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PROFESSOR: So, if you remember,
just before the

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break, as long ago as it was,
we had looked at the problem

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of fitting curves to data.

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00:00:35,520 --> 00:00:42,690
And the example we had seen, is
that it's often possible,

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00:00:42,690 --> 00:00:49,060
in fact, usually possible, to
find a good fit to old values.

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00:00:49,060 --> 00:00:51,460
What we looked at was, we looked
at a small number of

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00:00:51,460 --> 00:00:54,800
points, we took a high degree
polynomial, sure enough, we

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00:00:54,800 --> 00:00:57,970
got a great fit.

16
00:00:57,970 --> 00:01:03,210
The difficulty was, a great
fit to old values does not

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00:01:03,210 --> 00:01:24,840
necessarily imply a good
fit to new values.

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00:01:24,840 --> 00:01:31,150
And in general, that's
somewhat worrisome.

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00:01:31,150 --> 00:01:34,560
So now I want to spend a little
bit of time I'm looking

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00:01:34,560 --> 00:01:38,850
at some tools, that we can use
to better understand the

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00:01:38,850 --> 00:01:43,190
notion of, when we have a
bunch of points, what

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00:01:43,190 --> 00:01:45,220
do they look like?

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00:01:45,220 --> 00:01:47,610
How does the variation work?

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00:01:47,610 --> 00:01:51,680
This gets back to a concept that
we've used a number of

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00:01:51,680 --> 00:01:59,540
times, which is a notion
of a distribution.

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00:01:59,540 --> 00:02:04,860
Remember, the whole logic
behind our idea of using

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00:02:04,860 --> 00:02:11,340
simulation, or polling, or any
kind of statistical technique,

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00:02:11,340 --> 00:02:16,160
is the assumption that the
values we would draw were

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00:02:16,160 --> 00:02:21,650
representative of the values
of the larger population.

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00:02:21,650 --> 00:02:25,540
We're sampling some subset of
the population, and we're

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00:02:25,540 --> 00:02:30,150
assuming that that sample is
representative of the greater

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00:02:30,150 --> 00:02:31,700
population.

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00:02:31,700 --> 00:02:33,170
We talked about several
different

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00:02:33,170 --> 00:02:36,130
issues related to that.

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00:02:36,130 --> 00:02:40,400
I now want to look at that a
little bit more formally.

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00:02:40,400 --> 00:02:47,760
And we'll start with the very
old problem of rolling dice.

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00:02:47,760 --> 00:02:50,026
I presume you've all
seen what a pair of

38
00:02:50,026 --> 00:02:51,570
dice look like, right?

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00:02:51,570 --> 00:02:54,590
They've got the numbers 1
through 6 on them, you roll

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00:02:54,590 --> 00:02:57,730
them and something comes up.

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00:02:57,730 --> 00:03:01,700
If you haven't seen it, if you
look at the very back, at the

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00:03:01,700 --> 00:03:05,040
back page of the handout today,
you'll see a picture of

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00:03:05,040 --> 00:03:06,740
a very old die.

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00:03:06,740 --> 00:03:11,040
Some time from the fourth to
the second century BC.

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00:03:11,040 --> 00:03:15,260
Looks remarkably like a modern
dice, except it's not made out

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00:03:15,260 --> 00:03:18,610
of plastic, it's made
out of bones.

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00:03:18,610 --> 00:03:20,970
And in fact, if you were
interested in the history of

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00:03:20,970 --> 00:03:23,790
gambling, or if you happen to
play with dice, people do call

49
00:03:23,790 --> 00:03:25,610
them bones.

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00:03:25,610 --> 00:03:28,320
And that just dates back to
the fact that the original

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00:03:28,320 --> 00:03:30,770
ones were made that way.

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00:03:30,770 --> 00:03:35,505
And in fact, what we'll see
is, that in the history of

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00:03:35,505 --> 00:03:39,020
probability and statistics, an
awful lot of the math that we

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00:03:39,020 --> 00:03:42,240
take for granted today, came
from people's attempts to

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00:03:42,240 --> 00:03:46,290
understand various
games of chance.

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00:03:46,290 --> 00:03:50,230
So, let's look at it.

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00:03:50,230 --> 00:03:53,610
So we'll look at this program.

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00:03:53,610 --> 00:04:05,660
You should have this in the
front of the handout.

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00:04:05,660 --> 00:04:08,960
So I'm going to start
with a fair dice.

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00:04:08,960 --> 00:04:11,660
That is to say, when you roll
it, it's equally probable that

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00:04:11,660 --> 00:04:18,360
you get 1, 2, 3, 4, 5, or 6.

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And I'm going to throw a pair.

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You can see it's very simple.

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I'll take d 1, first die is
random dot choice from vals 1.

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00:04:30,910 --> 00:04:34,370
d 2 will be random dot
choice from vals 2.

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00:04:34,370 --> 00:04:40,100
So I'm going to pass it in two
sets of possible values, and

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00:04:40,100 --> 00:04:45,710
randomly choose one or the
other, and then return them.

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And the way I'll conduct a
trial is, I'll take some

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00:04:49,660 --> 00:04:53,940
number of throws, and two
different kinds of dice.

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00:04:53,940 --> 00:04:58,840
Throws will be the empty
set, actually, yeah.

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00:04:58,840 --> 00:05:01,070
And then I'll just do it.

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00:05:01,070 --> 00:05:06,270
For i in range number of throws,
d 1, d 2 is equal to

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00:05:06,270 --> 00:05:08,480
throw a pair, and then
I'll append it, and

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then I'll return it.

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Very simple, right?

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Could hardly imagine a simpler
little program.

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And then, we'll analyze it.

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And we're going to analyze it.

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Well, first let's analyze it one
way, and then we'll look

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00:05:25,430 --> 00:05:41,050
at something slightly
different.

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00:05:41,050 --> 00:05:43,200
I'm going to conduct some
number of trials

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00:05:43,200 --> 00:05:46,120
with two fair die.

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00:05:46,120 --> 00:05:50,270
Then I'm going to make a
histogram, because I happen to

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know that there are only
11 possible values,

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00:05:55,730 --> 00:06:01,600
I'll make 11 bins.

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00:06:01,600 --> 00:06:03,620
You may not have seen
this locution

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00:06:03,620 --> 00:06:06,710
here, Pylab dot x ticks.

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00:06:06,710 --> 00:06:10,680
That's telling it where to put
the markers on the x-axis, and

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00:06:10,680 --> 00:06:12,460
what they should be.

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00:06:12,460 --> 00:06:16,450
In this case 2 through 12,
and then I'll label it.

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So let's run this program.

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00:06:29,960 --> 00:06:34,760
And here we see the distribution
of values.

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00:06:34,760 --> 00:06:39,220
So we see that I get more 7s
than anything else, and fewer

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00:06:39,220 --> 00:06:42,320
2s and 12s.

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00:06:42,320 --> 00:06:47,540
Snake eyes and boxcars
to you gamblers.

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And it's a beautiful
distribution, in some sense.

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00:06:51,090 --> 00:06:53,150
I ran it enough trials.

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This kind of distribution
is called normal.

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00:07:01,240 --> 00:07:03,780
Also sometimes called
Gaussian, after the

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mathematician Gauss.

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Sometimes called the bell curve,
because in someone's

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imagination it looks
like a bell.

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00:07:16,650 --> 00:07:22,370
We see these things
all the time.

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00:07:22,370 --> 00:07:27,110
They're called normal, or
sometimes even natural,

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00:07:27,110 --> 00:07:30,800
because it's probably the most
commonly observed probability

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00:07:30,800 --> 00:07:35,460
distribution in nature.

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00:07:35,460 --> 00:07:40,240
First documented, although I'm
sure not first seen, by

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00:07:40,240 --> 00:07:44,470
deMoivre and Laplace
in the 1700s.

