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.