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The Normal Distribution
The Normal Distribution
Distribution – any collection of scores, from either a sample or
population
 Can be displayed in any form, but is usually represented as a
histogram
Normal Distribution – specific type of distribution that assumes
a characteristic bell shape and is perfectly symmetrical
The Normal Distribution
Can provide us with information on likelihood
of obtaining a given score


60 people scored a 6 – 60/350 = .17 = 17%
9 people scored a 1 – 3%
The Normal Distribution
Why is the Normal Distribution so important?


Almost all statistical tests that we will be covering
throughout the course assume that the population
distribution, that our sample is drawn from (but for the
variable we are looking at), is normally distributed
Many variables that psychologists and health professionals
look at are normally distributed
The Normal Distribution
Ordinate
Density – what is
measured on the
ordinate
Abscissa
The Normal Distribution
Mathematically defined as:
1
 X   2 / 2 2
f (X ) 
(e)
 2


Since  and e are constants, we only have
to determine μ (the population mean) and
σ (the population standard deviation) to
graph the mathematical function of any
variable we are interested in
No, this will not be on the test
The Normal Distribution
Using this formula, mathematicians have
determined the probabilities of obtaining
every score on a “standard normal
distribution”
To determine these probabilities for the
variable you’re interested in we must plug in
your variable to the formula

Note: This assumes that your variable fits a
normal distribution, if not, your results will be
inaccurate
The Normal Distribution
However, your data probably doesn’t
exactly fit a standard normal
distribution

μ = 0; σ = 1
How do you get your variable to fit?
z
X 

The Normal Distribution
Z-Scores

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Range from +∞ to -∞
Distribution: μ = 0; σ = 1
Represent the number of standard
deviations your score is from the mean
 i.e. z = +1 is a score that is 1 standard
deviation above the mean and z = -3 is a score
3 standard deviations below the mean
Normal Distribution
Cutoff at +1.645
1200
1000
800
600
400
200
0
50
3.
00
3.
50
2.
00
2.
50
1.
00
1.
0
.5
00
0.
0
-.5
0
.0
-1
0
.5
-1
0
.0
-2
0
.5
-2
0
.0
-3
0
.5
-3
0
.0
-4
z
Reminder: Z-Scores represent # of standard
deviations from the mean

For this distribution, if μ = 50 and σ = 10, what
score does z = -3 represent? z = +2.5?
The Normal Distribution
What are the scores that lie in the
middle 50% of a distribution of scores
with μ = 50 and σ = 10?
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

z = ± .67 = .2500
Solve for X using z-score formula
Scores = 56.7 and 43.3
The Normal Distribution
z
X 

X  50
 .67 
10
X  50  .67(10)
X  50  6.7
X  6.7  50
X  56.7 and 43.3
The Normal Distribution
We can do the same thing for the
middle 95% to identify the scores in the
extreme 2.5% of the distribution
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
z = 1.96 corresponds to p ≈ .025
Keep this in mind, we’ll come back to it
later
The Normal Distribution
Other uses for z-scores:
1.
Converting variables to a standard metric

You took two exams, you got an 80 in Statistics and a
50 in Biology – you cannot say which one you did
better in without knowing about the variability in
scores in each


If the class average in Stats was a 90 and the s = 15,
what would we conclude about your score now? How is it
different than just using the score itself?
If the mean in Bio was a 30 and the s = 5, you did 4 s’
above the mean (a z-score of +4) or much better than
everyone else
The Normal Distribution
Other uses for z-scores:
1.
Converting variables to a standard metric

This also allows us to compare two scores on different
metrics
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2.
i.e. two tests scored out of 100 = same metric
one test out of 50 vs. one out of 100 = two different
metrics
Is 20/50 better than 40/100? Is it better when compared
to the class average?
Allows for quick comparisons between a score
and the rest of the distribution it is a part of
The Normal Distribution
Standard Scores – scores with a
predetermined mean and standard deviation,
i.e. a z-score
Why convert to standard scores?
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You can compare performance on two different
tests with two different metrics
You can easily compute Percentile ranks
but they are population-relative!