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StatisticsorWhats normal about the normal

curve, whats standard about the standard

deviation,and whats co-relating in a

correlation?

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?

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?

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?

- 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?

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.)

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

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

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

What does this have to do with the normal

distribution?

What does this have to do with the normal

distribution?

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?

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

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.

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.

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

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.