Bhaskara's proof of the Pythagorean theorem

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:05.060
I will now do a proof for which
we credit the 12th century

00:00:05.060 --> 00:00:08.044
Indian mathematician, Bhaskara.

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So what we're going
to do is we're

00:00:09.460 --> 00:00:11.210
going to start with a square.

00:00:11.210 --> 00:00:13.750
So let me see if I
can draw a square.

00:00:13.750 --> 00:00:16.210
I'm going to draw it
tilted at a bit of an angle

00:00:16.210 --> 00:00:18.910
just because I think it'll make
it a little bit easier on me.

00:00:18.910 --> 00:00:22.600
So let me do my best
attempt at drawing something

00:00:22.600 --> 00:00:25.670
that reasonably
looks like a square.

00:00:25.670 --> 00:00:29.490
You have to bear with me if it's
not exactly a tilted square.

00:00:29.490 --> 00:00:33.389
So that looks pretty good.

00:00:33.389 --> 00:00:34.680
And I'm assuming it's a square.

00:00:34.680 --> 00:00:36.430
So this is a right angle.

00:00:36.430 --> 00:00:37.840
This is a right angle.

00:00:37.840 --> 00:00:38.850
That's a right angle.

00:00:38.850 --> 00:00:39.841
That's a right angle.

00:00:39.841 --> 00:00:42.340
I'm assuming the lengths of all
of these sides are the same.

00:00:42.340 --> 00:00:45.130
So let's just assume that
they're all of length, c.

00:00:45.130 --> 00:00:46.430
I'll write that in yellow.

00:00:46.430 --> 00:00:49.790
So all of the sides of the
square are of length, c.

00:00:49.790 --> 00:00:52.420
And now I'm going to construct
four triangles inside

00:00:52.420 --> 00:00:53.270
of this square.

00:00:53.270 --> 00:00:56.030
And the way I'm going to do it
is I'm going to be dropping.

00:00:56.030 --> 00:00:58.020
So here I'm going
to go straight down,

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and I'm going to drop a
line straight down and draw

00:01:03.560 --> 00:01:06.047
a triangle that looks like this.

00:01:06.047 --> 00:01:07.630
So I'm going to go
straight down here.

00:01:07.630 --> 00:01:09.964
Here, I'm going to
go straight across.

00:01:09.964 --> 00:01:11.380
And so since this
is straight down

00:01:11.380 --> 00:01:14.180
and this is straight across,
we know this is a right angle.

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Then from this
vertex on our square,

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I'm going to go straight up.

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And since this is straight up
and this is straight across,

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we know that this
is a right angle.

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And then from this
vertex right over here,

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I'm going to go
straight horizontally.

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I'm assuming that's
what I'm doing.

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And so we know that this is
going to be a right angle,

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and then we know this is
going to be a right angle.

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So we see that we've
constructed, from our square,

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we've constructed
four right triangles.

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And in between,
we have something

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that, at minimum, looks like a
rectangle or possibly a square.

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We haven't quite
proven to ourselves

00:01:49.090 --> 00:01:51.064
yet that this is a square.

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Now the next thing I
want to think about

00:01:52.730 --> 00:01:55.220
is whether these
triangles are congruent.

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So they definitely all
have the same length

00:01:57.860 --> 00:01:58.760
of their hypotenuse.

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All of the hypot-- I don't know
what the plural of hypotenuse

00:02:01.301 --> 00:02:03.250
is, hypoteni, hypotenuses.

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They have all length, c.

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The side opposite the right
angle is always length, c.

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So if we can show that all
the corresponding angles are

00:02:11.157 --> 00:02:12.740
the same, then we
know it's congruent.

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If you have something where
all the angles are the same

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and you have a
side that is also--

00:02:17.102 --> 00:02:18.810
the corresponding side
is also congruent,

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then the whole
triangles are congruent.

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And we can show
that if we assume

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that this angle is theta.

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Then this angle right over
here has to be 90 minus theta

00:02:29.930 --> 00:02:32.790
because together they
are complimentary.

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We know that because
they go combine

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to form this angle of the
square, this right angle.

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And this is 90 minus theta.

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We know this angle
and this angle

00:02:41.240 --> 00:02:42.760
have to add up to
90 because we only

00:02:42.760 --> 00:02:46.200
have 90 left when we subtract
the right angle from 180.

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So we know this has to be theta.

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And if that's theta, then
that's 90 minus theta.

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I think you see
where this is going.

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If that's 90 minus theta,
this has to be theta.

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And if that's theta, then
this is 90 minus theta.

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If this is 90 minus
theta, then this is theta,

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and then this would have
to be 90 minus theta.

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So we see in all four
of these triangles,

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the three angles are theta, 90
minus theta, and 90 degrees.

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So they all have the
same exact angle,

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so at minimum, they are
similar, and their hypotenuses

00:03:18.190 --> 00:03:19.040
are the same.

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So we know that all
four of these triangles

00:03:22.210 --> 00:03:24.950
are completely
congruent triangles.

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So with that
assumption, let's just

00:03:26.940 --> 00:03:32.320
assume that the longer
side of these triangles,

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that these are of length, b.

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So the longer side of
these triangles I'm

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just going to assume.

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So this length right over here,
I'll call that lowercase b.

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And let's assume that the
shorter side, so this distance

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right over here, this distance
right over here, this distance

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right over here, that these
are all-- this distance right

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over here, that these
are of length, a.

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So if I were to say this
height right over here,

00:04:01.740 --> 00:04:06.720
this height is of length--
that is of length, a.

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Now we will do
something interesting.

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Well, first, let's think about
the area of the entire square.

00:04:11.460 --> 00:04:14.772
What's the area of the
entire square in terms of c?

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Well, that's pretty
straightforward.

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It's a c by c square.

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So the area here is
equal to c squared.

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Now, what I'm going
to do is rearrange

00:04:29.960 --> 00:04:32.080
two of these triangles
and then come up

00:04:32.080 --> 00:04:35.850
with the area of that other
figure in terms of a's and b's,

00:04:35.850 --> 00:04:38.760
and hopefully it gets us
to the Pythagorean theorem.

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And to do that, just so we
don't lose our starting point

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because our starting
point is interesting,

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let me just copy and
paste this entire thing.

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So I don't want it to clip off.

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So let me just copy
and paste this.

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Copy and paste.

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So this is our original diagram.

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And what I will now
do-- and actually, let

00:04:59.310 --> 00:05:01.236
me clear that out.

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Edit clear.

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I'm now going to shift.

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This is the fun part.

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I'm going to shift this
triangle here in the top left.

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I'm going to shift it below this
triangle on the bottom right.

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And I'm going to attempt to do
that by copying and pasting.

00:05:14.440 --> 00:05:17.210
So let's see how much--
well, the way I drew it,

00:05:17.210 --> 00:05:20.450
it's not that-- well,
that might do the trick.

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I want to retain a little
bit of the-- so let me copy,

00:05:24.300 --> 00:05:27.930
or let me actually cut it,
and then let me paste it.

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So that triangle I'm going
to stick right over there.

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And let me draw in the
lines that I just erased.

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So just to be clear, we
had a line over there,

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and we also had this
right over here.

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And this was
straight up and down,

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and these were
straight side to side.

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Now, so I moved this
part over down here.

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So I moved that over down there.

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And now I'm going to move
this top right triangle down

00:06:01.070 --> 00:06:04.050
to the bottom left.

00:06:04.050 --> 00:06:07.570
So I'm just rearranging
the exact same area.

00:06:07.570 --> 00:06:11.290
So actually let me just
capture the whole thing

00:06:11.290 --> 00:06:13.660
as best as I can.

00:06:13.660 --> 00:06:18.850
So let me cut and
then let me paste.

00:06:18.850 --> 00:06:21.764
And I'm going to move
it right over here.

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While I went through
that process,

00:06:23.180 --> 00:06:26.510
I kind of lost its floor,
so let me redraw the floor.

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So I just moved it
right over here.

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So this thing,
this triangle-- let

00:06:31.660 --> 00:06:37.280
me color it in-- is
now right over there.

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And this triangle is
now right over here.

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That center square, it is a
square, is now right over here.

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So hopefully you can appreciate
how we rearranged it.

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Now my question for
you is, how can we

00:07:00.580 --> 00:07:02.470
express the area of
this new figure, which

00:07:02.470 --> 00:07:04.220
has the exact same
area as the old figure?

00:07:04.220 --> 00:07:06.160
I just shifted
parts of it around.

00:07:06.160 --> 00:07:11.030
How can we express this in
terms of the a's and b's?

00:07:11.030 --> 00:07:13.640
Well, the key insight
here is to recognize

00:07:13.640 --> 00:07:15.840
the length of this bottom side.

00:07:15.840 --> 00:07:19.970
What's the length of this
bottom side right over here?

00:07:19.970 --> 00:07:22.930
The length of this bottom
side-- well this length right

00:07:22.930 --> 00:07:26.450
over here is b, this length
right over here is a.

00:07:26.450 --> 00:07:31.590
So the length of this
entire bottom is a plus b.

00:07:31.590 --> 00:07:34.970
Well that by itself is
kind of interesting.

00:07:34.970 --> 00:07:41.369
But what we can realize is that
this length right over here,

00:07:41.369 --> 00:07:43.660
which is the exact same thing
as this length over here,

00:07:43.660 --> 00:07:45.560
was also a.

00:07:45.560 --> 00:07:47.785
So we can construct
an a by a square.

00:07:50.760 --> 00:07:54.310
So this square right
over here is a by a,

00:07:54.310 --> 00:07:57.110
and so it has area, a squared.

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Let me do that in a color
that you can actually see.

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So this has area of a squared.

00:08:03.710 --> 00:08:06.790
And then what's the area
of what's left over?

00:08:06.790 --> 00:08:12.410
Well if this is length, a, then
this is length, a, as well.

00:08:12.410 --> 00:08:15.340
If this entire
bottom is a plus b,

00:08:15.340 --> 00:08:17.920
then we know that
what's left over

00:08:17.920 --> 00:08:20.369
after subtracting
the a out has to b.

00:08:20.369 --> 00:08:21.910
If this whole thing
is a plus b, this

00:08:21.910 --> 00:08:24.830
is a, then this
right over here is b.

00:08:24.830 --> 00:08:29.060
And so the rest of this
newly oriented figure,

00:08:29.060 --> 00:08:33.690
this new figure, everything
that I'm shading in over here,

00:08:33.690 --> 00:08:36.809
this is just a b by b square.

00:08:36.809 --> 00:08:39.130
So the area here is b squared.

00:08:39.130 --> 00:08:40.950
So the entire area
of this figure

00:08:40.950 --> 00:08:45.150
is a squared plus b
squared, which lucky for us,

00:08:45.150 --> 00:08:48.930
is equal to the area of this
expressed in terms of c because

00:08:48.930 --> 00:08:51.250
of the exact same
figure, just rearranged.