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00:07:44,470 --> 00:07:48,340
And then in the 1800s, Gauss
used it to analyze

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00:07:48,340 --> 00:07:51,980
astronomical data.

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And it got to be called, in
that case, the Gaussian

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00:07:54,280 --> 00:07:57,250
distribution.

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So where do we see it occur?

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We see it occurring all
over the place.

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00:08:00,870 --> 00:08:04,000
We certainly see it
rolling dice.

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00:08:04,000 --> 00:08:09,180
We see it occur in things like
the distribution of heights.

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00:08:09,180 --> 00:08:12,700
If we were to take the height of
all the students at MIT and

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00:08:12,700 --> 00:08:17,100
plot the distribution, I would
be astonished if it didn't

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00:08:17,100 --> 00:08:18,700
look more or less like that.

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00:08:18,700 --> 00:08:22,280
It would be a normal
distribution.

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00:08:22,280 --> 00:08:24,050
A lot of things in
the same height.

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00:08:24,050 --> 00:08:26,980
Now presumably, we'd have to
round off to the nearest

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00:08:26,980 --> 00:08:29,380
millimeter or something.

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00:08:29,380 --> 00:08:35,250
And a few really tall people,
and a few really short people.

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It's just astonishing in nature
how often we look at

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00:08:39,010 --> 00:08:40,690
these things.

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00:08:40,690 --> 00:08:44,900
The graph looks exactly like
that, or similar to that.

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00:08:44,900 --> 00:08:47,880
The shape is roughly that.

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00:08:47,880 --> 00:08:53,460
The normal distribution can be
described, interestingly

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00:08:53,460 --> 00:08:57,700
enough, with just two numbers.

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The mean and the standard
deviation.

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00:09:13,400 --> 00:09:15,920
So if I give you those
two numbers, you

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00:09:15,920 --> 00:09:18,540
can draw that curve.

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Now you might not be able to
label, you couldn't label the

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00:09:21,920 --> 00:09:25,210
axes, because how would
you know how many

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00:09:25,210 --> 00:09:26,540
trials I did, right?

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00:09:26,540 --> 00:09:29,720
Whether I did 100, or 1,000 or a
million, but the shape would

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00:09:29,720 --> 00:09:34,970
always be the same.

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00:09:34,970 --> 00:09:39,180
And if I were to, instead of
doing, 100,000 throws of the

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00:09:39,180 --> 00:09:44,030
dice, as I did here, I did a
million, the label on the

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00:09:44,030 --> 00:09:47,310
y-axis would change,
but the shape would

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00:09:47,310 --> 00:09:51,370
be absolutely identical.

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00:09:51,370 --> 00:09:54,430
This is what's called a
stable distribution.

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00:09:54,430 --> 00:10:14,160
As you change the scale, the
shape doesn't change.

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00:10:14,160 --> 00:10:18,370
So the mean tells us where
it's centered, and the

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00:10:18,370 --> 00:10:23,500
standard deviation, basically,
is a measure of statistical

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dispersion.

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00:10:42,570 --> 00:10:45,400
It tells us how widely
spread the points of

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00:10:45,400 --> 00:10:48,460
the data set are.

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00:10:48,460 --> 00:10:52,420
If many points are going to be
very close to the mean, then

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the standard deviation is
what, big or small?

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00:10:57,650 --> 00:10:59,090
Pardon?

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00:10:59,090 --> 00:11:04,720
Small.

154
00:11:04,720 --> 00:11:08,950
If they're spread out, and it's
kind of a flat bell, then

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00:11:08,950 --> 00:11:13,750
the standard deviation
will be large.

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00:11:13,750 --> 00:11:16,980
And I'm sure you've all seen
standard deviations.

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We give exams, and we say here's
the mean, here's the

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00:11:19,500 --> 00:11:22,630
standard deviation.

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00:11:22,630 --> 00:11:25,170
And the notion is, that's trying
to tell you what the

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average score was, and how
spread out they are.

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00:11:29,280 --> 00:11:35,080
Now as it happens, rarely do
we have exams that actually

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00:11:35,080 --> 00:11:37,050
fall on a bell curve.

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00:11:37,050 --> 00:11:38,270
Like this.

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00:11:38,270 --> 00:11:44,710
So in a way, don't be deceived
by thinking that we're really

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00:11:44,710 --> 00:11:48,790
giving you a good measure of the
dispersion, in the sense,

166
00:11:48,790 --> 00:11:52,880
that we would get with
the bell curve.

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00:11:52,880 --> 00:11:58,420
So the standard deviation does
have a formal value, usually

168
00:11:58,420 --> 00:12:11,440
written sigma, and it's the
estimates of x squared minus

169
00:12:11,440 --> 00:12:18,840
the estimates of x, and then
I take all of this, the

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00:12:18,840 --> 00:12:26,790
estimates of x, right,
squared.

171
00:12:26,790 --> 00:12:29,300
So I don't worry much about
this, but what I'm basically

172
00:12:29,300 --> 00:12:36,200
doing is, x is all of the
values I have. And I can

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00:12:36,200 --> 00:12:44,500
square each of the values, and
then I subtract from that, the

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00:12:44,500 --> 00:12:50,670
sum of the values,
squaring that.

175
00:12:50,670 --> 00:12:54,730
What's more important than this
formula, for most uses,

176
00:12:54,730 --> 00:13:03,480
is what people think of as the
-- why didn't I write it down

177
00:13:03,480 --> 00:13:07,580
-- this is interesting.

178
00:13:07,580 --> 00:13:09,260
There is a some number --

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00:13:09,260 --> 00:13:10,930
I see why, I did write
it down, it just

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00:13:10,930 --> 00:13:12,280
got printed on two-sided.

181
00:13:12,280 --> 00:13:26,540
Is the Empirical Rule.

182
00:13:26,540 --> 00:13:35,370
And this applies for normal
distributions.

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00:13:35,370 --> 00:13:38,680
So anyone know how much of the
data that you should expect to

184
00:13:38,680 --> 00:13:44,120
fall within one standard
deviation of the mean?

185
00:13:44,120 --> 00:13:49,200
68.

186
00:13:49,200 --> 00:13:59,330
So 68% within one, 95% of the
data falls within two, and

187
00:13:59,330 --> 00:14:06,720
almost all of the data
within three.

188
00:14:06,720 --> 00:14:11,080
These values are approximations,
by the way.

189
00:14:11,080 --> 00:14:20,070
So, this is, really 95% falls
within 1.96 standard

190
00:14:20,070 --> 00:14:23,530
deviations, it's not two.

191
00:14:23,530 --> 00:14:28,330
But this gives you a sense
of how spread out it is.

192
00:14:28,330 --> 00:14:33,770
And again, this applies only
for a normal distribution.

193
00:14:33,770 --> 00:14:37,650
If you compute the standard
deviation this way, and apply

194
00:14:37,650 --> 00:14:40,910
it to something other than a
normal distribution, there's

195
00:14:40,910 --> 00:14:52,130
no reason to expect that
Empirical Rule will hold.

196
00:14:52,130 --> 00:14:55,330
OK people with me on this?

197
00:14:55,330 --> 00:14:58,560
It's amazing to me how many
people in society talk about

198
00:14:58,560 --> 00:15:01,190
standard deviations
without actually

199
00:15:01,190 --> 00:15:02,860
knowing what they are.

200
00:15:02,860 --> 00:15:06,010
And there's another way to
look at the same data, or

201
00:15:06,010 --> 00:15:07,120
almost the same data.

202
00:15:07,120 --> 00:15:08,990
Since it's a random experiment,
it won't be

203
00:15:08,990 --> 00:15:22,220
exactly the same.

