Title: Statistics or Whats normal about the normal curve, and whats standard about the standard deviation,
1StatisticsorWhats normal about the normal
curve, and whats standard about the standard
deviation,and what co-relates in a correlation
2Overview
- What are statistics?
- 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-relates in a correlation?
3What are statistics?
- Recall that a probability is a proportion a
ratio of the probability of an event to the space
of all possibilities - The sum of the probabilities of all possibilities
is 1
All possibilities
Possibility of some particular thing
4What are statistics?
- In the last few classes we have seen that some
events can occur in more ways than other events
they are more more probable - For example, it is more probable that we will
roll a 7 with two dice than that we will roll a 2
and more probable that we will get 5 heads out
ten coin flips than that we will get one, because
there are more ways for it to happen. - A better representation of probability space (the
space of all possible events) than our circle
might represent the probabilities of certain
events in a way that makes it more obvious that
they can happen in more ways than other events
it might represent the distribution of the
probabilities of events directly
5What are statistics?
- We ended last time considering this probability
distribution
6What are statistics?
- Statistics are methods for making calculations
about distributions of probabilities - In particular, we might ask How likely is some
set of events (e.g. lt 2 heads)? Is one
distribution different from another?
P 0.8
P 0.5
7Whats normal about the normal curve(s)?
- The normal curve is not a single curve, but a
class of curves of probability distributions,
that share properties in common - There are a number of ways of mathematically
defining and estimating the normal distribution - The actual definition (which you dont need to
know) is
8Whats normal about the normal curve(s)?
- The main questions I want to address today is
- What does that math actually 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?
9What is the normal curve?
- The normal curve has the following properties
- It is bell-shaped
- It is symmetric
- The total area under the curve is 1 (why?)
- The normal curve extends indefinitely in both
directions, getting infinitely close to zero in
either direction.
10Normal curve shapes
- The height of (non-infinite) normal
approximations simply reflects the most probable
event, which is a function of the number of
possible events and of their probability of
occurrence
10 flips
1000 flips
11From 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?
12- 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?
13Formal 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.) -
14Formal 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 outcomes are mutually exclusive
- 4. The events are randomly selected
15Lets try it out (Example 6.3 from our first
probability 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
16The 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 0.277
17What does this have to do with the normal
distribution?
18What does this have to do with the normal
distribution?
19Why does this normal distribution happen?
- See http//ccl.northwestern.edu/netlogo/
- for the NetLogo demo shown in class.
- Can you understand
- What effect changing the probabilities of each
event has? - What has to change to skew a normal curve?
20Why are so many things normally distributed?
- The normal curve is a picture of how randomness
distributes itself if it is left alone - That is Normality arises as a direct consequence
of randomness - This is itself a remarkable and almost holy fact
deep structure arises from that which is
unstructured! - But why are so many things (especially
psychometric things) unstructured and therefore
structured normally?
21The 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
22The standard deviation (SD)
- The definition of SD takes the second approach
it keeps the size of each division constant, but
it 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.
Why do we square the difference?
23Z-scores
- If we express differences by dividing them by
population SDs, we have z-scores standard units
of difference from the mean - THESE Z-SCORES COME IN EXTREMELY USEFUL IN
PSYCHOMETRICS! - 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 Wayne Gretzky was better hockey player than
Tiger Woods is a golfer (a prize for the person
who proves one or the other!) - If a person with a WAIS IQ of 140 is rarer (
less probable) than a person with a GPA of 3.9 - Etc.
24Z-scores
- Remember this Z-scores are just a way of talking
about distribution of probabilities- that is,
they are a shorthand way of talking about how
large a portion of probability space we are
lopping out of the distribution of probabilities
represented by our normal curve
25What 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
26We need first to know What is covariance?
