So the example that
I want to started off with involves books.
First we’ll think about books
as physical objects.
Here are 2 books on fractals,
and shapes and other things.
That I find myself consulting
as I prepare these lectures.
These are physical objects
and they have a mass.
I could put them on a scale and weigh them.
And this book,
this is sort of an average size book,
may be weighs pound.
This is little smaller
may be weighs half a pound.
So not all books are the same size.
But most are about this size.
Some are really small.
This is the smallest fractals book that I have.
It is Kenneth Falconer’s very short introduction.
It’s a great book.
And then the largest fractals book I have
is this one.
I can barely hold it with one hand.
So it’s much larger
and again, this is my biggest fractals book.
So books come in different sizes.
But there is an average size.
And there is not too much variation around that.
I could if I was curious
and want to push us farther,
I could take a bunch of books off my shelf.
I could weight them all and
I could make a histogram.
And I would find that
the distribution of book masses is centered
around the well-defined average.
And there is not too much of a variation
from side to side in either direction.
So let’s say actually do this experiment.
And I started weighing all of my books.
I would get a series of masses, weights.
And let’s say that may be I would get
one that would be point 9
imagine it’s in pounds
might get 0 point 6
1 point 1
1 point 05
0 point 8
and may be another 0 point 9
So the idea here is let’s just pretend,
because I really didn’t feel like weighing all my books
That I really do this 1,2,3,4,5,6
I weigh 6 books
And I could then present this data
in a histogram.
So just a quick reminder how that would go.
Let’s see here.
I’ll do 0.5
0 .75
1.0
and 1.25
So this is gonna be my mass.
And then I call frequency here.
So 0.9 I have got one
that’s between 0.75 and 1
0.6
1.1
this one adds in here
1.05
0.8 and 0.9 those both are in here.
So the picture here is as follows.
What this type of plot shows me
is that there are 3 books
that have a mass between 0.75 and 1
There is 1 book
that’s between half a pound and 0.75
And there are 2 books
that are between 1.0 and 1.25
So histogram is a way of just summarizing
of a bunch of data.
And this shows us what the frequency is.
We can also think about this
in terms of probability if I want it
higher this bar is here the more probable.
It is that I would observe a mass in this range.
And in this particular case this says
If I really choose one of these books at random
there is a 50 percent chance
that it would be between 0.75 and 1.0
Right, so far, I keep going and let’s say
I were to weigh hundred books
or two hundred books.
That would be enough fun for afternoon.
But I might expect this histogram
might end up looking something like this.
And what this would show me is that
there are a lot of books
concentrated around here
may be this is about pound.
It seems like most books are about pound.
And then there are few of many books
are a little bit on either side.
There aren’t too many books that are
really really heavy or really really light.
So there is a nice sort of well-defined range.
So this is just another way of saying
that books have a typical mass,
probably about a pound or so.
And a lot of thing in the world are like this.
Tomatoes are like this.
There is a typical size for tomatoes.
People’s heights are like this.
Most people are between 5 feet and 6 feet tall
and so on.
So this is a sort of thing we might get
if we are to weigh a bunch of books.