00:08:51.250 --> 00:08:54.045
So it's going to be
equal to c squared.

00:08:54.045 --> 00:08:57.110
And it all worked out,
and Bhaskara gave us

00:08:57.110 --> 00:09:02.320
a very cool proof of
the Pythagorean theorem.


Garfield's proof of the Pythagorean theorem

WEBVTT
Kind: captions
Language: en

00:00:00.421 --> 00:00:01.920
What we're going
to do in this video

00:00:01.920 --> 00:00:04.190
is study a proof of
the Pythagorean theorem

00:00:04.190 --> 00:00:07.590
that was first
discovered, or as far

00:00:07.590 --> 00:00:13.695
as we know first discovered,
by James Garfield in 1876,

00:00:13.695 --> 00:00:15.070
and what's exciting
about this is

00:00:15.070 --> 00:00:17.670
he was not a professional
mathematician.

00:00:17.670 --> 00:00:21.086
You might know James Garfield
as the 20th president

00:00:21.086 --> 00:00:21.960
of the United States.

00:00:21.960 --> 00:00:23.950
He was elected president.

00:00:23.950 --> 00:00:29.070
He was elected in 1880, and then
he became president in 1881.

00:00:29.070 --> 00:00:31.930
And he did this proof while
he was a sitting member

00:00:31.930 --> 00:00:33.889
of the United States
House of Representatives.

00:00:33.889 --> 00:00:35.264
And what's exciting
about that is

00:00:35.264 --> 00:00:38.390
that it shows that Abraham
Lincoln was not the only US

00:00:38.390 --> 00:00:41.330
politician or not the only
US President who was really

00:00:41.330 --> 00:00:43.280
into geometry.

00:00:43.280 --> 00:00:45.429
And what Garfield realized
is, if you construct

00:00:45.429 --> 00:00:46.970
a right triangle--
so I'm going to do

00:00:46.970 --> 00:00:49.310
my best attempt
to construct one.

00:00:49.310 --> 00:00:51.820
So let me construct
one right here.

00:00:51.820 --> 00:00:56.440
So let's say this side
right over here is length b.

00:00:56.440 --> 00:01:02.060
Let's say this side is
length a, and let's say

00:01:02.060 --> 00:01:05.269
that this side, the hypotenuse
of my right triangle,

00:01:05.269 --> 00:01:06.150
has length c.

00:01:06.150 --> 00:01:08.860
So I've just constructed
enough a right triangle,

00:01:08.860 --> 00:01:10.200
and let me make it clear.

00:01:10.200 --> 00:01:12.020
It is a right triangle.

00:01:12.020 --> 00:01:15.090
He essentially flipped and
rotated this right triangle

00:01:15.090 --> 00:01:18.750
to construct another one that
is congruent to the first one.

00:01:18.750 --> 00:01:19.900
So let me construct that.

00:01:19.900 --> 00:01:23.420
So we're going to have length b,
and it's collinear with length

00:01:23.420 --> 00:01:23.920
a.

00:01:23.920 --> 00:01:26.600
It's along the same
line, I should say.

00:01:26.600 --> 00:01:28.530
They don't overlap
with each other.

00:01:28.530 --> 00:01:33.070
So this is side of
length b, and then you

00:01:33.070 --> 00:01:35.340
have a side of
length-- let me draw

00:01:35.340 --> 00:01:39.460
a it so this will be a little
bit taller-- side of length b.

00:01:39.460 --> 00:01:44.210
And then, you have your side
of length a at a right angle.

00:01:44.210 --> 00:01:48.190
Your side of length a
comes in at a right angle.

00:01:48.190 --> 00:01:49.970
And then, you have
your side of length c.

00:01:56.232 --> 00:01:57.940
So the first thing we
need to think about

00:01:57.940 --> 00:02:01.190
is what's the angle
between these two sides?

00:02:01.190 --> 00:02:03.140
What's this mystery angle?

00:02:03.140 --> 00:02:06.787
What's that mystery
angle going to be?

00:02:06.787 --> 00:02:08.620
Well, it looks like
something, but let's see

00:02:08.620 --> 00:02:10.411
if we can prove to
ourselves that it really

00:02:10.411 --> 00:02:12.570
is what we think it looks like.

00:02:12.570 --> 00:02:15.340
If we look at this
original triangle

00:02:15.340 --> 00:02:18.030
and we call this
angle "theta," what's

00:02:18.030 --> 00:02:20.210
this angle over here,
the angle that's

00:02:20.210 --> 00:02:22.840
between sides of
length a and length c?

00:02:22.840 --> 00:02:26.000
What's the measure of
this angle going to be?

00:02:26.000 --> 00:02:28.650
Well, theta plus this
angle have to add up to 90.

00:02:28.650 --> 00:02:30.900
Because you add those two
together, they add up to 90.

00:02:30.900 --> 00:02:32.149
And then, you have another 90.

00:02:32.149 --> 00:02:35.080
You're going to get 180
degrees for the interior angles

00:02:35.080 --> 00:02:36.640
of this triangle.

00:02:36.640 --> 00:02:38.360
So these two have
to add up to 90.

00:02:38.360 --> 00:02:41.950
This angle is going
to be 90 minus theta.

00:02:41.950 --> 00:02:43.850
Well, if this triangle
appears congruent--

00:02:43.850 --> 00:02:46.110
and we've constructed
it so it is congruent--

00:02:46.110 --> 00:02:47.600
the corresponding
angle to this one

00:02:47.600 --> 00:02:48.850
is this angle right over here.

00:02:48.850 --> 00:02:51.430
So this is also going to be
theta, and this right over here

00:02:51.430 --> 00:02:53.385
is going to be 90 minus theta.

00:02:53.385 --> 00:02:54.760
So given that this
is theta, this

00:02:54.760 --> 00:02:57.550
is 90 minus theta, what
is our angle going to be?

00:02:57.550 --> 00:03:01.310
Well, they all collectively
go 180 degrees.

00:03:01.310 --> 00:03:05.010
So you have theta,
plus 90 minus theta,

00:03:05.010 --> 00:03:10.020
plus our mystery angle is going
to be equal to 180 degrees.

00:03:10.020 --> 00:03:12.770
The thetas cancel out.

00:03:12.770 --> 00:03:13.840
Theta minus theta.

00:03:13.840 --> 00:03:16.610
And you have 90 plus our
mystery angle is 180 degrees.

00:03:16.610 --> 00:03:19.840
Subtract 90 from
both sides, and you

00:03:19.840 --> 00:03:25.310
are left with your mystery
angle equaling 90 degrees.

00:03:25.310 --> 00:03:26.950
So that all worked out well.

00:03:26.950 --> 00:03:30.130
So let me make that
clear, and that's

00:03:30.130 --> 00:03:35.940
going to be useful
for us in a second.

00:03:35.940 --> 00:03:36.960
It's going to be useful.

00:03:36.960 --> 00:03:40.840
So we can now say definitively
that this is 90 degrees.

00:03:40.840 --> 00:03:43.260
This is a right angle.

00:03:43.260 --> 00:03:45.780
Now, what we are
going to do is we

00:03:45.780 --> 00:03:49.350
are going to
construct a trapezoid.

00:03:49.350 --> 00:03:52.490
This side a is parallel
to side b down here,

00:03:52.490 --> 00:03:55.190
the way it's been constructed,
and this is just one side right

00:03:55.190 --> 00:03:55.960
over here.

00:03:55.960 --> 00:03:58.760
This goes straight
up, and now let's just

00:03:58.760 --> 00:04:00.890
connect these two
sides right over there.

00:04:05.811 --> 00:04:07.310
So there's a couple
of ways to think

00:04:07.310 --> 00:04:10.940
about the area of
this trapezoid.

00:04:10.940 --> 00:04:13.090
One is we could just
think of it as a trapezoid

00:04:13.090 --> 00:04:15.970
and come up with its
area, and then we

00:04:15.970 --> 00:04:20.190
could think about it as the sum
of the areas of its components.

00:04:20.190 --> 00:04:23.900
So let's just first think
of it as a trapezoid

00:04:23.900 --> 00:04:26.410
so what do we know about
the area of a trapezoid?

00:04:26.410 --> 00:04:28.580
Well, the area of a
trapezoid is going

00:04:28.580 --> 00:04:31.590
to be the height of the
trapezoid, which is a plus b.

00:04:34.490 --> 00:04:36.930
That's the height
of the trapezoid.

00:04:36.930 --> 00:04:39.500
Times-- the way I
think of it-- the mean

00:04:39.500 --> 00:04:42.260
of the top and the
bottom, or the average

00:04:42.260 --> 00:04:43.405
of the top and the bottom.

00:04:45.990 --> 00:04:53.510
Since that's this times one
half times a plus a plus b.

00:04:59.239 --> 00:05:01.280
And the intuition there,
you're taking the height

00:05:01.280 --> 00:05:05.030
times the average of
this bottom and the top.

00:05:05.030 --> 00:05:07.530
The average of the
bottom and the top

00:05:07.530 --> 00:05:10.510
gives you the area
of the trapezoid.

00:05:10.510 --> 00:05:12.500
Now, how could we
also figure out

00:05:12.500 --> 00:05:15.260
the area with its
component parts?

00:05:15.260 --> 00:05:17.470
Regardless of how we
calculate the area, as long

00:05:17.470 --> 00:05:20.050
as we do correct things,
we should come up

00:05:20.050 --> 00:05:21.900
with the same result.

00:05:21.900 --> 00:05:24.260
So how else can we
come up with this area?

00:05:24.260 --> 00:05:30.170
Well, we could say it's the
area of the two right triangles.

00:05:30.170 --> 00:05:37.260
The area of each of them
is one half a times b,

00:05:37.260 --> 00:05:38.260
but there's two of them.

00:05:38.260 --> 00:05:41.160
Let me do that b in
that same blue color.

00:05:41.160 --> 00:05:43.500
But there's two of
these right triangle.

00:05:43.500 --> 00:05:45.710
So let's multiply by two.

00:05:45.710 --> 00:05:49.170
So two times one half ab.

00:05:49.170 --> 00:05:52.360
That takes into consideration
this bottom right triangle

00:05:52.360 --> 00:05:54.880
and this top one.

00:05:54.880 --> 00:05:57.400
And what's the area
of this large one

00:05:57.400 --> 00:06:01.150
that I will color in in green?

00:06:01.150 --> 00:06:04.200
What's the area
of this large one?

00:06:04.200 --> 00:06:05.920
Well, that's pretty
straightforward.

00:06:05.920 --> 00:06:07.800
It's just one half c times c.

00:06:07.800 --> 00:06:14.280
So plus one half c times c,
which is one half c squared.

00:06:14.280 --> 00:06:17.560
Now, let's simplify this thing
and see what we come up with,

00:06:17.560 --> 00:06:20.360
and you might guess where
all of this is going.

