value.
 As we’ll see, the z table actually
provides a number of areas relevant
to any specific z-score.
Chapter 3
Slide 19
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z-scores
 What percent of students scored better
than 9.2 out of 10 on the quiz, given
that the mean was 7.6 and the standard
deviation was 1.6?
Chapter 3
Slide 20
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z-scores
 I have found the following online applet
which you can use to see this process a little
more directly.
 It allows you to find the area between two
points on the “standard normal” distribution.
 Try it by clicking here – does this help your
understanding?
Chapter 3
Slide 21