204
00:15:22,220 --> 00:15:28,670
So as before, we had the
distribution, and fortunately

205
00:15:28,670 --> 00:15:30,890
it looks pretty much
like the last one.

206
00:15:30,890 --> 00:15:32,950
We would've expected that.

207
00:15:32,950 --> 00:15:35,940
And I've now done something,
another way of looking at the

208
00:15:35,940 --> 00:15:42,050
same information, really, is,
I printed, I plotted, the

209
00:15:42,050 --> 00:15:46,900
probabilities of different
values.

210
00:15:46,900 --> 00:15:52,950
So we can see here that the
probability of getting a 7 is

211
00:15:52,950 --> 00:15:59,120
about 0.17 or something
like that.

212
00:15:59,120 --> 00:16:03,200
Now, since I threw 100,000 die,
it's not surprising that

213
00:16:03,200 --> 00:16:07,050
the probability of 0.17
looks about the same

214
00:16:07,050 --> 00:16:11,540
as 17,000 over here.

215
00:16:11,540 --> 00:16:15,770
But it's just a different way
of looking at things.

216
00:16:15,770 --> 00:16:21,480
Right, had I thrown some
different number, it might

217
00:16:21,480 --> 00:16:23,900
have been harder to visualize
what the probability

218
00:16:23,900 --> 00:16:26,000
distribution looked like.

219
00:16:26,000 --> 00:16:28,140
But we often do talk
about that.

220
00:16:28,140 --> 00:16:35,380
How probable is a
certain value?

221
00:16:35,380 --> 00:16:39,210
People who design games of
chance, by the way, something

222
00:16:39,210 --> 00:16:39,960
I've been meaning to say.

223
00:16:39,960 --> 00:16:42,690
You'll notice down here there's
just, when we want to

224
00:16:42,690 --> 00:16:45,840
save these things, there's
this little icon that's a

225
00:16:45,840 --> 00:16:48,800
floppy disk, to indicate
store.

226
00:16:48,800 --> 00:16:51,260
And I thought maybe many of
you'd never seen a floppy

227
00:16:51,260 --> 00:16:53,730
disk, so I decided
to bring one in.

228
00:16:53,730 --> 00:16:55,500
You've seen the icons.

229
00:16:55,500 --> 00:16:58,590
And probably by the time most of
you got, any you ever had a

230
00:16:58,590 --> 00:17:01,310
machine with a quote
floppy drive?

231
00:17:01,310 --> 00:17:04,030
Did it actually flop the disk?

232
00:17:04,030 --> 00:17:07,540
No, they were pretty rigid, but
the old floppy disks were

233
00:17:07,540 --> 00:17:09,840
really floppy.

234
00:17:09,840 --> 00:17:12,670
Hence they got the name.

235
00:17:12,670 --> 00:17:16,580
And, you know it's kind of like
a giant size version.

236
00:17:16,580 --> 00:17:19,800
And it's amazing how people will
probably continue to talk

237
00:17:19,800 --> 00:17:21,770
about floppy disks as
long as they talk

238
00:17:21,770 --> 00:17:24,270
about dialing a telephone.

239
00:17:24,270 --> 00:17:27,190
And probably none of you've ever
actually dialed a phone,

240
00:17:27,190 --> 00:17:29,820
for that matter, just
pushed buttons .

241
00:17:29,820 --> 00:17:33,040
But they used to have dials
that you would twirl.

242
00:17:33,040 --> 00:17:35,060
Anyway, I just thought everyone
should at least see a

243
00:17:35,060 --> 00:17:37,250
floppy disk once.

244
00:17:37,250 --> 00:17:40,140
This is, by the way, a very good
way to get data security.

245
00:17:40,140 --> 00:17:42,510
There's probably no way in the
world to read the information

246
00:17:42,510 --> 00:17:48,870
on this disk anymore.

247
00:17:48,870 --> 00:17:55,060
All right, as I said people
who design games of chance

248
00:17:55,060 --> 00:17:58,430
understand these probabilities
very well.

249
00:17:58,430 --> 00:18:02,010
So I'm gonna now look at, show
how we can understand these

250
00:18:02,010 --> 00:18:06,950
things in some other ways
of popular example.

251
00:18:06,950 --> 00:18:09,410
A game of dice.

252
00:18:09,410 --> 00:18:13,520
Has anyone here ever played
the game called craps?

253
00:18:13,520 --> 00:18:16,390
Did you win or lose money?

254
00:18:16,390 --> 00:18:18,050
You won.

255
00:18:18,050 --> 00:18:21,880
All right, you beat the odds.

256
00:18:21,880 --> 00:18:23,780
Well, it's a very popular
game, and I'm going to

257
00:18:23,780 --> 00:18:25,020
explain it to you.

258
00:18:25,020 --> 00:18:28,150
As you will see, this is not
an endorsement of gambling,

259
00:18:28,150 --> 00:18:30,370
because one of the things you
will notice is, you are likely

260
00:18:30,370 --> 00:18:32,590
to lose money if you do this.

261
00:18:32,590 --> 00:18:35,800
So I tend not, I don't do it.

262
00:18:35,800 --> 00:18:45,830
All right, so how does
a game of craps work?

263
00:18:45,830 --> 00:18:50,630
You start by rolling two dice.

264
00:18:50,630 --> 00:18:55,840
If you get a 7 or an 11, the
roller, we'll call that the

265
00:18:55,840 --> 00:19:01,490
shooter, you win.

266
00:19:01,490 --> 00:19:08,870
If you get a 2, 3, or
a 12, you lose.

267
00:19:08,870 --> 00:19:10,720
I'm assuming here, you're
betting what's

268
00:19:10,720 --> 00:19:12,690
called the pass line.

269
00:19:12,690 --> 00:19:14,720
There are different ways to bet,
this is the most common

270
00:19:14,720 --> 00:19:17,820
way to bet, we'll just
deal with that.

271
00:19:17,820 --> 00:19:27,150
If it's not any of these, what
you get is, otherwise, the

272
00:19:27,150 --> 00:19:37,330
number becomes what's
called the point.

273
00:19:37,330 --> 00:19:41,470
Once you've got the point, you
keep rolling the dice until 1

274
00:19:41,470 --> 00:19:44,540
of o things happens.

275
00:19:44,540 --> 00:19:56,430
You get a 7, in which case you
lose, or you get the point, in

276
00:19:56,430 --> 00:20:04,250
which case you win.

277
00:20:04,250 --> 00:20:07,420
So it's a pretty simple game.

278
00:20:07,420 --> 00:20:12,740
Very popular game.

279
00:20:12,740 --> 00:20:14,430
So I've implemented.

280
00:20:14,430 --> 00:20:18,400
So one of the interesting things
about this is, if you

281
00:20:18,400 --> 00:20:22,160
try and actually figure out
what the probabilities are

282
00:20:22,160 --> 00:20:25,700
using pencil and paper,
you can, but it gets

283
00:20:25,700 --> 00:20:33,670
a little bit involved.

284
00:20:33,670 --> 00:20:35,960
Gets involved because you have
to, all right, what are the

285
00:20:35,960 --> 00:20:38,890
odds of winning or losing
on the first throw?

286
00:20:38,890 --> 00:20:41,160
Well, you can compute those
pretty easily, and you can see

287
00:20:41,160 --> 00:20:43,550
that you'd actually win
more than you lose,

288
00:20:43,550 --> 00:20:44,920
on the first throw.

289
00:20:44,920 --> 00:20:48,030
But if you look at the
distribution of 7s and 11s and

290
00:20:48,030 --> 00:20:51,690
2s, 3s, and 12s, you add them
up, you'll see, well, this is

291
00:20:51,690 --> 00:20:53,890
more likely than this.

292
00:20:53,890 --> 00:20:55,900
But then you say, suppose
I don't get those.

