Download PSY 360 z Scores and Normal Distributions

z Scores and
Normal Distributions
PSY 360
Introduction to Statistics for
the Behavioral Sciences
z Scores
The aspect of the data we want to
describe/measure is relative position
z scores tell us how many standard deviations above
or below the mean a given score falls
z scores are statistics that describe the relative
position of something in its distribution
Verbal formula: z is something minus its mean
divided by its standard deviation
Formulas:
For X in sample, z = (X-X)
s
For X in population, z = (X-µ)
σ
z Scores
Characteristics:
The mean of a distribution of z scores is zero
The variance of a distribution of z scores is one
The shape of a distribution of z scores is reflective, the
shape is the same as the shape of the distribution of the
Xs
Example: Compute z for a sample when
X=34, X=40, and s²=9
z= (X-X) = (34-40) = -6 = -2
s
3
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Compute z for a population when X=10,
µ=8, and σ²=16
z = (X-µ)/σ = (10-8)/4 = 2/4 = .5
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Normal Distributions
Family of theoretical distributions
There are many different normal distributions
Characteristics:
Symmetric, continuous, unimodal
Bell-shaped
Scores range from -∞ to +∞
Mean, median, and mode are all the same
value
Each distribution has two parameters, µ and
σ²
Normal Distributions
Examples:
IQ is normally distributed with µ=100 and σ²=225
Height of American males is normally distributed with
µ=69 and σ²=9
The standard normal (or unit normal)
distribution has µ=0 and σ²=1
We can transform any normal distribution
to the standard normal distribution by
computing z scores: the resulting
distribution of z scores will have a shape
that is normal. WHY?
Standard Normal Distribution
We use this distribution to get probabilities
associated with a z score (probability,
proportion, and area under the curve are
synonymous)
Example:
If Joe is 73 inches tall, what is
the probability that any randomly
selected man is his height or
taller?
For height, µ=69 and σ²=9, so
z =(X-µ)/σ=(73-69)/3=4/3=1.33
From Table B.1 (pp. 687-690),
-3
p(z>1.33)=.0918
-2
-1
0
1
2
3
2
Standard Normal Distribution
Using Table B.1, there are two key
facts:
Total area equals one
Symmetry
Steps
Example: p(z>2)
Tail
Small
p(z>2)=.0228 (from Table B.1)
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Draw a picture with zero and z
Locate desired area: is it small or
large?
Is area in tail or middle?
Use symmetry and/or total area=1
-3
-2
-1
0
1
2
3
Picture -3
-2
-1
0
1
2
3
Standard Normal
Distribution
More examples: if z is normal and
p(z>1.645)=.05
Compute p(z<1.645)
Want all the area to the left of 1.645, a large area.
The area is in the middle and left tail.
Use total area=1 to get
p(z<1.645) = 1-.05 = .95
0
1.645
Standard Normal
Distribution
More examples: if z is normal and
p(z>1.645)=.05
Compute p(z<-1.645)
Want all the area to the left of -1.645, a
small area.
The area is in the left tail.
Use symmetry to get
p(z<-1.645) = .05
-1.645
0
3
Standard Normal
Distribution
More examples: if z is normal and
p(z>1.645)=.05
Compute p(z>-1.645)
Want all the area to the right of -1.645, a large area
The area is in the middle
and right tail
Use symmetry and total
area=1 to get
p(z>-1.645) = .95
-1.645
0
Standard Normal
Distribution
More examples: if z is normal and
p(z>1.645)=.05
Compute p(-1.645<z<1.645)
Want all the area between -1.645 and 1.645, a large
area
The area is in the middle
Use symmetry and total
area=1 to get
p(-1.645<z<1.645) =
1-2(.05) = 1-.10 = .90
-1.645
0
1.645
Standard Normal
Distribution
More examples: if z is normal and
p(z>1.1)=.1357
Compute p(z<1.1)
Want all the area to the left of 1.1, a large area
The area is in the middle and left tail
Use total area=1 to get p(z<1.1) = 1-.1357 = .8643
0
1.1
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Standard Normal
Distribution
More examples: if z is normal and
p(z>1.1)=.1357
Compute p(z<-1.1)
Want all the area to the left of -1.1, a small
area
The area is in the left
tail
Use symmetry to get
p(z<-1.1) = .1357
-1.1
0
Standard Normal
Distribution
More examples: if z is normal and
p(z>1.1)=.1357
Compute p(z>-1.1)
Want all the area to the right of -1.1, a large area
The area is in the middle and right tail
Use symmetry and total area=1 to get p(z>-1.1) =
.8643
-1.1
0
Standard Normal
Distribution
More examples: if z is normal and
p(z>1.1)=.1357
Compute p(-1.1<z<1.1)
Want all the area between –1.1 and 1.1, a
large area
The area is in the
middle
Use symmetry and
total area=1 to get
p(-1.1<z<1.1) =
-1.1
0
1.1
1-2(.1357) = .7286
5
Standard Normal
Distribution
Problem: Joe took a standardized test for which
the norms are µ = 24 and σ2 = 4. His score
was X = 22. When he computed his z score,
he got z = (22–24)/4. Knowing that he is not
doing as well in statistics as you are, Joe
asked you to check his work. What did Joe
get for his z score? Was he correct? Why?
Solution: He should have divided by σ rather
than σ2. His z score was really -1, not -.5
Standard Normal
Distribution
Data: 4, 3, 6, 5, 8, 3, 2, 4, 7, 8
Compute X
Compute s2
Compute the z score for the person who has X=6
If you computed all ten z scores, what would the
mean be for these ten z scores?
If you computed all ten z scores, what would the
variance be for these ten z scores?
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