Transcript Number 4 - University of Notre Dame

N.D. Analysis
We are NOT going to analyze Notre Dame, just
Numerical (Quantitative) Data.
There are essentially two ways to represent and
analyze quantitative data,
1. visually
2. computationally
We already have seen the four main ways to
represent numerical data visually.
° dot plots
° pie charts
° histograms
° stem and leaf
For the sake of completeness we look at an
example of each:
Dot Plot:
Histogram
Pie Chart
Stem and Leaf
There is a fifth way, very inventive, called a
box plot
we will learn it a little later.
We have to learn first how to work
Computationally
with a collection of numerical data.
First of all we want to distinguish results computed
from collections that
come from a whole population
from results computed from collections that
come from a sample
1. Results from populations are called
parameters
2. Results from samples are called
statistics
The rule is easy to remember:
Population  Parameter
Sample  Statistic
(the PP-SS rule)
With one exception, the calculations one does are
the same, whether one is computing a parameter
or a statistic. The exception will come later.
What kind of results will we be looking for? In other
words, what calculations can you do on a given set
of numbers?
The list is endless, we have seen the following
mentioned in class:
Range, maximum, minimum, average, bidding
method, variance, standard deviation, …
A long standing convention stipulates that
symbols used for parameters are Greek letters
symbols used for statistics are latin letters
A wonderful opportunity to learn some Greek
letters! We will mostly use two:
and
Important Calculations
Time to start computing things. From now on,
until stated otherwise, we assume we have a
given set of numerical data
x1, x2, x3,…, xn
We also assume that we have already taken the
trouble to order the data, that is
x1 ≤ x2 ≤ x3 ≤ … ≤ xn
We will be interested in “measuring” two
concepts:
“Centrality” and “Spread”
We measure “centrality” by trying to compute
the “center” of the data. The trouble is that we
don’t know what we mean by
that’s right, the center !
Intuitively the center of two data, say 7 and 21,
should be the midpoint,
14 (you don’t need your TI84 plus for this, I
hope! )
But what if the data are 7, 7 and 21 ?
Most of you already know the answer ….
That’s right, the average !
By the way, another word for average (more
impressive!) is “mean”. Notation:
The symbol for the sample mean is x
The symbol for the population mean is
Using the
notation we learned in the last class
we can write:
The mean has one serious defect:
It is extremely sensitive to extreme values. The
following set of data
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 1,000,000
ought to have a center kind of close to 8 (there are
15 of them!)
Instead your TI84 plus will tell you that the mean is
62,507.50 (wow !)
A more faithful (but harder to compute in general,
and computationally less useful) measure of
centrality is
the median
You know the definition, essentially it is a number
that cuts the ordered data into two equal halves.
More precisely,
If you have an odd number of data (think of 37),
you pick the one smack in the middle
(the 19-th, think about it!)
If you have an even number of data (think of 46)
you pick any number you please ….
between the 23-rd and 24-th entries
(if they are equal you got no choice!)
We do a couple of examples on the board. Then
one on the computer.
Distributional Symmetry
(or lack thereof !)
What’s the relationship, if any, between
the median and the mean
??
Just for fun let’s call the median Md and the mean Mn (clever, huh ?).
Then either
Mn < Md or
Mn = Md
or
Mn > Md
Now, by its very definition Md is in the Middle !
On the other hand, Mn is the average, the middle value.
When the data are distributed in perfect symmetry we must therefore
have
Mn = Md
What if Mn < Md ?
It means that somehow the average is pulled down to the left of the
middle. There are more extreme values to the left.
We say that the data are skewed to the left.
In the same way, if Mn > Md
it means that somehow the average is pulled up to
the right of the middle. There are more extreme
values to the right.
We say that the data are skewed to the right.
Of course, there can’t be too many of these
extreme values, on either side, because in that
case the median is shifted.
In terms of histograms, we get these four cases,
shown on the next slides.
Symmetric distributions
Distributions like the this one
(symmetric and bunched around the middle) are said
to be
“nicely mound-shaped.”
We’ll say more about them later.