00:06:20.360 --> 00:06:22.520
So let's see what we get.

00:06:22.520 --> 00:06:25.680
So we can rearrange this.

00:06:25.680 --> 00:06:26.670
Let me rearrange this.

00:06:26.670 --> 00:06:36.370
So one half times
a plus b squared

00:06:36.370 --> 00:06:40.740
is going to be equal
to 2 times one half.

00:06:40.740 --> 00:06:42.280
Well, that's just
going to be one.

00:06:42.280 --> 00:06:48.855
So it's going to be equal
to a times b, plus one half

00:06:48.855 --> 00:06:49.355
c squared.

00:06:52.309 --> 00:06:54.350
Well, I don't like these
one halfs laying around,

00:06:54.350 --> 00:06:58.770
so let's multiply both
sides of this equation by 2.

00:06:58.770 --> 00:07:01.930
I'm just going to multiply both
sides of this equation by 2.

00:07:01.930 --> 00:07:05.386
On the left-hand side, I'm just
left with a plus b squared.

00:07:05.386 --> 00:07:06.260
So let me write that.

00:07:11.080 --> 00:07:14.395
And on the right-hand
side, I am left with 2ab.

00:07:19.340 --> 00:07:21.530
Trying to keep the
color coding right.

00:07:21.530 --> 00:07:23.220
And then, 2 times
one half c squared,

00:07:23.220 --> 00:07:27.250
that's just going to be
c squared plus c squared.

00:07:27.250 --> 00:07:28.870
Well, what happens
if you multiply out

00:07:28.870 --> 00:07:30.260
a plus b times a plus b?

00:07:30.260 --> 00:07:32.740
What is a plus b squared?

00:07:32.740 --> 00:07:45.590
Well, it's going to be a
squared plus 2ab plus 2ab plus b

00:07:45.590 --> 00:07:46.230
squared.

00:07:46.230 --> 00:07:51.080
And then, our
right-hand side it's

00:07:51.080 --> 00:07:52.930
going to be equal to
all of this business.

00:07:52.930 --> 00:07:56.390
And changing all the
colors is difficult for me,

00:07:56.390 --> 00:07:59.009
so let me copy and
let me paste it.

00:07:59.009 --> 00:08:01.300
So it's still going to be
equal to the right-hand side.

00:08:01.300 --> 00:08:03.440
Well, this is interesting.

00:08:03.440 --> 00:08:04.660
How can we simplify this?

00:08:04.660 --> 00:08:07.427
Is there anything that we
can subtract from both sides?

00:08:07.427 --> 00:08:08.260
Well, sure there is.

00:08:08.260 --> 00:08:09.790
You have a 2ab on
the left-hand side.

00:08:09.790 --> 00:08:11.373
You have a 2ab on
the right-hand side.

00:08:11.373 --> 00:08:14.830
Let's subtract 2ab
from both sides.

00:08:14.830 --> 00:08:19.650
If you subtract 2ab from both
sides, what are you left with?

00:08:19.650 --> 00:08:23.340
You are left with the
Pythagorean theorem.

00:08:23.340 --> 00:08:31.040
So you're left with a squared
plus b squared is equal

00:08:31.040 --> 00:08:33.120
to c squared.

00:08:33.120 --> 00:08:35.130
Very, very exciting.

00:08:35.130 --> 00:08:38.799
And for that, we have to
thank the 20th president

00:08:38.799 --> 00:08:42.919
of the United States,
James Garfield.

00:08:42.919 --> 00:08:44.890
This is really exciting.

00:08:44.890 --> 00:08:47.490
The Pythagorean theorem, it was
around for thousands of years

00:08:47.490 --> 00:08:49.160
before James
Garfield, and he was

00:08:49.160 --> 00:08:51.900
able to contribute just
kind of fiddling around

00:08:51.900 --> 00:08:55.206
while he was a member of the
US House of Representatives.


Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.620
Here is a simulation created
by Khan Academy user Justin

00:00:03.620 --> 00:00:07.150
Helps that once again tries to
give us an understanding of why

00:00:07.150 --> 00:00:11.710
we divide by n minus 1 to get an
unbiased estimate of population

00:00:11.710 --> 00:00:14.450
variance when we're trying to
calculate the sample variance.

00:00:14.450 --> 00:00:16.280
So what he does
here, the simulation,

00:00:16.280 --> 00:00:19.430
it has a population that
has a uniform distribution.

00:00:19.430 --> 00:00:21.690
So he says, I used a flat
probabilistic distribution

00:00:21.690 --> 00:00:24.870
from 0 to 100 for my population.

00:00:24.870 --> 00:00:26.900
Then we start sampling
from that population.

00:00:26.900 --> 00:00:29.510
We're going to use
samples of size 50.

00:00:29.510 --> 00:00:31.720
And what we do is for
each of those samples,

00:00:31.720 --> 00:00:37.330
we calculate the sample
variance based on dividing by n,

00:00:37.330 --> 00:00:40.440
by dividing by n
minus 1 and n minus 2.

00:00:40.440 --> 00:00:43.210
And as we keep having more
and more and more samples,

00:00:43.210 --> 00:00:45.950
we take the mean of the
variances calculated

00:00:45.950 --> 00:00:46.701
in different ways.

00:00:46.701 --> 00:00:48.658
And we figure out what
those means converge to.

00:00:48.658 --> 00:00:49.460
So that's a sample.

00:00:49.460 --> 00:00:50.376
Here's another sample.

00:00:50.376 --> 00:00:51.380
Here's another sample.

00:00:51.380 --> 00:00:53.910
If I sample here then
I'm now adding a bunch

00:00:53.910 --> 00:00:56.030
and I'm sampling continuously.

00:00:56.030 --> 00:00:59.210
And he saw something
very interesting happen.

00:00:59.210 --> 00:01:06.020
When I divide by n, I get
my sample variance is still,

00:01:06.020 --> 00:01:08.650
even when I'm taking the mean
of many, many, many, many sample

00:01:08.650 --> 00:01:10.066
variances that
I've already taken,

00:01:10.066 --> 00:01:12.570
I'm still underestimating
the true variance.

00:01:12.570 --> 00:01:14.200
When I divide by n
minus 1, it looks

00:01:14.200 --> 00:01:15.920
like I'm getting a
pretty good estimate,

00:01:15.920 --> 00:01:17.830
the mean of all of
my sample variances

00:01:17.830 --> 00:01:20.160
is really converged
to the true variance.

00:01:20.160 --> 00:01:22.720
When I divided by n
minus 2 just for kicks,

00:01:22.720 --> 00:01:27.510
it's pretty clear
that I overestimated

00:01:27.510 --> 00:01:30.780
with my mean of my
sample variances,

00:01:30.780 --> 00:01:32.729
I overestimated
the true variance.

00:01:32.729 --> 00:01:34.770
So this gives us a pretty
good sense of n minus 1

00:01:34.770 --> 00:01:36.310
is the right thing to do.

00:01:36.310 --> 00:01:40.140
Now this is another interesting
way of visualizing it.

00:01:40.140 --> 00:01:42.370
In the horizontal
axis right over here,

00:01:42.370 --> 00:01:45.900
we're comparing each plot
is one of our samples,

00:01:45.900 --> 00:01:48.100
and how far to the right
is how much more is

00:01:48.100 --> 00:01:50.294
that sample mean
than the true mean?

00:01:50.294 --> 00:01:52.210
And when we go to the
left, it's how much less

00:01:52.210 --> 00:01:54.230
is a sample mean
than the true mean?

00:01:54.230 --> 00:01:56.120
So for example, this
sample right over here,

00:01:56.120 --> 00:01:57.870
it's all the way
over to the right.

00:01:57.870 --> 00:02:00.720
It's the sample mean there was
a lot more than the true mean.

00:02:00.720 --> 00:02:03.160
Sample mean here was a lot
less than the true mean.

00:02:03.160 --> 00:02:06.210
Sample mean here only a little
bit more than the true mean.

00:02:06.210 --> 00:02:10.979
In the vertical axis, using
this denominator, dividing by n,

00:02:10.979 --> 00:02:13.570
we calculate two
different variances.

00:02:13.570 --> 00:02:16.170
One variance, we
use the sample mean.

00:02:16.170 --> 00:02:19.430
The other variance, we
use the population mean.

00:02:19.430 --> 00:02:21.510
And this, in the
vertical axis, we

00:02:21.510 --> 00:02:24.610
compare the difference
between the mean calculated

00:02:24.610 --> 00:02:26.850
with the sample mean
versus the mean calculated

00:02:26.850 --> 00:02:28.490
with the population mean.

00:02:28.490 --> 00:02:30.600
So for example, this
point right over here,

00:02:30.600 --> 00:02:33.690
when we calculate our mean
with our sample mean, which

00:02:33.690 --> 00:02:36.224
is the normal way we
do it, it significantly

00:02:36.224 --> 00:02:37.640
underestimates
what the mean would

00:02:37.640 --> 00:02:40.470
have been if somehow we knew
what the population mean was

00:02:40.470 --> 00:02:42.670
and we could
calculate it that way.

00:02:42.670 --> 00:02:44.480
And you get this really
interesting shape.

00:02:44.480 --> 00:02:45.896
And it's something
to think about.

00:02:45.896 --> 00:02:48.870
And he recommends some
thinking about why or what kind

00:02:48.870 --> 00:02:51.350
of a shape this actually is.

00:02:51.350 --> 00:02:54.250
The other interesting thing is
when you look at it this way,

00:02:54.250 --> 00:02:55.770
it's pretty clear
this entire graph

00:02:55.770 --> 00:02:58.850
is sitting below
the horizontal axis.

00:02:58.850 --> 00:03:01.810
So we're always, when we
calculate our sample variance

00:03:01.810 --> 00:03:06.380
using this formula, when we use
of the sample mean to do it,

00:03:06.380 --> 00:03:10.670
which we typically do, we're
always getting a lower variance

00:03:10.670 --> 00:03:14.980
than when we use
the population mean.

00:03:14.980 --> 00:03:18.120
Now this over here, when
we divide by n minus 1,

00:03:18.120 --> 00:03:19.970
we're not always
underestimating.

00:03:19.970 --> 00:03:21.840
Sometimes we are
overestimating it.

00:03:21.840 --> 00:03:24.089
And when you take the mean
of all of these variances,

00:03:24.089 --> 00:03:24.630
you converge.

00:03:24.630 --> 00:03:27.670
And here we're overestimating
it a little bit more.

00:03:27.670 --> 00:03:30.170
And just to be clear what we're
talking about in these three

00:03:30.170 --> 00:03:31.940
graphs, let me take
a screen shot of it

00:03:31.940 --> 00:03:34.610
and explain it in a
little bit more depth.

