Download The Normal Distribution

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 – 6/350 = .17 = 17%
9 people scored a 1 – 3%
The Normal Distribution
Why is the Normal Distribution so important?



Almost all of the statistical tests that we will be covering (ZTests, T-Tests, ANOVA, etc.) 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
Also, many variables that psychologists and health
professionals look at are normally distributed
Why this is requires a detailed examination of the derivation
of our statistics, that involves way more detail than you need
to use the statistic.
The Normal Distribution
Ordinate
Density – what is
measured on the
ordinate (more on
this in Ch. 7)
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
Don’t worry, understanding this is not necessary
to understanding the normal distribution, only a
helpful aside for the mathematically inclined
The Normal Distribution
Using this formula, mathematicians have
determined the probabilities of obtaining
every score on a “standard normal
distribution” (see Table E.10 in your book)
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, this table refers to a Standard
Normal Distribution

Μ = 0; σ = 1
How do you get your variable to fit?
z
X 

The Normal Distribution
Z-Scores


Range from +∞ to -∞
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
Now we can begin to use the table to
determine the probability that our z score will
occur using table E.10
The Normal Distribution
z
Mean to
z
Larger
Portion
Smaller
Portion
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
The Normal Distribution
Mean to Z
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
The Normal Distribution
The Normal Distribution
Larger Portion
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
The Normal Distribution
Smaller Portion
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
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?
z = -.1
z = 1.645
(z = -.1, “Mean to Z”) + (z = 1.645, “Mean to Z”)
.0398 + .4500 = .4898 = 49%
z = -1.00, “Smaller Portion” = Red + Blue
z = -1.645, “Smaller Portion” = Blue
(Red + Blue) - Blue = Red
z = -1.645
z = -1.00
(z = -1.00, “Smaller Portion”) – (z = -1.645, “Smaller
Portion”)
.1587 - .0500 = .1087 = 11%
The Normal Distribution
What are the scores that lie in the
middle 50% of a distribution of scores
with μ = 50 and σ = 10?




Look for “Smaller Portion” = .2500 on
Table E.10
z = .67
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
Other uses for z-scores:
1.
Converting two variable 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.d. 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.d. was a 5, you did
4 s.d’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


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?



You can compare performance on two different
tests with two different metrics
You can easily compute Percentile ranks
but they are population-relative!
Percentile – the point below which a certain
percent of scores fall

i.e. If you are at the 75th%ile (percentile), then
75% of the scores are at or below your score
The Normal Distribution
How do you compute %ile?



Convert your raw score into a z-score
Look at Table E.10, and find the “Smaller Portion” if your zscore is negative and the “Larger Portion” if it is positive
Normal Distribution
Multiply by 100
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
The Normal Distribution
New Score = New s.d. (z) + New Mean
New IQ Score = 15 (2) + 100 = 130
T-Score – commonly used standardized
normal distribution w/ mean = 50 and s.d. =
10
T-Score = 10 (2) + 50 = 70