Standardized Distributions

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Standardized Distributions

• Statistics 2126

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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?

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So, what did you get?

• Say you are comparing your quiz marks with other
  people in the class
• Lets 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

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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
• Lets say ? 1.5

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So compare

• How many standard deviations are you from the
  mean?
• We call this a z score

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x8 ?7 ?1.5
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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
• Lets say you get 8 on the next quiz
• But now the class mean is 7.5

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Change it up a little

• Now lets 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?

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x8 ?7.5 ?.5
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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

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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?

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You can answer these questions by looking at the
formula
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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

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x 118 ? 100 ? 15
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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 ?

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Now just sub into the formula and cross multiply
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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.

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So it looks like this