00:03:34.610 --> 00:03:39.240
So just to be clear, in this
red graph right over here,

00:03:39.240 --> 00:03:40.270
let me do this.

00:03:40.270 --> 00:03:42.160
A color close to at least.

00:03:42.160 --> 00:03:44.290
So this orange,
what this distance

00:03:44.290 --> 00:03:46.820
is for each of
these samples, we're

00:03:46.820 --> 00:03:54.260
calculating the sample
variance using, so let me,

00:03:54.260 --> 00:03:56.160
using the sample mean.

00:03:56.160 --> 00:03:59.830
And in this case, we are
using n as our denominator.

00:03:59.830 --> 00:04:01.470
In this case right over here.

00:04:01.470 --> 00:04:04.595
And from that we're subtracting
the sample variance,

00:04:04.595 --> 00:04:06.970
or I guess you could call this
some kind of pseudo sample

00:04:06.970 --> 00:04:11.130
variance, if we somehow
knew the population mean.

00:04:11.130 --> 00:04:13.520
This isn't something that
you see a lot in statistics.

00:04:13.520 --> 00:04:19.540
But it's a gauge of how much we
are underestimating our sample

00:04:19.540 --> 00:04:22.960
variance given that we don't
have the true population mean

00:04:22.960 --> 00:04:24.660
at our disposal.

00:04:24.660 --> 00:04:26.165
And so this is the distance.

00:04:26.165 --> 00:04:27.790
This is the distance
we're calculating.

00:04:27.790 --> 00:04:31.860
And you see we're
always underestimating.

00:04:31.860 --> 00:04:33.830
Here we overestimate
a little bit.

00:04:33.830 --> 00:04:35.460
And we also underestimate.

00:04:35.460 --> 00:04:38.130
But when you take the mean,
when you average them all out,

00:04:38.130 --> 00:04:39.920
it converges to
the actual value.

00:04:39.920 --> 00:04:41.770
So here we're
dividing by n minus 1,

00:04:41.770 --> 00:04:44.802
here we're dividing
by n minus 2.


Simulation providing evidence that (n-1) gives us unbiased estimate

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:04.522
Here's a simulation created
by Khan Academy user TETF.

00:00:04.522 --> 00:00:07.327
I can assume that's
pronounced tet f.

00:00:07.327 --> 00:00:08.910
And what it allows
us to do is give us

00:00:08.910 --> 00:00:12.610
an intuition as to why
we divide by n minus 1

00:00:12.610 --> 00:00:15.120
when we calculate
our sample variance

00:00:15.120 --> 00:00:18.310
and why that gives us an
unbiased estimate of population

00:00:18.310 --> 00:00:19.330
variance.

00:00:19.330 --> 00:00:21.750
So the way this starts
off, and I encourage

00:00:21.750 --> 00:00:23.650
you to go try this
out yourself, is

00:00:23.650 --> 00:00:25.420
that you can construct
a distribution.

00:00:25.420 --> 00:00:28.970
It says build a population
by clicking in the blue area.

00:00:28.970 --> 00:00:32.810
So here, we are actually
creating a population.

00:00:32.810 --> 00:00:37.450
So every time I click, it
increases the population size.

00:00:37.450 --> 00:00:39.100
And I'm just
randomly doing this,

00:00:39.100 --> 00:00:42.890
and I encourage you to go
onto this scratch pad-- it's

00:00:42.890 --> 00:00:44.940
on the Khan Academy
Computer Science--

00:00:44.940 --> 00:00:47.260
and try to do it yourself.

00:00:47.260 --> 00:00:50.680
So here I could
stop at some point.

00:00:50.680 --> 00:00:52.460
So I've constructed
a population.

00:00:52.460 --> 00:00:54.790
I can throw out some
random points up here.

00:00:54.790 --> 00:00:56.484
So this is our
population, and as you

00:00:56.484 --> 00:00:58.400
saw while I was doing
that, it was calculating

00:00:58.400 --> 00:00:59.650
parameters for the population.

00:00:59.650 --> 00:01:01.600
It was calculating
the population

00:01:01.600 --> 00:01:05.360
mean at 204.09 and
also the population

00:01:05.360 --> 00:01:08.540
standard deviation, which is
derived from the population

00:01:08.540 --> 00:01:09.120
variance.

00:01:09.120 --> 00:01:11.720
This is the square root of
the population variance,

00:01:11.720 --> 00:01:13.600
and it's at 63.8.

00:01:13.600 --> 00:01:16.780
It was also pop plotting the
population variance down here.

00:01:16.780 --> 00:01:19.630
You see it's 63.8, which
is the standard deviation,

00:01:19.630 --> 00:01:23.280
and it's a little harder to
see, but it says it's squared.

00:01:23.280 --> 00:01:24.850
These are these numbers squared.

00:01:24.850 --> 00:01:32.894
So essentially, 63.8 squared
is the population variance.

00:01:32.894 --> 00:01:34.810
So that's interesting
by itself, but it really

00:01:34.810 --> 00:01:36.490
doesn't tell us a
lot so far about

00:01:36.490 --> 00:01:38.790
why we divide by n minus 1.

00:01:38.790 --> 00:01:40.540
And this is the
interesting part.

00:01:40.540 --> 00:01:42.209
We can now start
to take samples,

00:01:42.209 --> 00:01:44.250
and we can decide what
sample size we want to do.

00:01:44.250 --> 00:01:47.020
I'll start with really small
samples, so the smallest

00:01:47.020 --> 00:01:49.699
possible sample that
makes any sense.

00:01:49.699 --> 00:01:51.740
So I'm going to start with
a really small sample.

00:01:51.740 --> 00:01:53.740
And what they're going to do--
what the simulation is going

00:01:53.740 --> 00:01:55.520
to do-- is every
time I take a sample,

00:01:55.520 --> 00:01:57.430
it's going to
calculate the variance.

00:01:57.430 --> 00:02:00.300
So the numerator is going to
be the sum of each of my data

00:02:00.300 --> 00:02:03.190
points in my sample
minus my sample mean,

00:02:03.190 --> 00:02:04.700
and I'm going to square it.

00:02:04.700 --> 00:02:08.590
And then it's going to
divide it by n plus a,

00:02:08.590 --> 00:02:09.810
and it's going to vary a.

00:02:09.810 --> 00:02:12.540
It's going to divide it
by anywhere between n

00:02:12.540 --> 00:02:18.274
plus negative 3, so n minus
3, all the way to n plus a.

00:02:18.274 --> 00:02:20.690
And we're going to do it in
many, many, many, many, times.

00:02:20.690 --> 00:02:23.148
We're going to essentially take
the mean of those variances

00:02:23.148 --> 00:02:26.990
for any a and figure out which
gives us the best estimate.

00:02:26.990 --> 00:02:28.910
So if I just generate
one sample right

00:02:28.910 --> 00:02:35.490
over there, when we see kind
of this curve, when we have

00:02:35.490 --> 00:02:37.840
high values of a, we
are underestimating.

00:02:37.840 --> 00:02:40.130
When we have lower
values of a, we

00:02:40.130 --> 00:02:43.290
are overestimating the
population variance,

00:02:43.290 --> 00:02:45.910
but that was just
for one sample,

00:02:45.910 --> 00:02:47.260
not really that meaningful.

00:02:47.260 --> 00:02:48.840
It's one sample of size two.

00:02:48.840 --> 00:02:50.690
Let's generate a
bunch of samples

00:02:50.690 --> 00:02:53.475
and then average them
over many of them.

00:02:53.475 --> 00:02:55.850
And you see when you look at
many, many, many, many, many

00:02:55.850 --> 00:02:58.890
examples, something
interesting is happening.

00:02:58.890 --> 00:03:01.170
When you look at the
mean of those samples,

00:03:01.170 --> 00:03:03.260
when you average together
those curves from all

00:03:03.260 --> 00:03:06.400
of those samples, you see
that our best estimate is

00:03:06.400 --> 00:03:09.210
when a is pretty
close to negative 1,

00:03:09.210 --> 00:03:13.400
is when this is n plus
negative 1 or n minus 1.

00:03:13.400 --> 00:03:15.350
Anything less than
negative 1-- if we

00:03:15.350 --> 00:03:19.960
did negative n minus
1.05 or n minus 1.5--

00:03:19.960 --> 00:03:22.810
we start overestimating
the variance.

00:03:22.810 --> 00:03:27.760
Anything less than
negative 1, so if we

00:03:27.760 --> 00:03:33.750
have n plus 0, if we divide
by n or if we have n plus 0.05

00:03:33.750 --> 00:03:37.620
or whatever it
might be, we start

00:03:37.620 --> 00:03:40.075
underestimating the
population variance.

00:03:40.075 --> 00:03:42.200
And you can do this for
samples of different sizes.

00:03:42.200 --> 00:03:44.690
Let me try a sample size 6.

00:03:44.690 --> 00:03:46.970
And here you go once
again, as I press--

00:03:46.970 --> 00:03:49.520
I'm just keeping Generate
Sample pressed down--

00:03:49.520 --> 00:03:52.350
as we generate more and
more and more samples--

00:03:52.350 --> 00:03:55.030
and for all the a's we
essentially take the average

00:03:55.030 --> 00:03:57.670
across those samples
for the variance

00:03:57.670 --> 00:03:59.480
depending on how
we calculate it--

00:03:59.480 --> 00:04:05.320
you'll see that once again, our
best estimate is pretty darn

00:04:05.320 --> 00:04:06.430
close to negative 1.

00:04:06.430 --> 00:04:10.970
And if you were to get this to
millions of samples generated,

00:04:10.970 --> 00:04:13.270
you'll see that your
best estimate is

00:04:13.270 --> 00:04:20.430
when a is negative 1 or when
you're dividing by n minus 1.

00:04:20.430 --> 00:04:23.620
So once again, thanks
TETF, tet f, for this.

00:04:23.620 --> 00:04:25.580
I think it's a really
interesting way

00:04:25.580 --> 00:04:29.770
to think about why we
divide by n minus 1.


Simulation showing bias in sample variance

WEBVTT
Kind: captions
Language: en

00:00:01.003 --> 00:00:03.844
Voiceover: This right here is a
simulation that was created

00:00:03.859 --> 00:00:06.732
by Peter Collingridge using the

00:00:06.732 --> 00:00:09.397
Khan Academy computer science scratch pad

00:00:09.397 --> 00:00:13.763
to better understand why
we divide by n minus one

00:00:13.763 --> 00:00:18.067
when we calculate an
unbiased sample variance.

00:00:18.067 --> 00:00:20.529
When we are in an unbiased
way trying to estimate

00:00:20.529 --> 00:00:22.545
the true population variance.

00:00:22.545 --> 00:00:24.627
So what this simulation does is first

00:00:24.627 --> 00:00:27.404
it constructs a population distribution,

00:00:27.404 --> 00:00:28.428
a random one,

00:00:28.428 --> 00:00:29.788
and every time you go to it,

00:00:29.788 --> 00:00:31.860
it will be a different
population distribution.