293
00:20:55,900 --> 00:20:58,610
What's the likelihood of getting
each other possible

294
00:20:58,610 --> 00:21:02,070
point value, and then given that
point value, what's the

295
00:21:02,070 --> 00:21:04,900
likelihood of getting
that before a 7?

296
00:21:04,900 --> 00:21:07,720
And you can do it, but
it gets very tedious.

297
00:21:07,720 --> 00:21:11,450
So those of us who are
inclined to think

298
00:21:11,450 --> 00:21:15,430
computationally, and I hope by
now that's all of you as well

299
00:21:15,430 --> 00:21:19,270
as me, say well, instead of
doing the probabilities by

300
00:21:19,270 --> 00:21:22,420
hand, I'm just going to write
a little program.

301
00:21:22,420 --> 00:21:27,330
And it's a program that took me
maybe 10 minutes to write.

302
00:21:27,330 --> 00:21:34,240
You can see it's quite small,
I did it yesterday.

303
00:21:34,240 --> 00:21:39,620
So the first function here is
craps, it returns true if the

304
00:21:39,620 --> 00:21:43,350
shooter wins by betting
the pass line.

305
00:21:43,350 --> 00:21:46,380
And it's just does
what I said.

306
00:21:46,380 --> 00:21:48,860
Rolls them, if the total
is 1 or 11, it

307
00:21:48,860 --> 00:21:51,270
returns true, you win.

308
00:21:51,270 --> 00:21:54,750
If it's 2, 3, or 12, it returns
false, you lose.

309
00:21:54,750 --> 00:21:58,190
Otherwise the point
becomes the total.

310
00:21:58,190 --> 00:22:01,830
And then while true, I'll
just keep rolling.

311
00:22:01,830 --> 00:22:07,290
Until either, if the total gets
the point, I return true.

312
00:22:07,290 --> 00:22:09,930
Or if the total's equal
7, I return false.

313
00:22:09,930 --> 00:22:12,110
And that's it.

314
00:22:12,110 --> 00:22:15,590
So essentially I just took these
rules, typed them down,

315
00:22:15,590 --> 00:22:20,270
and I had my game.

316
00:22:20,270 --> 00:22:24,620
And then I'll simulate it will
some number of bets.

317
00:22:24,620 --> 00:22:28,980
Keeping track of the numbers
of wins and losses.

318
00:22:28,980 --> 00:22:31,950
Just by incrementing 1 or the
other, depending upon whether

319
00:22:31,950 --> 00:22:34,500
I return true or false.

320
00:22:34,500 --> 00:22:37,220
I'm going to, just to show what
we do, print the number

321
00:22:37,220 --> 00:22:39,410
of wins and losses.

322
00:22:39,410 --> 00:22:44,210
And then compute, how
does the house do?

323
00:22:44,210 --> 00:22:48,120
Not the gambler, but the person
who's running the game,

324
00:22:48,120 --> 00:22:49,610
the casino.

325
00:22:49,610 --> 00:22:53,720
Or in other circumstances,
other places.

326
00:22:53,720 --> 00:22:56,910
And then we'll see
how that goes.

327
00:22:56,910 --> 00:23:00,570
And I'll try it with
100,000 games.

328
00:23:00,570 --> 00:23:03,940
Now, this is more than 100,000
rolls of the dice, right?

329
00:23:03,940 --> 00:23:08,730
Because I don't get a 7
or 11, I keep rolling.

330
00:23:08,730 --> 00:23:15,790
So before I do it, I'll as the
easy question first. Who

331
00:23:15,790 --> 00:23:21,990
thinks the casino wins more
often than the player?

332
00:23:21,990 --> 00:23:26,150
Who thinks the player wins more
often than the casino?

333
00:23:26,150 --> 00:23:30,090
Well, very logical, casinos are
not in business of giving

334
00:23:30,090 --> 00:23:32,870
away money.

335
00:23:32,870 --> 00:23:34,800
So now the more interesting
question.

336
00:23:34,800 --> 00:23:38,770
How steep do you think the odds
are in the house's favor?

337
00:23:38,770 --> 00:23:43,980
Anyone want to guess?

338
00:23:43,980 --> 00:23:45,040
Actually pretty thin.

339
00:23:45,040 --> 00:23:57,850
Let's run it and see.

340
00:23:57,850 --> 00:24:02,800
So what we see is, the house
wins 50, in this case 50.424%

341
00:24:02,800 --> 00:24:04,770
of the time.

342
00:24:04,770 --> 00:24:06,360
Not a lot.

343
00:24:06,360 --> 00:24:08,570
On the other hand, if
people bet 100,000,

344
00:24:08,570 --> 00:24:14,200
the house wins 848.

345
00:24:14,200 --> 00:24:23,010
Now, 100,000 is actually
a small number.

346
00:24:23,010 --> 00:24:25,780
Let's get rid of these, should
have gotten rid of these

347
00:24:25,780 --> 00:24:53,560
figures, you don't need to
see them every time.

348
00:24:53,560 --> 00:24:58,240
We'll keep one figure,
just for fun.

349
00:24:58,240 --> 00:25:04,850
Let's try it again.

350
00:25:04,850 --> 00:25:06,300
Probably get a little
different answer.

351
00:25:06,300 --> 00:25:11,650
Considerable different, but
still, less than 51% of the

352
00:25:11,650 --> 00:25:13,270
time in this trial.

353
00:25:13,270 --> 00:25:16,070
But you can see that the house
is slowly but surely going to

354
00:25:16,070 --> 00:25:20,570
get rich playing this game.

355
00:25:20,570 --> 00:25:24,650
Now let's ask the other
interesting question.

356
00:25:24,650 --> 00:25:29,330
Just for fun, suppose
we want to cheat.

357
00:25:29,330 --> 00:25:32,670
Now, I realize none of you
would never do that.

358
00:25:32,670 --> 00:25:36,750
But let's consider using
a pair of loaded dice.

359
00:25:36,750 --> 00:25:39,870
So there's a long history, well
you can imagine when you

360
00:25:39,870 --> 00:25:43,480
looked at that old bone I showed
you, that it wasn't

361
00:25:43,480 --> 00:25:44,960
exactly fair.

362
00:25:44,960 --> 00:25:47,940
That some sides were a little
heavier than others, and in

363
00:25:47,940 --> 00:25:52,970
fact you didn't get, say, a
5 exactly 1/6 of the time.

364
00:25:52,970 --> 00:25:55,430
And therefore, if you were using
your own dice, instead

365
00:25:55,430 --> 00:25:57,950
of somebody else's, and you knew
what was most likely, you

366
00:25:57,950 --> 00:26:00,070
might do better.

367
00:26:00,070 --> 00:26:02,560
Well, the modern version of that
is, people do cheat by

368
00:26:02,560 --> 00:26:06,430
putting little weights in
dice, to just make tiny

369
00:26:06,430 --> 00:26:09,210
changes in the probability
of one number or

370
00:26:09,210 --> 00:26:11,240
another coming up.

371
00:26:11,240 --> 00:26:14,570
So let's do that.

372
00:26:14,570 --> 00:26:16,560
And let's first ask the
question, well, what would be

373
00:26:16,560 --> 00:26:20,250
a nice way to do that?

374
00:26:20,250 --> 00:26:24,540
It's very easy here.

375
00:26:24,540 --> 00:26:37,290
If we look at it, all I've
done is, I've changed the

376
00:26:37,290 --> 00:26:41,210
distribution of values, so
instead of here being 1, 2, 3,

377
00:26:41,210 --> 00:26:46,270
4, 5, and 6, it's 1, 2,
3, 4, 5, 5, and 6.

378
00:26:46,270 --> 00:26:51,520
I snuck in an extra 5 on
one of the two dice.

