standardized distributions statistics 2126. introduction last time we talked about measures of...
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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 −μ
σ
x=8 =7 =1.5
€
z =x −μ
σ
z =8 − 7
1.5
z =1
1.5z = .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 −μ
σ
z =8 − 7.5
.5
z =.5
.5z =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 −μ
σ
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 −μ
σ
z =118 −100
15
z =18
15z =1.2
You could go the other way too
• 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 −μ
σ
1.62 =x −100
15x −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