00:00:31.860 --> 00:00:34.131
This one has a population of 383,

00:00:34.131 --> 00:00:35.891
and then it calculates the parameters

00:00:35.891 --> 00:00:37.716
for that population directly from it.

00:00:37.716 --> 00:00:41.515
the mean is 10 point nine the
variance is 25 point five.

00:00:41.515 --> 00:00:43.868
and then it uses that population

00:00:43.868 --> 00:00:47.437
and samples from it and it does samples

00:00:47.437 --> 00:00:49.907
of size two, three, four,
five, all the way up to 10,

00:00:49.907 --> 00:00:51.219
and it keeps sampling from it,

00:00:51.219 --> 00:00:54.451
calculates the statistics
for those samples

00:00:54.451 --> 00:00:56.427
so the sample mean and
the sample variance,

00:00:56.427 --> 00:00:58.332
in particular the biased sample variance

00:00:58.332 --> 00:01:00.044
It starts telling us some things about us

00:01:00.044 --> 00:01:01.659
that give us some intuition.

00:01:01.777 --> 00:01:04.513
You can actually click on
each of these and zoom in

00:01:04.513 --> 00:01:08.522
to really be able to study
these graphs in detail.

00:01:08.522 --> 00:01:10.442
I have already taken a screen shot of this

00:01:10.442 --> 00:01:13.281
and put it on my little doodle pad,

00:01:13.281 --> 00:01:15.169
so you can really delve
into some of the math

00:01:15.169 --> 00:01:19.056
and the intuition of what
this is actually showing us.

00:01:19.117 --> 00:01:22.135
So here I took a screen shot,

00:01:22.135 --> 00:01:25.697
and you see for this case right over here,

00:01:25.697 --> 00:01:27.808
the population was 529.

00:01:27.808 --> 00:01:31.456
Population mean was 10 point six,

00:01:31.456 --> 00:01:33.431
and down here in this chart,

00:01:33.431 --> 00:01:38.375
he plots the population mean
right here at 10 point six,

00:01:38.375 --> 00:01:40.080
right over there, and you see that

00:01:40.080 --> 00:01:43.088
the population variance
is at 36 point eight,

00:01:43.088 --> 00:01:47.399
and right here he plots that
right over here, 36 point eight.

00:01:47.399 --> 00:01:49.343
This first chart on the bottom left

00:01:49.343 --> 00:01:51.247
tells us a couple of interesting things.

00:01:51.247 --> 00:01:56.417
Just to be clear, this is
the biased sample variance

00:01:56.417 --> 00:01:57.512
that he is calculating.

00:01:57.512 --> 00:02:01.119
This is the biased sample variance.

00:02:01.119 --> 00:02:02.856
So he is calculating it.

00:02:02.856 --> 00:02:06.456
That is being calculated
for each of our data points.

00:02:06.456 --> 00:02:09.183
So starting with our first data
point in each of our samples,

00:02:09.183 --> 00:02:11.512
going to our nth data point in the sample.

00:02:11.512 --> 00:02:13.368
You're taking that data point,

00:02:13.368 --> 00:02:16.576
subtracting out the
sample mean, squaring it,

00:02:16.576 --> 00:02:20.025
and then dividing the whole
thing, not by n minus one,

00:02:20.025 --> 00:02:22.330
but by lower case n.

00:02:22.330 --> 00:02:25.266
This tells us several interesting things.

00:02:25.266 --> 00:02:29.122
The first thing it shows us is that

00:02:29.122 --> 00:02:31.937
the cases where we are
significantly underestimating

00:02:31.937 --> 00:02:33.001
the sample variance,

00:02:33.001 --> 00:02:35.114
and we are getting sample
variances close to zero,

00:02:35.114 --> 00:02:40.419
these are also the cases, or
they are disproportionately

00:02:40.419 --> 00:02:43.921
the cases where the
means for those samples

00:02:43.921 --> 00:02:47.386
are way far off from the true sample mean,

00:02:47.386 --> 00:02:49.226
or we could do that the other way around.

00:02:49.226 --> 00:02:52.586
The cases where the mean is way
far off from the sample mean

00:02:52.586 --> 00:02:56.138
it seems like you're much
more likely to underestimate

00:02:56.138 --> 00:02:58.403
the sample variance in those situations.

00:02:58.403 --> 00:03:00.386
The other thing that might pop out at you

00:03:00.386 --> 00:03:02.938
is the realization that the pinker dots

00:03:02.938 --> 00:03:05.322
are the ones for smaller sample size,

00:03:05.322 --> 00:03:08.706
while the bluer dots are the
ones of a larger sample size.

00:03:08.706 --> 00:03:10.961
You see here these two little,

00:03:10.961 --> 00:03:16.636
I guess the tails ,so
to speak, of this hump,

00:03:16.636 --> 00:03:21.649
that these ends, are
more of a reddish color.

00:03:21.649 --> 00:03:24.818
that most of the blueish
or the purplish dots

00:03:24.818 --> 00:03:27.945
are focused right in the
middle right over here,

00:03:27.945 --> 00:03:29.944
that they are giving us better estimates.

00:03:29.944 --> 00:03:30.808
There are some red ones here,

00:03:30.808 --> 00:03:32.646
and that's why it gives
us that purplish color,

00:03:32.646 --> 00:03:35.359
but out here on these tails,

00:03:35.359 --> 00:03:37.735
it's almost purely some of these red.

00:03:37.735 --> 00:03:39.175
Every now and then by happenstance

00:03:39.175 --> 00:03:40.227
you get a little blue one,

00:03:40.227 --> 00:03:42.886
but it's disproportionately far more red,

00:03:42.886 --> 00:03:43.855
which really makes sense when you

00:03:43.855 --> 00:03:45.469
have a smaller sample size,

00:03:45.469 --> 00:03:48.606
you are more likely to get a sample mean

00:03:48.606 --> 00:03:51.502
that is a bad estimate
of the population mean,

00:03:51.502 --> 00:03:53.503
that's far from the population mean,

00:03:53.503 --> 00:03:54.862
and you're more likely

00:03:54.862 --> 00:03:59.022
to significantly underestimate
the sample variance.

00:03:59.022 --> 00:04:03.806
Now this next chart really
gets to the meat of the issue,

00:04:03.806 --> 00:04:04.977
because what this is telling us is

00:04:04.977 --> 00:04:06.638
that for each of these sample sizes,

00:04:06.638 --> 00:04:09.391
so this right over here
for sample size two,

00:04:09.391 --> 00:04:11.455
if we keep taking sample size two,

00:04:11.455 --> 00:04:14.032
and we keep calculating
the biased sample variances

00:04:14.032 --> 00:04:16.023
and dividing that by
the population variance,

00:04:16.023 --> 00:04:18.471
and finding the mean over all of those,

00:04:18.471 --> 00:04:20.942
you see that over many, many, many trials,

00:04:20.942 --> 00:04:22.895
and many, many samples of size two,

00:04:22.895 --> 00:04:25.895
that that biased sample variance
over population variance,

00:04:25.895 --> 00:04:30.444
it's approaching half of the
true population variance.

00:04:30.444 --> 00:04:34.342
When sample size is three,
it's approaching 2/3,

00:04:34.342 --> 00:04:38.254
66 point six percent, of the
true population variance.

00:04:38.254 --> 00:04:39.559
When sample size is four,

00:04:39.559 --> 00:04:43.992
it's approaching 3/4 of the
true population variance.

00:04:43.992 --> 00:04:47.855
So we can come up with the
general theme that's happening.

00:04:47.855 --> 00:04:50.230
When we use the biased estimate,

00:04:50.230 --> 00:04:55.175
we're not approaching
the population variance.

00:04:55.175 --> 00:05:04.440
We're approaching n minus one over n

00:05:04.440 --> 00:05:06.250
times the population variance.

00:05:06.250 --> 00:05:09.584
When n was two, this approached 1/2.

00:05:09.584 --> 00:05:12.416
When n is three, this is 2/3.

00:05:12.416 --> 00:05:15.176
When n is four, this is 3/4.

00:05:15.176 --> 00:05:18.049
So this is giving us a biased estimate.

00:05:18.049 --> 00:05:20.577
So how would we unbias this?

00:05:20.577 --> 00:05:24.537
Well, if we really want
to get our best estimate

00:05:24.537 --> 00:05:27.288
of the true population variance,

00:05:27.288 --> 00:05:30.403
not n minus one over n times
the population variance,

00:05:30.403 --> 00:05:32.155
we would want to multiply,

00:05:32.155 --> 00:05:34.340
I'll do this in a color
I haven't used yet,

00:05:34.340 --> 00:05:41.313
we would want to multiply
times n over n minus one.

00:05:41.836 --> 00:05:43.899
to get an unbiased estimate.

00:05:43.899 --> 00:05:46.587
Here, these cancel out
and you are just left

00:05:46.587 --> 00:05:50.316
with your population variance.

00:05:50.316 --> 00:05:51.788
That's what we want to estimate.

00:05:51.788 --> 00:05:59.815
Over here you are left
with our unbiased estimate

00:06:00.030 --> 00:06:01.560
of population variance,

00:06:01.560 --> 00:06:03.408
our unbiased sample variance,

00:06:03.408 --> 00:06:06.057
which is equal to, and this is what we saw

00:06:06.057 --> 00:06:09.240
in the last several videos, what
you see in statistics books,

00:06:09.240 --> 00:06:11.088
and sometimes it's confusing why,

00:06:11.088 --> 00:06:14.816
hopefully Peter's simulation
gives you a good idea of why,

00:06:14.816 --> 00:06:18.334
or at least convinces
you that it is the case.

00:06:18.334 --> 00:06:22.334
So you would want to
divide by n minus one.


Illinois pension obligations

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:02.730
In the last video, we
talked about pensions

00:00:02.730 --> 00:00:04.880
and how they're
defined benefit plans

00:00:04.880 --> 00:00:07.310
and how they could
to get underfunded

00:00:07.310 --> 00:00:10.910
or how there could be temptation
for people to underfund them.

00:00:10.910 --> 00:00:13.160
In this video, I want to
make things a little bit more

00:00:13.160 --> 00:00:15.190
concrete by looking
at actual numbers,

00:00:15.190 --> 00:00:17.150
especially at the state level.

00:00:17.150 --> 00:00:21.560
So right over here is a map of,
obviously, the United States.

00:00:21.560 --> 00:00:26.100
And what it shows is how funded
the pension liabilities are

00:00:26.100 --> 00:00:27.610
in the different states.

00:00:27.610 --> 00:00:31.040
So for example-- actually,
Texas, for example,

00:00:31.040 --> 00:00:34.740
83% of their pension
liabilities are funded.