379
00:26:51,520 --> 00:26:59,300
So this has changed the odds of
rolling a 5 from 1 in 6 to

380
00:26:59,300 --> 00:27:04,150
roughly 3 in 12.

381
00:27:04,150 --> 00:27:09,510
Now 1/6, which is 2/12, vs.
3/12, it's not a big

382
00:27:09,510 --> 00:27:10,790
difference.

383
00:27:10,790 --> 00:27:12,950
And you can imagine, if you were
sitting there watching

384
00:27:12,950 --> 00:27:17,130
it, you wouldn't notice that 5
was coming up a little bit

385
00:27:17,130 --> 00:27:19,830
more often than you expected.

386
00:27:19,830 --> 00:27:21,150
Normally.

387
00:27:21,150 --> 00:27:23,750
Close enough that you
wouldn't notice it.

388
00:27:23,750 --> 00:27:28,800
But let's see if, what
difference it makes?

389
00:27:28,800 --> 00:27:30,590
What difference do you think
it will make here?

390
00:27:30,590 --> 00:27:32,840
First of all, is it going
to be better or

391
00:27:32,840 --> 00:27:36,110
worse for the player?

392
00:27:36,110 --> 00:27:39,830
Who thinks better?

393
00:27:39,830 --> 00:27:42,500
Who thinks worse?

394
00:27:42,500 --> 00:27:45,650
Who thinks they haven't
a clue?

395
00:27:45,650 --> 00:27:48,640
All right, we have
an honest man.

396
00:27:48,640 --> 00:27:52,130
Where is Diogenes when
we we need him?

397
00:27:52,130 --> 00:27:55,650
The reward for honesty.

398
00:27:55,650 --> 00:27:58,030
I could reward you and wake
him up at the same time.

399
00:27:58,030 --> 00:28:01,340
It's good.

400
00:28:01,340 --> 00:28:16,810
All right, well, let's
see what happens.

401
00:28:16,810 --> 00:28:21,095
All right, so suddenly, the odds
have swung in favor of

402
00:28:21,095 --> 00:28:23,090
the player.

403
00:28:23,090 --> 00:28:28,030
This tiny little change has now
made it likely that the

404
00:28:28,030 --> 00:28:33,440
player win money, instead
of the house.

405
00:28:33,440 --> 00:28:34,300
So what's the point?

406
00:28:34,300 --> 00:28:36,730
The point is not, you should
go out and try and cheat

407
00:28:36,730 --> 00:28:41,070
casinos, because you'll probably
find an unpleasant

408
00:28:41,070 --> 00:28:42,880
consequence of that.

409
00:28:42,880 --> 00:28:47,700
The point is that, once I've
written this simulation, I can

410
00:28:47,700 --> 00:28:52,480
play thought experiments
in a very easy way.

411
00:28:52,480 --> 00:28:53,870
So-called what if games.

412
00:28:53,870 --> 00:28:54,820
What if we did this?

413
00:28:54,820 --> 00:28:56,690
What if we did that?

414
00:28:56,690 --> 00:28:59,680
And it's trivial to do those
kinds of things.

415
00:28:59,680 --> 00:29:03,000
And that's one of the reasons we
typically do try and write

416
00:29:03,000 --> 00:29:04,640
these simulations.

417
00:29:04,640 --> 00:29:09,080
So that we can experiment
with things.

418
00:29:09,080 --> 00:29:11,070
Are there any other experiments
people would like

419
00:29:11,070 --> 00:29:13,500
to perform while we're here?

420
00:29:13,500 --> 00:29:18,280
Any other sets of die you
might like to try?

421
00:29:18,280 --> 00:29:22,060
All right, someone give me a
suggestion of something that

422
00:29:22,060 --> 00:29:24,320
might work in the
house's favor.

423
00:29:24,320 --> 00:29:27,120
Suppose a casino wanted
to cheat.

424
00:29:27,120 --> 00:29:28,430
What do you think would
help them out?

425
00:29:28,430 --> 00:29:28,690
Yeah?

426
00:29:28,690 --> 00:29:32,594
STUDENT: Increase prevalence
of 1, instead of 5?

427
00:29:32,594 --> 00:29:34,760
PROFESSOR: All right, so let's
see if we increase the

428
00:29:34,760 --> 00:29:52,210
probability of 1,
what it does?

429
00:29:52,210 --> 00:29:58,490
Yep, clearly helped the
house out, didn't it?

430
00:29:58,490 --> 00:30:04,750
So that would be a good
thing for the house.

431
00:30:04,750 --> 00:30:08,840
Again, you know, three key
strokes and we get to try it.

432
00:30:08,840 --> 00:30:15,350
It's really a very nice kind
of thing to be able to do.

433
00:30:15,350 --> 00:30:19,320
OK, this works nicely.

434
00:30:19,320 --> 00:30:21,350
We'll get normal
distributions.

435
00:30:21,350 --> 00:30:24,210
We can look at some things.

436
00:30:24,210 --> 00:30:26,130
There are two other kinds
of distributions I

437
00:30:26,130 --> 00:30:27,190
want to talk about.

438
00:30:27,190 --> 00:30:40,680
We can get rid of this
distraction.

439
00:30:40,680 --> 00:30:43,492
As you can imagine, I played a
lot with these things, just

440
00:30:43,492 --> 00:30:52,310
cause it was fun
once I had it.

441
00:30:52,310 --> 00:30:57,170
You have these in
your handout.

442
00:30:57,170 --> 00:31:01,800
So the one on the upper right,
is the Gaussian, or normal,

443
00:31:01,800 --> 00:31:07,800
distribution we've been
talking about.

444
00:31:07,800 --> 00:31:12,080
As I said earlier, quite common,
we see it a lot.

445
00:31:12,080 --> 00:31:17,090
The upper left is what's called
a, and these, by the

446
00:31:17,090 --> 00:31:19,550
way, all of these distributions
are symmetric,

447
00:31:19,550 --> 00:31:23,110
just in this particular
picture.

448
00:31:23,110 --> 00:31:26,750
How do you spell symmetric,
one or two m's?

449
00:31:26,750 --> 00:31:29,420
I help here.

450
00:31:29,420 --> 00:31:30,290
That right?

451
00:31:30,290 --> 00:31:32,720
OK, thank you.

452
00:31:32,720 --> 00:31:35,720
And they're symmetric in the
sense that, if you take the

453
00:31:35,720 --> 00:31:40,280
mean, it looks the same on
both sides of the mean.

454
00:31:40,280 --> 00:31:43,370
Now in general, you can
have asymmetric

455
00:31:43,370 --> 00:31:48,260
distributions as well.

456
00:31:48,260 --> 00:31:51,830
But for simplicity, we'll here
look at symmetric ones.

457
00:31:51,830 --> 00:31:54,950
So we've seen the bell curve,
and then on the upper left is

458
00:31:54,950 --> 00:32:04,290
what's called the uniform.

459
00:32:04,290 --> 00:32:09,690
In a uniform distribution, each
value in the range is

460
00:32:09,690 --> 00:32:15,810
equally likely.

461
00:32:15,810 --> 00:32:18,960
So to characterize it,
you only need to give

462
00:32:18,960 --> 00:32:21,930
the range of values.

463
00:32:21,930 --> 00:32:26,250
I say the values range from
0 to 100, and it tells me

464
00:32:26,250 --> 00:32:30,050
everything I know about the
uniform distribution.

465
00:32:30,050 --> 00:32:35,850
Each value in that will occur
the same number of times.

466
00:32:35,850 --> 00:32:39,970
Have we seen a uniform
distribution?

467
00:32:39,970 --> 00:32:46,350
What have we seen that's
uniform here?

468
00:32:46,350 --> 00:32:46,650
Pardon?

469
00:32:46,650 --> 00:32:48,620
STUDENT: Playing dice.