- Covariance is closely related to variance (which
is, recall, the average of the squared deviations
from a mean) - The covariance of two features X and Y measures
their tendency to vary together, i.e., to
co-vary. - It is defined as the average of (differences from
the mean for X multiplied by the differences from
the mean for Y) - That is the average of the products of the
deviations from the mean of X and Y - In variance (one variables), we square the
differences of each data point from their mean - In covariance (two variables), we multiply the
difference from one mean by the difference from
the other mean
27We need first to know What is covariance?
- Covariance is the average of the products of the
deviations from the mean of X and Y - Properties
- If X and Y tend to increase together, then c(X,Y)
gt 0 - If one tends to decrease when the other
increases, then c(X,Y) lt 0 - If X and Y are independent, then c(X,Y) 0
- c(X,Y) lt the product of the standard
deviations of X and Y
28We need first to know What is covariance?
- Covariance is the average of the products of the
deviations from the mean of X and Y - Properties
- If X and Y tend to increase together, then c(X,Y)
gt 0 - Why? Because negative distances from the X mean
will be likely to paired with negative distances
from the Y mean (so their product will be
positive) and positive distances from the X mean
will be likely to paired with positive distances
from the Y mean (so their product will also be
positive)
29We need first to know What is covariance?
- Covariance is the average of the products of the
deviations from the mean of X and Y - Properties
- If one tends to decrease when the other
increases, then c(X,Y) lt 0 - Why? Because negative distances from the X mean
will be likely to paired with positive distances
from the Y mean (so their product will be
negative) and positive distances from the X mean
will be likely to paired with negative distances
from the Y mean (so their product will also be
negative)
30We need first to know What is covariance?
- Covariance is the average of the products of the
deviations from the mean of X and Y - Properties
- If X and Y are independent, then c(X,Y) 0
- Why? Because positive distances from the X mean
will as just as likely to be paired with positive
and negative distances from the Y mean, and short
distances from the X mean are as likely to be
paired with short as long distances from theY
mean, so their product will be as likely to be
positive as negative, and as likely large as
small, and will tend to average out.
31We need first to know What is covariance?
- Covariance is the average of the products of the
deviations from the mean of X and Y - Properties
- c(X,Y) lt the product of the standard
deviations of X and Y - Why? The SD is a measure of average deviation
(difference) from the mean, so it is a measure of
how far away from the mean X and Y are - If every X is exactly the same distance from its
means as every Y, c(X,Y) the product of the
standard deviations of X and Y
32What is a correlation?
- r The covariance of x and y / the product of
the SDs of X and Y - It is standardized measure of covariance,
standardized as a fraction of the total possible
deviation from the mean - When X and Y are related, covariance will close
to the product of the SDs of X and Y, so R will
be close to 1. - When X and Y are unrelated, the differences from
the means by item will depart from the average
differences from the mean c(x,y) lt SD(x) SD(y)
33Significance tests for correlation
- It is possible to transform a correlation into a
t-score, so we calculate how reliable it is (that
is, test the null hypothesis that r 0) - t r / sqrt( 1 - r2)/(N -2) df N - 2
- Note that, with r constant, t increases as N
increases - What are the implications of this?
- For on-line calculation http//faculty.vassar.e
du/lowry/rsig.html
34Significance tests for correlation
- It is also possible to calculate how reliable a
difference between two correlations is, using a
z-transformation - Z ln(r1)/r-1)/2
- ln Natural logarithm log to the base e,
where e 2.718281828459, for reasons that need
not concern us here.
35Correlation squared
- The square of a correlation tells us the
percentage of total variance that is accounted
for in one variable by the other - We will not attempt to understand why in this
class - The distribution is symmetric (for linear
correlations only) - If r 0.1, then 0.01 (1) of the variance in one
variable is accounted for by the other - If r 0.5, then 0.25 (25) of the variance in
one variable is accounted for by the other
36Visual help
- Check out the normal curve and correlation
real-time demos (as well as many 2-dice
problems!) at - http//noppa5.pc.helsinki.fi/koe/