00:00:34.740 --> 00:00:38.330
They've set aside 83% of
the right amount of money

00:00:38.330 --> 00:00:41.170
to fund their pension
obligations, not 100%.

00:00:41.170 --> 00:00:43.790
It is underfunded,
but it's not crazy.

00:00:43.790 --> 00:00:46.619
California, pretty high, 78%.

00:00:46.619 --> 00:00:48.910
But one of these states is
probably jumping out at you,

00:00:48.910 --> 00:00:51.990
probably because it
has been shaded in red.

00:00:51.990 --> 00:00:55.090
And that is the
state of Illinois,

00:00:55.090 --> 00:00:58.100
and Illinois is in
trouble because it's only

00:00:58.100 --> 00:01:02.030
funded 45% of its
pension obligations.

00:01:02.030 --> 00:01:04.767
And Illinois really jumps
out because it's in red,

00:01:04.767 --> 00:01:07.350
but there's a lot of states that
are pretty close to Illinois.

00:01:07.350 --> 00:01:09.080
Louisiana, 56%.

00:01:09.080 --> 00:01:10.550
Oklahoma, 56%.

00:01:10.550 --> 00:01:12.380
Kentucky, 54%.

00:01:12.380 --> 00:01:14.952
West Virginia, 58%.

00:01:14.952 --> 00:01:16.910
And this is an issue
because they've set aside,

00:01:16.910 --> 00:01:19.272
in the past, very little
money for the pension

00:01:19.272 --> 00:01:20.980
obligations that are
starting to hit now,

00:01:20.980 --> 00:01:23.450
especially that you have
a retiring baby boomer

00:01:23.450 --> 00:01:24.390
population.

00:01:24.390 --> 00:01:26.520
And in order to meet
those obligations,

00:01:26.520 --> 00:01:28.300
those promised
obligations, they're

00:01:28.300 --> 00:01:31.680
going to have to dig into
money that was being spent

00:01:31.680 --> 00:01:33.650
other places, that
going in the past when

00:01:33.650 --> 00:01:35.520
they were underfunding
the pension,

00:01:35.520 --> 00:01:38.570
they were able to fund
other things nicely, but not

00:01:38.570 --> 00:01:41.690
fund the pension and kind of
kick the can down the road.

00:01:41.690 --> 00:01:44.924
But now that the can can't
be kicked any further,

00:01:44.924 --> 00:01:46.840
it's going to have to
go the other way around.

00:01:46.840 --> 00:01:49.006
You're going to have to
take money from other things

00:01:49.006 --> 00:01:50.330
to fund your pensions.

00:01:50.330 --> 00:01:54.550
And to make it clear, let's
focus on the state of Illinois.

00:01:54.550 --> 00:01:56.090
So this right over here.

00:01:56.090 --> 00:01:57.890
There's a couple
of things going on.

00:01:57.890 --> 00:02:01.110
In this kind of yellow ochre
color-- and I'll circle it

00:02:01.110 --> 00:02:03.080
in yellow ochre--
they were talking

00:02:03.080 --> 00:02:04.514
about the total liabilities.

00:02:04.514 --> 00:02:05.930
And just to make
this graph clear,

00:02:05.930 --> 00:02:08.471
it's not just the yellow ochre
part that's total liabilities.

00:02:08.471 --> 00:02:10.789
The entire height of
each of these bars

00:02:10.789 --> 00:02:15.990
is the total
liabilities, and you

00:02:15.990 --> 00:02:20.070
see how it has just
completely blossomed here.

00:02:20.070 --> 00:02:23.350
And there's a lot of things that
go into the total liabilities,

00:02:23.350 --> 00:02:26.240
the same things that we talked
about in the last video.

00:02:26.240 --> 00:02:28.075
There are things like
return on investment.

00:02:31.350 --> 00:02:33.420
If you are in a low
interest rate environment,

00:02:33.420 --> 00:02:36.250
like we are now-- for example,
my money in my savings account,

00:02:36.250 --> 00:02:39.020
I think, is getting
like 0.4% interest.

00:02:39.020 --> 00:02:40.870
It's getting pretty
much no interest.

00:02:40.870 --> 00:02:43.330
If you're in a low
interest rate environment,

00:02:43.330 --> 00:02:45.890
if you're not getting good
returns-- and a lot of pensions

00:02:45.890 --> 00:02:47.800
tend to go into
very safe assets,

00:02:47.800 --> 00:02:49.720
but those are getting
very low returns.

00:02:49.720 --> 00:02:52.670
You're going to have to
set aside more money,

00:02:52.670 --> 00:02:56.930
and so you see these
obligations essentially just

00:02:56.930 --> 00:02:58.950
growing dramatically.

00:02:58.950 --> 00:03:01.520
On top of that, you
have things like cost

00:03:01.520 --> 00:03:04.320
of living adjustments.

00:03:04.320 --> 00:03:07.779
These are attempts at kind
of factoring in inflation,

00:03:07.779 --> 00:03:09.570
how much things are
costing in that region.

00:03:09.570 --> 00:03:11.195
But they are also
sometimes negotiated.

00:03:11.195 --> 00:03:13.670
And sometimes, and especially
in the case of Illinois,

00:03:13.670 --> 00:03:17.580
they've grown faster than
the rate of inflation.

00:03:17.580 --> 00:03:19.130
And so you have
these liabilities,

00:03:19.130 --> 00:03:21.700
and you see that they're getting
less and less well funded.

00:03:21.700 --> 00:03:23.560
So if we go right
over here, this

00:03:23.560 --> 00:03:25.990
is what this green line
is, the funding ratio.

00:03:25.990 --> 00:03:28.220
So how well funded
are these liabilities?

00:03:28.220 --> 00:03:32.210
Say the red part of the bar is
the part that is not paid for.

00:03:32.210 --> 00:03:36.430
And the green is
the ratio of the red

00:03:36.430 --> 00:03:38.680
or is the ratio of what
is funded, essentially

00:03:38.680 --> 00:03:40.170
this higher part.

00:03:40.170 --> 00:03:42.820
It's the ratio of
this part right

00:03:42.820 --> 00:03:46.230
over here to the entire bar.

00:03:46.230 --> 00:03:51.170
And you see right over here,
Illinois is in a bad situation.

00:03:51.170 --> 00:03:54.860
Their total liabilities
are 138 billion.

00:03:54.860 --> 00:03:57.560
This is in millions, so
it's 138,000 million.

00:03:57.560 --> 00:03:59.390
So it's 138 billion.

00:03:59.390 --> 00:04:00.780
This is for one state.

00:04:00.780 --> 00:04:05.719
And 85 or 86 billion
of that is unfunded,

00:04:05.719 --> 00:04:07.260
that they have to
figure out some way

00:04:07.260 --> 00:04:09.920
to get the money because
the right amount of money

00:04:09.920 --> 00:04:12.010
was not being set aside.

00:04:12.010 --> 00:04:13.860
And to do that,
they're going to have

00:04:13.860 --> 00:04:16.350
to dig in into other things.

00:04:16.350 --> 00:04:19.839
So this right over here, this
is the pension contribution.

00:04:19.839 --> 00:04:21.269
Let me circle this.

00:04:21.269 --> 00:04:22.850
So in this yellow
color, once again,

00:04:22.850 --> 00:04:24.790
this is the pension
contribution.

00:04:24.790 --> 00:04:27.640
And now the state, they're
going to have to-- in order

00:04:27.640 --> 00:04:29.270
to get to a funded
position, they're

00:04:29.270 --> 00:04:32.160
going to make up for all of
the underfunding of the past

00:04:32.160 --> 00:04:35.300
and also the other factors that
are making this obligation even

00:04:35.300 --> 00:04:35.990
larger.

00:04:35.990 --> 00:04:39.070
They're going to have to
dig into other things.

00:04:39.070 --> 00:04:42.324
So you see right
over here in yellow,

00:04:42.324 --> 00:04:43.990
these are the
contributions that they're

00:04:43.990 --> 00:04:45.930
going to have to
make for the pension.

00:04:45.930 --> 00:04:47.250
And you see that growing.

00:04:47.250 --> 00:04:51.900
It's growing to in excess by
2018 of $6 billion a year.

00:04:51.900 --> 00:04:54.510
But what's really
fascinating about this graph

00:04:54.510 --> 00:04:58.260
is it's passing up total
education funding in the state.

00:04:58.260 --> 00:05:03.200
So the cost of
funding retirements

00:05:03.200 --> 00:05:05.714
for people who have already
done service for the state

00:05:05.714 --> 00:05:07.380
but aren't in service
to the state right

00:05:07.380 --> 00:05:12.450
now is going to pass up-- and
this is happening very soon--

00:05:12.450 --> 00:05:14.220
is going to pass
up actual spending

00:05:14.220 --> 00:05:17.320
on a state-wide
basis on education.

00:05:17.320 --> 00:05:19.310
And at the state
level, education

00:05:19.310 --> 00:05:21.687
is a major, major,
major expenditure.

00:05:21.687 --> 00:05:24.020
So it's going to be passing
up a major expenditure, very

00:05:24.020 --> 00:05:27.680
important expenditure for
the future of the state

00:05:27.680 --> 00:05:30.720
based on past obligations.

00:05:30.720 --> 00:05:33.060
And to understand
where this is going--

00:05:33.060 --> 00:05:35.360
and just to understand
Illinois' situation,

00:05:35.360 --> 00:05:38.700
there's 750,000
Illinois-- I don't

00:05:38.700 --> 00:05:41.190
know how to say this--
Illinoians, Illinoisians,

00:05:41.190 --> 00:05:44.500
who are members of the
state's five pension system.

00:05:44.500 --> 00:05:46.760
So this is the Teachers
Retirement System.

00:05:46.760 --> 00:05:49.760
This is the State Universities
Retirement System.

00:05:49.760 --> 00:05:52.280
This is the State Employees
Retirement System.

00:05:52.280 --> 00:05:53.900
This is the Judges
Retirement System.

00:05:53.900 --> 00:05:55.880
You see there's a lot
fewer judges than that.

00:05:55.880 --> 00:05:57.930
There's many more teachers
who've been retired,

00:05:57.930 --> 00:06:00.479
many who have been in
the state universities.

00:06:00.479 --> 00:06:02.520
And this right over here
is the General Assembly.

00:06:02.520 --> 00:06:05.650
Very few people who are in
the Illinois state assembly.

00:06:05.650 --> 00:06:07.700
And you can see kind
of comparable salaries.

00:06:07.700 --> 00:06:10.510
This is the retirement
benefit, not just the salaries.

00:06:10.510 --> 00:06:12.570
This is how much on
average these folks

00:06:12.570 --> 00:06:15.350
are getting once they
retire on an annual basis.

00:06:15.350 --> 00:06:17.100
So you see that they're
pretty reasonable,

00:06:17.100 --> 00:06:21.420
especially for the
judges, although they

00:06:21.420 --> 00:06:22.690
are a small fraction.