470
00:32:48,620 --> 00:32:49,990
PROFESSOR: Playing dice.

471
00:32:49,990 --> 00:32:51,580
Exactly right.

472
00:32:51,580 --> 00:32:57,780
Each roll of the die
was equally likely.

473
00:32:57,780 --> 00:32:59,900
Between 1 and 6.

474
00:32:59,900 --> 00:33:03,610
So, we got a normal distribution
when I summed

475
00:33:03,610 --> 00:33:09,710
them, but if I gave you the
distribution of a single die,

476
00:33:09,710 --> 00:33:11,870
it would have been
uniform, right?

477
00:33:11,870 --> 00:33:14,340
So there's an interesting
lesson there.

478
00:33:14,340 --> 00:33:18,060
One die, the distribution was
uniform, but when I summed

479
00:33:18,060 --> 00:33:27,700
them, I ended up getting
a normal distribution.

480
00:33:27,700 --> 00:33:29,870
So where else do we see them?

481
00:33:29,870 --> 00:33:34,070
In principle, lottery winners
are uniformly distributed.

482
00:33:34,070 --> 00:33:38,480
Each number is equally
likely to come up.

483
00:33:38,480 --> 00:33:41,640
To a first approximation,
birthdays are uniformly

484
00:33:41,640 --> 00:33:44,670
distributed, things like that.

485
00:33:44,670 --> 00:33:49,710
But, in fact, they rarely
arise in nature.

486
00:33:49,710 --> 00:33:52,350
You'll hardly ever run a physics
experiment, or a

487
00:33:52,350 --> 00:33:55,440
biology experiment, or anything
like that, and come

488
00:33:55,440 --> 00:33:58,100
up with a uniform
distribution.

489
00:33:58,100 --> 00:34:04,810
Nor do they arise very often in
complex systems. So if you

490
00:34:04,810 --> 00:34:07,970
look at what happens in
financial markets, none of the

491
00:34:07,970 --> 00:34:10,500
interesting distributions
are uniform.

492
00:34:10,500 --> 00:34:13,680
You know, the prices of stocks,
for example, are

493
00:34:13,680 --> 00:34:16,250
clearly not uniformly
distributed.

494
00:34:16,250 --> 00:34:19,820
Up days and down days in the
stock market are not uniformly

495
00:34:19,820 --> 00:34:22,780
distributed.

496
00:34:22,780 --> 00:34:28,700
Winners of football games are
not uniformly distributed.

497
00:34:28,700 --> 00:34:32,160
People seem to like to use
them in games of chance,

498
00:34:32,160 --> 00:34:35,120
because they seem fair, but
mostly you see them only in

499
00:34:35,120 --> 00:34:39,750
invented things, rather
than real things.

500
00:34:39,750 --> 00:34:41,800
The third kind of distribution,
the one in the

501
00:34:41,800 --> 00:34:55,010
bottom, is the exponential
distribution.

502
00:34:55,010 --> 00:34:59,360
That's actually quite common
in the real world.

503
00:34:59,360 --> 00:35:05,560
It's often used, for example,
to model arrival times.

504
00:35:05,560 --> 00:35:08,170
If you want to model the
frequency at which, say,

505
00:35:08,170 --> 00:35:16,040
automobiles arrive, get on the
Mass Turnpike, you would find

506
00:35:16,040 --> 00:35:20,750
that the arrivals
are exponential.

507
00:35:20,750 --> 00:35:25,180
We see with an exponential is,
things fall off much more

508
00:35:25,180 --> 00:35:34,700
steeply around the mean than
with the normal distribution.

509
00:35:34,700 --> 00:35:40,520
All right, that make sense?

510
00:35:40,520 --> 00:35:43,960
What else is exponentially
distributed?

511
00:35:43,960 --> 00:35:48,090
Requests for web pages are
often exponentially

512
00:35:48,090 --> 00:35:49,170
distributed.

513
00:35:49,170 --> 00:35:52,890
The amount of traffic
at a website.

514
00:35:52,890 --> 00:35:55,290
How frequently they arrive.

515
00:35:55,290 --> 00:35:59,640
We'll see much more starting
next week, or maybe even

516
00:35:59,640 --> 00:36:03,260
starting Thursday, about
exponential distributions, as

517
00:36:03,260 --> 00:36:06,370
we go on with a final case study
that we'll be dealing

518
00:36:06,370 --> 00:36:09,960
with in the course.

519
00:36:09,960 --> 00:36:15,460
You can think of each of
these, by the way, as

520
00:36:15,460 --> 00:36:19,350
increasing order of
predictability.

521
00:36:19,350 --> 00:36:23,590
Uniform distribution means the
result is most unpredictable,

522
00:36:23,590 --> 00:36:27,390
it could be anything.

523
00:36:27,390 --> 00:36:34,750
A normal distribution says,
well, it's pretty predictable.

524
00:36:34,750 --> 00:36:37,170
Again, depending on the
standard deviation.

525
00:36:37,170 --> 00:36:44,090
If you guess the mean, you're
pretty close to right.

526
00:36:44,090 --> 00:36:47,730
The exponential is
very predictable.

527
00:36:47,730 --> 00:36:55,100
Most of the answers are
right around the mean.

528
00:36:55,100 --> 00:36:57,220
Now there are many other
distributions, there are

529
00:36:57,220 --> 00:37:01,330
Pareto distributions which
have fat tails, there are

530
00:37:01,330 --> 00:37:05,920
fractal distributions, there
are all sorts of things.

531
00:37:05,920 --> 00:37:09,580
We won't go into to
those details.

532
00:37:09,580 --> 00:37:13,090
Now, I hope you didn't find
this short excursion into

533
00:37:13,090 --> 00:37:17,610
statistics either too boring
or too confusing.

534
00:37:17,610 --> 00:37:21,350
The point was not to teach you
statistics, probability, we

535
00:37:21,350 --> 00:37:24,040
have multiple courses
to do that.

536
00:37:24,040 --> 00:37:27,040
But to give you some tools that
would help improve your

537
00:37:27,040 --> 00:37:31,450
intuition in thinking
about data.

538
00:37:31,450 --> 00:37:36,400
In closing, I want to give
a few words about

539
00:37:36,400 --> 00:37:38,830
the misuse of data.

540
00:37:38,830 --> 00:37:43,400
Since I think we misuse
data an awful lot.

541
00:37:43,400 --> 00:37:57,960
So, point number 0, as in the
most important, is beware of

542
00:37:57,960 --> 00:38:25,100
people who give you properties
of data, but not the data.

543
00:38:25,100 --> 00:38:29,130
We see that sort of thing
all the time.

544
00:38:29,130 --> 00:38:33,940
Where people come in, and they
say, OK, here it is, here's

545
00:38:33,940 --> 00:38:37,120
the mean value of the quiz,
and here's the standard

546
00:38:37,120 --> 00:38:42,460
deviation of the quiz, and that
just doesn't really tell

547
00:38:42,460 --> 00:38:44,840
you where you stand,
in some sense.

548
00:38:44,840 --> 00:38:47,630
Because it's probably not
normally distributed.

549
00:38:47,630 --> 00:38:50,030
You want to see the data.

550
00:38:50,030 --> 00:38:53,310
At the very least, if you see
the data, you can then say,

551
00:38:53,310 --> 00:38:57,060
yeah, it is normally
distributed, so the standard

552
00:38:57,060 --> 00:39:01,530
deviation is meaningful,
or not meaningful.

553
00:39:01,530 --> 00:39:05,880
So, whenever you can,
try and get, at

554
00:39:05,880 --> 00:39:14,840
least, to see the data.

555
00:39:14,840 --> 00:39:18,500
So that's 1, or 0.

556
00:39:18,500 --> 00:39:24,830
1 is, well, all right.

