Statistics or What

1
StatisticsorWhats normal about the normal
curve, whats standard about the standard
deviation,and whats co-relating in a
correlation?
2
Overview

• Whats normal about the normal curve?
• The nature of the confusion
• One formal answer
• An intuitive answer (real-time demo)
• Whats standard about a standard deviation?
• Z-scores
• Whats co-relating in a correlation?

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Whats normal about the normal curve(s)?

• There are a number of ways of mathematically
  defining and estimating the normal distribution
  (which defines a class of curves, not one single
  curve)
• The main question I want to address today is
  what does that math mean? Why are so many things
  normally distributed? What makes sure that those
  things stay distributed normally? What stops
  other things from being normally distributed at
  all?

4
From Wilensky, U., (1997). What is Normal
Anyway? Therapy for Epistemological Anxiety.
Educational Studies in Mathematics. Special Issue
on Computational Environments in Mathematics
Education. Noss R. (Ed.) Volume 33, No. 2. pp.
171-202.

• U Why do you think height is distributed
  normally?
• L Come again? (sarcastic)
• U Why is it that women's height can be graphed
  using a normal curve?
• L That's a strange question.
• U Strange?
• L No one's ever asked me that before.....
  (thinking to herself for a while) I guess there
  are 2 possible theories Either it's just a fact
  about the world, some guy collected a lot of
  height data and noticed that it fell into a
  normal shape.....
• U Or?
• L Or maybe it's just a mathematical trick.
• U A trick? How could it be a trick?

5

• L Well... Maybe some mathematician somewhere
  just concocted this crazy function, you know, and
  decided to say that height fit it.
• U You mean...
• L You know the height data could probably be
  graphed with lots of different functions and the
  normal curve was just applied to it by this one
  guy and now everybody has to use his function.
• U So youre saying that in the one case, it's a
  fact about the world that height is distributed
  in a certain way, and in the other case, it's a
  fact about our descriptions but not about height?
  
• L Yeah.
• U Well, if you had to commit to one of these
  theories, which would it be?
• L If I had to choose just one?
• U Yeah.
• L I don't know. That's really interesting. Which
  theory do I really believe? I guess I've always
  been uncertain which to believe and it's been
  there in the background you know, but I don't
  know. I guess if I had to choose, if I have to
  choose one, I believe it's a mathematical trick,
  a mathematician's game. ....What possible reason
  could there be for height, ....for nature, to
  follow some weird bizarro function?

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Formal answer 1 The binomial distribution I

• The chance of an event of probability p happening
  r times out of n tries
• P(r) n!/(r! (n - r)!) pr (1 - p) n-r
• (Recall We wondered about this generalization
  last class.)

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Formal answer 1 The binomial distribution II

• Why is it called the binomial distribution?
• Bi 2 Nom thing
• the two-thing distribution
• It can be used wherever
• 1.Each trial has two possible outcomes (say,
  success and failure or heads and tails)
• 2.The trials are independent the outcome of one
  trial has no influence over the outcome of
  another trial.
• 3. The trials are mutually exclusive
• 4. The events are randomly selected

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Lets try it out (Example 6.3 from last class)

• What are the odds of there being exactly one
  seven out of two rolls?

• one way is to roll 7 first, but not second
• - the odds of this are 1/6 5/6 (independent
  events) 0.138
• - the odds of rolling 7 second are 5/6 1/6
  (independent events) 0.138
• - since these two outcomes are mutually
  exclusive, we can add them to get 0.138 0.138
  0.277

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The generalization (Example 6.3 from last class)

• What are the odds of there being exactly one
  seven out of two rolls?

An event of probability p happens r times out of
n tries P(r) n!/(r! (n - r)!) pr (1 - p)
n-r p 1/6 N 2 r 1 2!/(1!1!)1/615/
61 0.277777778
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What does this have to do with the normal
distribution?
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What does this have to do with the normal
distribution?
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Why does this normal distribution happen?

• See http//ccl.northwestern.edu/cm/index.html
• for the StarLogoT demo used in class.
• Can you understand
• What effect changing the probabilities of each
  event has?
• What has to change to skew a normal curve?

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The standard deviation
From http//www.psychstat.smsu.edu/introbook/sbk0
0.htm

• Given the non-linear shape of the normal
  distribution, one has two choices
• A.) Keep the amount of variation in each division
  constant, but vary the size of the divisions
• B.) Keep the size of each division constant, but
  vary the the amount of variation in each division

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The standard deviation (SD)

• The SD takes the second approach it keeps the
  size of each division constant, but varies the
  the amount of variation in each division
• The SD is a measure of average deviation
  (difference) from the mean
• It is the square root of the variance, which is
  the average squared difference from the mean.

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Z-scores

• If we express differences by dividing them by
  SDs, we have z-scores standard units of
  difference from the mean
• THESE Z-SCORES WILL COME IN EXTREMELY USEFUL!
• For example, we might want to know
• If a 12-foot elephant is taller (compared to the
  height of average elephants) than a 230 pound
  man is heavy (compared to weight of average men)
• If a person with a WAIS IQ of 140 is rarer than a
  person with a GPA of 3.9
• Etc.

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What co-relates in a correlation?

• In a correlation, we want to find the equation
  for the (one and only) line (the line of
  regression) which describes the relation between
  variables with the least error.
• This is done mathematically, but the idea is
  simply that we draw a line such that the squared
  distances on two (or more) dimensions of points
  from the line would not be less for any other line

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What co-relates in a correlation?

• R The covariance of x and y / the product of
  the SDs of X and Y
• Covariance is related to variance the mean
  value of all the pairs of differences from the
  mean for X multiplied by the differences from the
  mean for Y (the mean product of differences from
  the means)
• When X and Y are related, large numbers will be
  systematically multiplied by large numbers with
  the same sign (for differences on both sides of
  the mean) covariance will be large close to
  the product of the SDs of X and Y, so R will be
  close to 1.