00:06:22.690 --> 00:06:25.370
But in all fairness, this
was promised to these people.

00:06:25.370 --> 00:06:26.020
They planned.

00:06:26.020 --> 00:06:28.070
The probably took
lower compensation

00:06:28.070 --> 00:06:30.560
while they were working
with the expectation

00:06:30.560 --> 00:06:33.510
that they would be able to
get these benefits once they

00:06:33.510 --> 00:06:34.440
retired.

00:06:34.440 --> 00:06:36.840
They also probably stayed
in the jobs longer.

00:06:36.840 --> 00:06:38.470
This is a way of
retaining employees,

00:06:38.470 --> 00:06:41.420
because they knew that they
were going to get this benefit.

00:06:41.420 --> 00:06:44.020
So you might say, oh, these are
really, really great benefits.

00:06:44.020 --> 00:06:45.910
But at the same time,
these people probably

00:06:45.910 --> 00:06:49.630
sacrificed other things in
order to get these benefits now.

00:06:49.630 --> 00:06:51.570
But it's a very hard question.

00:06:51.570 --> 00:06:54.980
When you look at this, you say,
well, these people, they've

00:06:54.980 --> 00:06:55.780
done service.

00:06:55.780 --> 00:06:57.144
They put these expectations.

00:06:57.144 --> 00:06:58.560
But at the same
time, you're like,

00:06:58.560 --> 00:07:00.277
well, this is
really cutting in--

00:07:00.277 --> 00:07:02.110
and this is just one
thing that I'm showing.

00:07:02.110 --> 00:07:05.120
It's really cutting into
very important areas

00:07:05.120 --> 00:07:07.410
of investment for
the entire state.

00:07:07.410 --> 00:07:09.285
So the whole reason of
really just surfacing

00:07:09.285 --> 00:07:11.905
this, this whole pension
issue, is just to put this in.

00:07:11.905 --> 00:07:14.270
And hopefully people
understand what the issues are,

00:07:14.270 --> 00:07:18.180
because that's the only way that
fairly hard decisions are going

00:07:18.180 --> 00:07:22.000
to have to be made, decisions
on cutting necessary investment

00:07:22.000 --> 00:07:25.500
or restructuring or who
knows what it might be.

00:07:25.500 --> 00:07:28.990
I don't envy the people who
have to make these decisions.


Pension obligations

WEBVTT
Kind: captions
Language: en

00:00:00.436 --> 00:00:03.209
Voiceover: Talking about
pensions isn't always

00:00:03.209 --> 00:00:06.026
viewed as the most
interesting thing to do,

00:00:06.026 --> 00:00:08.204
but hopefully this video and the next one

00:00:08.204 --> 00:00:10.590
might convince you that
it is at least worth

00:00:10.590 --> 00:00:12.667
talking about and
understanding because it has

00:00:12.667 --> 00:00:15.207
major implications for
the fiscal situation

00:00:15.207 --> 00:00:18.820
of many, many, many states in the U.S.

00:00:18.820 --> 00:00:22.024
A pension is essentially
a defined benefit plan.

00:00:22.024 --> 00:00:27.371
Defined benefit plan for retirement.

00:00:29.540 --> 00:00:32.406
We can compare that to a
defined contribution plan.

00:00:32.406 --> 00:00:37.036
Defined contribution plan.

00:00:39.622 --> 00:00:41.855
The defined contribution plan, and this is

00:00:41.855 --> 00:00:44.456
more typical in a lot of
private companies right now,

00:00:44.456 --> 00:00:48.022
every year that you work,
let's plot a little graph here.

00:00:48.022 --> 00:00:50.125
This is years that you're working,

00:00:50.125 --> 00:00:54.974
and this is kind of compensation.

00:00:54.974 --> 00:00:56.898
This is compensation right over here.

00:00:56.898 --> 00:00:59.392
In a defined contribution,
every year that you work,

00:00:59.392 --> 00:01:01.333
you're obviously going
to make your salary.

00:01:01.333 --> 00:01:02.927
Let's say you're making $60,000 a year.

00:01:02.927 --> 00:01:04.134
You're obviously going to make

00:01:04.134 --> 00:01:05.959
your salary every year that you work.

00:01:05.959 --> 00:01:07.787
In a defined contribution plan,

00:01:07.787 --> 00:01:09.055
let's say this is when you retire,

00:01:09.055 --> 00:01:10.770
let's say for simplicity, we're assuming

00:01:10.770 --> 00:01:12.393
that you're going to retire at 65.

00:01:12.393 --> 00:01:14.391
At 65, you're no longer going to be

00:01:14.391 --> 00:01:16.130
making your salary anymore.

00:01:16.130 --> 00:01:17.961
You're not working for
that company anymore.

00:01:17.961 --> 00:01:20.300
What the company will
do is set up some plan,

00:01:20.300 --> 00:01:23.213
and a lot of these are 401(k)s, IRAs,

00:01:23.213 --> 00:01:25.308
where some combination of you and

00:01:25.308 --> 00:01:27.417
the company will set some money aside,

00:01:27.417 --> 00:01:29.338
and it's usually done
in a tax-deferred way,

00:01:29.338 --> 00:01:31.525
so you don't have to pay
taxes on it in that year.

00:01:31.525 --> 00:01:33.795
Every year, you're going
to set some money aside,

00:01:33.795 --> 00:01:35.010
and I'll do that in green.

00:01:35.010 --> 00:01:37.991
You're going to set maybe
10% of your income aside

00:01:37.991 --> 00:01:40.834
in every year, so these are every year.

00:01:40.834 --> 00:01:42.121
Actually, the years are
going to be much smaller

00:01:42.121 --> 00:01:44.639
than that, if we're thinking
this is about 30 years.

00:01:44.639 --> 00:01:48.062
Over the 30 years,
you're setting some level

00:01:48.062 --> 00:01:50.055
of money aside, and you're investing it:

00:01:50.055 --> 00:01:51.640
you're putting it in the
stock market; you're buying

00:01:51.640 --> 00:01:53.426
bonds with it; you're buying mutual funds.

00:01:53.426 --> 00:01:54.922
Who knows what you might be doing with it?

00:01:54.922 --> 00:01:58.128
You're setting this money
aside, so that when you

00:01:58.128 --> 00:02:02.011
retire, it will hopefully have grown.

00:02:02.011 --> 00:02:04.260
Well, one, it's there,
and you've invested it.

00:02:04.260 --> 00:02:05.856
Hopefully, if you've invested
it well and the stock

00:02:05.856 --> 00:02:07.994
market didn't do anything
crazy, it will have grown,

00:02:07.994 --> 00:02:10.813
and it will be just a
big lump sum of money.

00:02:10.813 --> 00:02:13.922
Let's say that you set aside $6,000 a year

00:02:13.922 --> 00:02:19.151
on average for 30 years,
so you set aside $180,000.

00:02:19.151 --> 00:02:22.755
Let's say you invested it
pretty well, and now that has

00:02:22.755 --> 00:02:25.735
grown to, I don't know,
let's say it's grown

00:02:25.735 --> 00:02:28.198
to $1 million, because
it was invested well.

00:02:28.198 --> 00:02:29.436
We could do the math to figure out what

00:02:29.436 --> 00:02:31.132
it could grow based on
different growth rates.

00:02:31.132 --> 00:02:34.745
You have this huge lump sum of money now.

00:02:34.745 --> 00:02:36.945
I'm not even drawing it to scale.

00:02:36.945 --> 00:02:39.142
You have this huge lump sum of money,

00:02:39.142 --> 00:02:41.195
$1 million, assuming it was invested well.

00:02:41.195 --> 00:02:42.634
If it was invested badly,

00:02:42.634 --> 00:02:47.892
maybe that $180,000 is still $180,000.

00:02:47.892 --> 00:02:49.794
In theory, maybe if it
was invested really badly,

00:02:49.794 --> 00:02:52.142
it could be even less than $180,000.

00:02:52.142 --> 00:02:54.492
Whatever that number is,
whether it's $1 million

00:02:54.492 --> 00:02:57.060
or whether it's $200,000, or whether it's

00:02:57.060 --> 00:02:59.726
something smaller, that's
essentially the money

00:02:59.726 --> 00:03:01.886
that you have to live
on for your retirement.

00:03:01.886 --> 00:03:03.659
Regardless of how long you live,

00:03:03.659 --> 00:03:06.144
regardless of what the
cost of living might be,

00:03:06.144 --> 00:03:08.327
regardless of what your needs might be,

00:03:08.327 --> 00:03:12.965
regardless of how expensive
your healthcare might be,

00:03:12.965 --> 00:03:17.326
this is going to be the money
that you have to live on.

00:03:17.326 --> 00:03:19.267
It might be more than enough money, if you

00:03:19.267 --> 00:03:21.344
invest it well and you
put enough money aside;

00:03:21.344 --> 00:03:22.839
it might be a lot less than you need,

00:03:22.839 --> 00:03:24.665
in which case you're
going to be in trouble.

00:03:24.665 --> 00:03:27.591
A defined benefit plan,
and this is typical;

00:03:27.591 --> 00:03:30.258
a lot of state employees
have defined benefit plan,

00:03:30.258 --> 00:03:33.063
a lot of more traditional industries

00:03:33.063 --> 00:03:34.664
oftentimes that are unionized.

00:03:34.664 --> 00:03:37.394
You also have a defined
benefit plan or a pension,

00:03:37.394 --> 00:03:39.664
and the situation is a little different.

00:03:39.664 --> 00:03:43.964
Just like any organization,
you will get your salary

00:03:43.964 --> 00:03:46.453
every year that you
work; and let's just say

00:03:46.453 --> 00:03:48.943
that this is over the course of 30 years.

00:03:48.943 --> 00:03:52.037
Once again, you retire at age 65.

00:03:54.866 --> 00:03:58.547
In defined benefit plan,
the employer is going to

00:03:58.562 --> 00:04:00.321
set aside some money, and sometimes

00:04:00.321 --> 00:04:02.531
the employee sets some
money aside as well,

00:04:02.531 --> 00:04:04.481
so some money is set aside.

00:04:04.481 --> 00:04:06.461
Once again, the money is set aside,

00:04:07.999 --> 00:04:12.269
and it is invested,
hopefully in a safe way.

00:04:13.130 --> 00:04:16.106
But regardless of what that
money and regardless of

00:04:16.106 --> 00:04:18.123
what that turns into
through an investment,

00:04:18.123 --> 00:04:21.056
you are guaranteed a
certain degree of benefits.

00:04:21.056 --> 00:04:24.992
In this case, you are
guaranteed; let's say that

00:04:24.992 --> 00:04:27.527
if you'd done more than
20 years of service here,

00:04:27.527 --> 00:04:30.989
you get 60% of your
last five years' salary.

