Transcript Standardized Distributions

Standardized Distributions
Statistics 2126
Introduction
• Last time we talked about measures of
spread
• Specifically the variance and the
standard deviation
• s and s2
• You might ask yourself “Why is this
useful?”
So, what did you get?
• Say you are comparing your quiz marks
with other people in the class
• Let’s say you got 8
• And the class average was 7
• That is a population mean, we are
considering the class to be a population
so  = 7
What did you get, in relation to
others
• By how much are you better than the
class average
• By 1….
• If everyone got say below you, you rock
• This is where the population standard
deviation or  comes into play
• Let’s say  = 1.5
So compare
• How many standard deviations are you
from the mean?
• We call this a z score
z
x 

x=8 =7 =1.5
x 
z

87
z
1.5
1
z
1.5
z  .67
So what does that mean?
• It means you are .67 standard
deviations away from the mean.
• We now have a measure of how far
away you are from a mean
• We call this a standard score
• Let’s say you get 8 on the next quiz
• But now the class mean is 7.5
Change it up a little
• Now let’s say the standard deviation is
.5
• So now on this quiz the scores were
packed much more tightly
• Did you do relatively better on the first
quiz or on the second one?
x=8 =7.5 =.5
z
x 

8  7.5
z
.5
.5
z
.5
z  1.00
So compare the two
• You did better on the second quiz than
you did on the first one
• You are 1 standard deviation from the
mean
• You are simply comparing the two z
scores
Properties of z
• It can be negative or positive
• If you are off to the left of the mean you will
get a negative score
• If you are off to the right, your z score will be
positive
• What is the shape?
• What is the average z score?
• What is the standard deviation?
You can answer these
questions by looking at the
formula
z
x 

An example
• IQ has a mean of 100 and a standard
deviation of 15
• N(100,15)
• That just means it is normal with a
mean of 100 and a sd of 15
• So what is the z score of someone with
an IQ of 118
x = 118  = 100  = 15
z
x 

118100
z
15
18
z
15
z  1.2
You could go the other way
too
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•
•
•
•
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•
So say someone had a z score of 1.62
What is their IQ?
Well again just list what you know
z = 1.62
 = 100
 = 15
x=?
Now just sub into the formula
and cross multiply
z
x 

x 100
1.62 
15
x 100 15(1.62)
x 100 24.3
x  124.3
Well this must all have a point
• Using a z table
• Or this VERY cool website:
• http://davidmlane.com/hyperstat/z_table
.html
• So if you know the z, you can find out
what the probability of getting a z score
at a certain level is.
So it looks like this