557
00:39:24,830 --> 00:39:38,060
I'm going to test your Latin.

558
00:39:38,060 --> 00:39:41,060
Cum hoc ergo propter hoc.

559
00:39:41,060 --> 00:39:43,920
All right.

560
00:39:43,920 --> 00:39:49,170
I need a Latin scholar
to translate this.

561
00:39:49,170 --> 00:39:53,870
Did not one of you take
Latin in high school?

562
00:39:53,870 --> 00:39:55,110
We have someone who did.

563
00:39:55,110 --> 00:39:55,570
Go ahead.

564
00:39:55,570 --> 00:39:59,119
STUDENT: I think it
means, with this,

565
00:39:59,119 --> 00:40:01,150
therefore, because of this.

566
00:40:01,150 --> 00:40:03,070
PROFESSOR: Exactly right.

567
00:40:03,070 --> 00:40:08,090
With this, therefore,
because of this.

568
00:40:08,090 --> 00:40:09,660
I'm glad that at least
one person

569
00:40:09,660 --> 00:40:12,800
has a classical education.

570
00:40:12,800 --> 00:40:16,660
I don't, by the way.

571
00:40:16,660 --> 00:40:19,610
Essentially what this is
telling us, is that

572
00:40:19,610 --> 00:40:39,970
correlation does not
imply causation.

573
00:40:39,970 --> 00:40:42,830
So sometimes two things
go together.

574
00:40:42,830 --> 00:40:45,310
They both go up, they
both go down.

575
00:40:45,310 --> 00:40:46,990
And people jump to the
conclusion that

576
00:40:46,990 --> 00:40:49,590
one causes the other.

577
00:40:49,590 --> 00:40:53,250
That there's a cause and
effect relationship.

578
00:40:53,250 --> 00:40:57,660
That is just not true.

579
00:40:57,660 --> 00:41:01,100
It's what's called a
logical fallacy.

580
00:41:01,100 --> 00:41:04,310
So we see some examples
of this.

581
00:41:04,310 --> 00:41:05,710
And you can get into
big trouble.

582
00:41:05,710 --> 00:41:08,230
So here's a very interesting
one.

583
00:41:08,230 --> 00:41:11,110
There was a very widely reported
epidemiological

584
00:41:11,110 --> 00:41:14,830
study, that's a medical study
where you get statistics about

585
00:41:14,830 --> 00:41:16,870
large populations.

586
00:41:16,870 --> 00:41:20,570
And it showed that women, who
are taking hormone replacement

587
00:41:20,570 --> 00:41:24,850
therapy, were found to have a
lower incidence of coronary

588
00:41:24,850 --> 00:41:28,570
heart disease than
women who didn't.

589
00:41:28,570 --> 00:41:32,320
This was a big study
of a lot of women.

590
00:41:32,320 --> 00:41:38,230
This led doctors to propose
that hormone replacement

591
00:41:38,230 --> 00:41:41,430
therapy for middle aged women
was protective against

592
00:41:41,430 --> 00:41:44,210
coronary heart disease.

593
00:41:44,210 --> 00:41:48,720
And in fact, in response to
this, a large number of

594
00:41:48,720 --> 00:41:50,640
medical societies recommended
this.

595
00:41:50,640 --> 00:41:55,310
And a large number of women
were given this therapy.

596
00:41:55,310 --> 00:42:00,810
Later, controlled trials showed
that in fact, hormone

597
00:42:00,810 --> 00:42:04,960
replacement therapy in women
caused a small and significant

598
00:42:04,960 --> 00:42:10,430
increase in coronary
heart disease.

599
00:42:10,430 --> 00:42:13,350
So they had taken the fact that
these were correlated,

600
00:42:13,350 --> 00:42:17,760
said one causes the other, made
a prescription, and it

601
00:42:17,760 --> 00:42:20,840
turned out to be
the wrong one.

602
00:42:20,840 --> 00:42:25,100
Now, how could this be?

603
00:42:25,100 --> 00:42:27,960
How could this be?

604
00:42:27,960 --> 00:42:33,320
It turned out that the women in
the original study who were

605
00:42:33,320 --> 00:42:39,120
taking the hormone replacement
therapy, tended to be from a

606
00:42:39,120 --> 00:42:43,760
higher socioeconomic group
than those who didn't.

607
00:42:43,760 --> 00:42:46,460
Because the therapy was not
covered by insurance, so the

608
00:42:46,460 --> 00:42:48,960
women who took it
were wealthy.

609
00:42:48,960 --> 00:42:52,000
Turns out wealthy people do a
lot of other things that are

610
00:42:52,000 --> 00:42:54,010
protective of their hearts.

611
00:42:54,010 --> 00:42:57,520
And, therefore, are in general
healthier than poor people.

612
00:42:57,520 --> 00:42:59,370
This is not a surprise.

613
00:42:59,370 --> 00:43:02,590
Rich people are healthier
than poor people.

614
00:43:02,590 --> 00:43:09,090
And so in fact, it was this
third variable that was

615
00:43:09,090 --> 00:43:13,150
actually the meaningful one.

616
00:43:13,150 --> 00:43:14,460
This is what is called in

617
00:43:14,460 --> 00:43:30,500
statistics, a lurking variable.

618
00:43:30,500 --> 00:43:34,390
Both of the things they were
looking at in this study, who

619
00:43:34,390 --> 00:43:39,180
took the therapy, and who had a
heart disease, each of those

620
00:43:39,180 --> 00:43:42,320
was correlated with the
lurking variable of

621
00:43:42,320 --> 00:43:46,680
socioeconomic position.

622
00:43:46,680 --> 00:43:49,960
And so, in effect, there
was no cause and effect

623
00:43:49,960 --> 00:43:51,370
relationship.

624
00:43:51,370 --> 00:43:54,990
And once they did another study,
in which the lurking

625
00:43:54,990 --> 00:43:59,400
variable was controlled, and
they looked at heart disease

626
00:43:59,400 --> 00:44:02,620
among rich women separately
from poor women, with this

627
00:44:02,620 --> 00:44:07,060
therapy, they discovered that
therapy was not good.

628
00:44:07,060 --> 00:44:10,700
It was, in fact, harmful.

629
00:44:10,700 --> 00:44:16,290
So this is a very important
moral to remember.

630
00:44:16,290 --> 00:44:21,790
When you look at correlations,
don't assume cause and effect.

631
00:44:21,790 --> 00:44:25,540
And don't assume that there
isn't a lurking variable that

632
00:44:25,540 --> 00:44:29,940
really is the dominant factor.

633
00:44:29,940 --> 00:44:35,310
So that's one statistical, a
second statistical worry.

634
00:44:35,310 --> 00:44:48,260
Number 2 is, beware of what's
called, non-response bias.

635
00:44:48,260 --> 00:44:51,030
Which is another fancy
way of saying,

636
00:44:51,030 --> 00:45:05,120
non-representative samples.

637
00:45:05,120 --> 00:45:08,130
No one doing a study beyond
the trivial can sample

638
00:45:08,130 --> 00:45:11,590
everybody or everything.

639
00:45:11,590 --> 00:45:17,080
And only mind readers can be
sure of what they've missed.

640
00:45:17,080 --> 00:45:18,640
Unless, of course,
people choose to

641
00:45:18,640 --> 00:45:20,600
miss things on purpose.

642
00:45:20,600 --> 00:45:22,030
Which you also see.

643
00:45:22,030 --> 00:45:24,720
And that brings me to
my next anecdote.

644
00:45:24,720 --> 00:45:31,240
A former professor at the
University of Nebraska, who

645
00:45:31,240 --> 00:45:34,300
later headed a group called The
Family Research Institute,

646
00:45:34,300 --> 00:45:37,340
which some of you may have heard
about, claimed that gay

647
00:45:37,340 --> 00:45:43,610
men have an average life
expectancy of 43 years.

648
00:45:43,610 --> 00:45:46,570
And they did a study full of
statistics showing that this

649
00:45:46,570 --> 00:45:49,550
was the case.

650
00:45:49,550 --> 00:45:53,690
And the key was, they calculated
the figure by

651
00:45:53,690 --> 00:45:58,160
checking gay newspapers for
obituaries and news about

652
00:45:58,160 --> 00:46:00,170
stories of death.

653
00:46:00,170 --> 00:46:04,410
So they went through the gay
newspapers, took a list of

654
00:46:04,410 --> 00:46:08,190
everybody whose obituary
appeared, how old they were

655
00:46:08,190 --> 00:46:13,080
when they died, took the
average, and said it was 43.

656
00:46:13,080 --> 00:46:15,430
Then they did a bunch of
statistics, with all sorts of

657
00:46:15,430 --> 00:46:19,190
tests, showing how, you know,
what the curves look like, the

658
00:46:19,190 --> 00:46:21,310
distributions, and
the significance.

659
00:46:21,310 --> 00:46:25,130
All the math was valid.

660
00:46:25,130 --> 00:46:28,890
The problem was, it was a very
unrepresentative sample.

661
00:46:28,890 --> 00:46:29,680
What was the most

662
00:46:29,680 --> 00:46:32,990
unrepresentative
thing about it?

663
00:46:32,990 --> 00:46:33,480
Somebody?

664
00:46:33,480 --> 00:46:38,595
STUDENT: Not all deaths
have obituaries.

665
00:46:38,595 --> 00:46:40,930
PROFESSOR: Well, that's
one thing.

666
00:46:40,930 --> 00:46:41,820
That's certainly true.

667
00:46:41,820 --> 00:46:44,420
But what else?

668
00:46:44,420 --> 00:46:48,170
Well, not all gay people
are dead, right?

669
00:46:48,170 --> 00:46:51,370
So if you're looking at
obituaries, you're in fact

670
00:46:51,370 --> 00:46:52,690
only getting --

671
00:46:52,690 --> 00:46:56,310
I'm sure that's what you were
planning to say -- sorry.

672
00:46:56,310 --> 00:46:59,400
You're only getting the people
who are dead, so it's clearly

673
00:46:59,400 --> 00:47:02,970
going to make the number
look smaller, right?

674
00:47:02,970 --> 00:47:05,460
Furthermore, you're only getting
the ones that were

675
00:47:05,460 --> 00:47:09,720
reported in newspapers, the
newspapers are typically

676
00:47:09,720 --> 00:47:17,140
urban, rather than out in rural
areas, so it turns out,

677
00:47:17,140 --> 00:47:22,230
it's also biased against gays
who chose not come out of the

678
00:47:22,230 --> 00:47:26,400
closet, and therefore didn't
appear in these.

679
00:47:26,400 --> 00:47:29,410
Lots and lots of things
with the problems.

680
00:47:29,410 --> 00:47:31,260
Believe it or not, this
paper was published

681
00:47:31,260 --> 00:47:34,040
in a reputable journal.

682
00:47:34,040 --> 00:47:38,330
And someone checked all the
math, but missed the fact that

683
00:47:38,330 --> 00:47:49,100
all of that was irrelevant
because the sample was wrong.

684
00:47:49,100 --> 00:47:50,790
Data enhancement.

685
00:47:50,790 --> 00:47:53,500
It even sounds bad, right?

686
00:47:53,500 --> 00:47:56,820
You run an experiment, you get
your data, and you enhance it.

687
00:47:56,820 --> 00:47:59,460
It's kind of like when you ran
those physics experiments in

688
00:47:59,460 --> 00:48:02,640
high school, and you've got
answers that you knew didn't

689
00:48:02,640 --> 00:48:05,340
match the theory, so you
fudged the data?

690
00:48:05,340 --> 00:48:08,910
I know none of you would have
ever done that, but some

691
00:48:08,910 --> 00:48:09,850
people been known to do.

692
00:48:09,850 --> 00:48:12,380
That's not actually
what this means.

693
00:48:12,380 --> 00:48:16,260
What this means is, reading
more into the data than it

694
00:48:16,260 --> 00:48:19,390
actually implies.

695
00:48:19,390 --> 00:48:23,410
So well-meaning people are
often the guiltiest here.

696
00:48:23,410 --> 00:48:26,010
So here's another one
of my favorites.

697
00:48:26,010 --> 00:48:29,730
For example, there are people
who try to scare us into

698
00:48:29,730 --> 00:48:31,330
driving safely.

699
00:48:31,330 --> 00:48:33,920
Driving safely is
a good thing.

700
00:48:33,920 --> 00:48:36,280
By telling holiday deaths.

701
00:48:36,280 --> 00:48:39,790
So you'll read things like,
400 killed on the highways

702
00:48:39,790 --> 00:48:42,240
over long weekend.

703
00:48:42,240 --> 00:48:46,600
It sounds really bad, until
you observe the fact that

704
00:48:46,600 --> 00:48:51,940
roughly 400 people are killed
on any 3-day period.

705
00:48:51,940 --> 00:48:56,790
And in fact, it's no higher
on the holiday weekends.

706
00:48:56,790 --> 00:48:58,360
I'll bet you all thought
more people got

707
00:48:58,360 --> 00:48:59,580
killed on holiday weekends.

708
00:48:59,580 --> 00:49:02,330
Well, typically not.

709
00:49:02,330 --> 00:49:05,210
They just report how many died,
but they don't tell you

710
00:49:05,210 --> 00:49:12,440
the context, say, oh, by the
way, take any 3-day period.

711
00:49:12,440 --> 00:49:17,260
So the moral there is, you
really want to place the data

712
00:49:17,260 --> 00:49:20,250
in context.

713
00:49:20,250 --> 00:49:25,530
Data taken out of context
without comparison is usually

714
00:49:25,530 --> 00:49:27,460
meaningless.

715
00:49:27,460 --> 00:49:39,540
Another variance of this
is extrapolation.

716
00:49:39,540 --> 00:49:42,150
A commonly quoted statistic.

717
00:49:42,150 --> 00:49:45,300
Most auto accidents happen
within 10 miles of home.

718
00:49:45,300 --> 00:49:48,750
Anyone here heard
that statistic?

719
00:49:48,750 --> 00:49:53,080
It's true, but what
does it mean?

720
00:49:53,080 --> 00:49:55,730
Well, people tend to say it
means, it's dangerous to drive

721
00:49:55,730 --> 00:49:57,620
when you're near home.

722
00:49:57,620 --> 00:50:02,430
But in fact, most driving is
done within 10 miles of home.

723
00:50:02,430 --> 00:50:06,090
Furthermore, we don't actually
know where home is.

724
00:50:06,090 --> 00:50:09,370
Home is where the car is
supposedly garaged on the

725
00:50:09,370 --> 00:50:14,800
state registration forms. So,
data enhancements would

726
00:50:14,800 --> 00:50:18,650
suggest that I should register
my car in Alaska.

727
00:50:18,650 --> 00:50:20,980
And then I would never be
driving within 10 miles of

728
00:50:20,980 --> 00:50:24,100
home, and I would
be much safer.

729
00:50:24,100 --> 00:50:32,330
Well, it's probably
not a fact.

730
00:50:32,330 --> 00:50:37,020
So there are all sorts
of things on that.

731
00:50:37,020 --> 00:50:39,090
Well, I think I will come back
to this, because I have a

732
00:50:39,090 --> 00:50:42,090
couple more good stories which
I hate not to give you.

733
00:50:42,090 --> 00:50:45,750
So we'll come back on Thursday
and look at a couple of more

734
00:50:45,750 --> 00:50:48,180
things that can go wrong
with statistics.