00:04:30.989 --> 00:04:33.621
There's different ways of
defining that defined benefit.

00:04:33.621 --> 00:04:37.052
It could be like that, it
could be you get $100 per month

00:04:37.052 --> 00:04:39.473
for every year that you
worked at the organization.

00:04:39.473 --> 00:04:42.919
You get $100 per month
extra when you retire.

00:04:43.672 --> 00:04:46.470
But they tend to be for life,
for the rest of your life.

00:04:46.470 --> 00:04:48.737
So from 65 until you pass away,

00:04:48.737 --> 00:04:55.898
you are guaranteed this defined benefit.

00:04:57.533 --> 00:05:01.138
If the money is set
aside, if it was set aside

00:05:01.138 --> 00:05:03.410
and invested well and
happen to be a lot more

00:05:03.410 --> 00:05:06.682
money than necessary,
that's great, but all

00:05:06.682 --> 00:05:09.022
the employee would get is
this kind of guarantee.

00:05:09.022 --> 00:05:10.732
If the money is less than necessary,

00:05:10.732 --> 00:05:14.270
then the company is still promising that

00:05:14.270 --> 00:05:17.520
they are going to pay this benefit.

00:05:17.520 --> 00:05:19.730
They'll probably have to
put more, or the state,

00:05:19.730 --> 00:05:21.686
or whoever is doing
this, would have to put

00:05:21.686 --> 00:05:25.868
more money in in order
to pay this compensation.

00:05:25.868 --> 00:05:28.446
Now, what are the things
that you would have to

00:05:28.446 --> 00:05:30.174
estimate if you are the
person setting aside

00:05:30.174 --> 00:05:33.266
this money, to figure out
what you have to set aside

00:05:33.266 --> 00:05:36.748
in order to give this defined benefit?

00:05:36.748 --> 00:05:39.258
You're going to have to hire
a bunch of statisticians,

00:05:39.258 --> 00:05:41.006
essentially actuaries,

00:05:41.006 --> 00:05:43.458
to say how long are people going to live.

00:05:43.458 --> 00:05:45.940
You're going to have to
care about life span.

00:05:45.940 --> 00:05:48.424
Obviously, you can't predict
any one person's life span,

00:05:48.424 --> 00:05:49.836
but if you're doing this for hundreds

00:05:49.836 --> 00:05:52.064
of thousands of employees, maybe you can

00:05:52.064 --> 00:05:54.335
figure out what a likely life span is.

00:05:54.335 --> 00:05:56.270
You're going to have to
figure out cost of living.

00:05:56.270 --> 00:05:58.948
Inflation is a measure of cost of living,

00:05:58.948 --> 00:06:00.917
but it might be more
specific to the region, or it

00:06:00.917 --> 00:06:03.206
might be negotiated in
some ways with the union.

00:06:03.206 --> 00:06:06.605
You're going to have cost of living.

00:06:06.605 --> 00:06:09.068
This is a cost of living adjustment.

00:06:09.068 --> 00:06:10.475
When people talk about COLAs,

00:06:10.475 --> 00:06:12.424
if they're not talking about soda,

00:06:12.424 --> 00:06:15.585
they're talking about cost
of living adjustments.

00:06:15.585 --> 00:06:16.869
You're going to have to think about

00:06:16.869 --> 00:06:19.930
this money that you set aside.

00:06:19.945 --> 00:06:22.446
What is the assumed growth rate?

00:06:22.446 --> 00:06:26.730
What is the assumed growth rate?

00:06:26.730 --> 00:06:30.331
If you make very optimistic
estimates of how well your

00:06:30.331 --> 00:06:32.912
investments will do, you
can set aside less money.

00:06:32.912 --> 00:06:35.153
If you think that your
money isn't going to

00:06:35.153 --> 00:06:37.135
do well investment-wise, you're going to

00:06:37.135 --> 00:06:40.203
have to set aside even more money.

00:06:40.203 --> 00:06:43.395
This is one of the cruxes of the issue,

00:06:43.395 --> 00:06:44.875
because you could imagine,

00:06:44.875 --> 00:06:46.653
let's say that we're
talking at a state level,

00:06:46.653 --> 00:06:50.453
and people are, right now let's say that

00:06:50.453 --> 00:06:53.116
your current actuaries or
statisticians are saying,

00:06:53.116 --> 00:06:55.548
"Look, for this person,
in order to guarantee them

00:06:55.548 --> 00:06:57.866
60% of their salary when they retire,"

00:06:57.866 --> 00:07:01.123
so that's $36,000, "In
order to guarantee that,

00:07:01.123 --> 00:07:03.454
we have to put aside ..." and I'm just

00:07:03.454 --> 00:07:06.457
estimating these number right over here.

00:07:06.457 --> 00:07:09.223
Let's say we have to
set aside $6,000 a year,

00:07:10.346 --> 00:07:11.787
especially when we're 30 years in advance.

00:07:11.787 --> 00:07:13.281
Actually, let me do a
little more than that.

00:07:13.281 --> 00:07:16.075
Let's say $10,000 per year.

00:07:16.075 --> 00:07:18.146
Let's say that the person in charge,

00:07:18.146 --> 00:07:20.056
the state official, goes
to those actuaries and say,

00:07:20.056 --> 00:07:21.735
"What are you assuming
about how much we're

00:07:21.735 --> 00:07:23.797
going to get on our investments here?"

00:07:23.797 --> 00:07:25.148
The actuaries are saying,

00:07:25.148 --> 00:07:28.672
"We're going to assume
a fairly conservative.

00:07:28.672 --> 00:07:30.429
"We're going to assume that

00:07:30.429 --> 00:07:35.794
we're going to get 3%
return on our money."

00:07:35.794 --> 00:07:36.996
But then the state official says,

00:07:36.996 --> 00:07:39.398
ideally, they would want some of this

00:07:39.398 --> 00:07:41.965
$10,000 per year to spend on other things,

00:07:41.965 --> 00:07:44.397
and so they would like this to be lower.

00:07:44.397 --> 00:07:47.266
They say, "That seems very conservative.

00:07:47.266 --> 00:07:49.174
"In the last 10 years in the stock market,

00:07:49.174 --> 00:07:51.860
"we've gotten 10% return," or, "I know an

00:07:51.860 --> 00:07:54.535
"endowment that's
recently gotten 6% return.

00:07:54.535 --> 00:07:56.662
"Why don't we assume a higher return here?

00:07:56.662 --> 00:07:58.890
"If we assume a higher return,

00:07:58.890 --> 00:08:04.200
"why don't we assume a 5% return?"

00:08:04.200 --> 00:08:06.263
All of a sudden, if we're
assuming a 5% return,

00:08:06.263 --> 00:08:09.559
then we'll have to set
aside less money that year.

00:08:09.559 --> 00:08:11.554
$8,000 a year.

00:08:11.554 --> 00:08:13.222
Sometimes, it's not even this,

00:08:13.222 --> 00:08:15.210
it's not even this playing
with the assumptions,

00:08:15.210 --> 00:08:17.100
making more optimistic
assumptions that allow you

00:08:17.100 --> 00:08:18.867
to spend less money in that current year,

00:08:18.867 --> 00:08:22.605
sometimes you might know
that you have to spend

00:08:22.605 --> 00:08:26.267
$10,000 a year to kind of be able to

00:08:26.267 --> 00:08:28.823
properly fund these
pensions in the future.

00:08:28.823 --> 00:08:30.669
You do have some type
of unfunded pensions;

00:08:30.669 --> 00:08:33.217
but, in theory, a
responsible party should try

00:08:33.217 --> 00:08:36.307
to fund these as much as possible.

00:08:36.307 --> 00:08:39.460
You might know that you
have to fund $10,000 a year

00:08:39.460 --> 00:08:42.810
in order to credibly
give this defined benefit

00:08:42.810 --> 00:08:45.336
for this employee 30 years in the future.

00:08:45.336 --> 00:08:47.474
But 30 years in the future is a long time.

00:08:47.474 --> 00:08:49.610
You have present difficuties;

00:08:49.610 --> 00:08:52.045
you have present shortfalls
in your budget, and say,

00:08:52.045 --> 00:08:54.315
"I recognize that I have
to put $10,000 a year,"

00:08:54.315 --> 00:08:55.908
but you still don't do that,

00:08:55.908 --> 00:08:57.595
so you underfund the pension.

00:08:57.595 --> 00:09:00.346
Even if you recognize this,
or if you recognize this,

00:09:00.346 --> 00:09:03.693
you still only put $5,000 a year.

00:09:03.693 --> 00:09:06.202
Really just kind of kicking
the can down the road,

00:09:06.202 --> 00:09:08.800
hoping that the next guy or
gal who's in your position

00:09:08.800 --> 00:09:11.130
is going to figure out
something; or maybe you'll just

00:09:11.130 --> 00:09:13.003
be very optimistic that
the growth will turn out,

00:09:13.003 --> 00:09:17.145
or that the state will
eventually work things out.

00:09:17.145 --> 00:09:19.404
What we'll see over the next video,

00:09:19.404 --> 00:09:27.793
this notion of underfunding
pensions is a big, big,

00:09:27.793 --> 00:09:30.199
big, big problem because we've had decades

00:09:30.199 --> 00:09:32.397
of underfunded pensions, and it's been

00:09:32.397 --> 00:09:34.464
especially pronounced
in particular states.

00:09:34.464 --> 00:09:37.202
Because of that, those
states, in order to fund

00:09:37.202 --> 00:09:40.269
the pension obligations
that are hitting now,

00:09:40.269 --> 00:09:43.468
those expenses for
employees that are retired

00:09:43.468 --> 00:09:46.465
are starting to grow beyond their budgets

00:09:46.465 --> 00:09:48.696
for the employees that
are working right now.

00:09:48.696 --> 00:09:49.797
It's a tough issue.

00:09:49.797 --> 00:09:51.538
You can't cut these things very easily.

00:09:51.538 --> 00:09:52.743
People expected these.

00:09:52.743 --> 00:09:53.538
These are retirees.

00:09:53.538 --> 00:09:54.661
These are people who've been working

00:09:54.661 --> 00:09:56.862
their whole life based on this assumption.

00:09:56.862 --> 00:09:58.555
But, at the same time, they're starting to

00:09:58.555 --> 00:10:01.744
squeeze out key services
that the state is doing.

00:10:01.744 --> 00:10:05.744
It turns into a major, major hairy issue.


Review and intuition why we divide by n-1 for the unbiased sample

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:02.940
What I want to do in
this video is review much

00:00:02.940 --> 00:00:05.510
of what we've already talked
about and then hopefully build

00:00:05.510 --> 00:00:09.730
some of the intuition on
why we divide by n minus 1

00:00:09.730 --> 00:00:13.180
if we want to have an unbiased
estimate of the population

00:00:13.180 --> 00:00:17.290
variance when we're calculating
the sample variance.

00:00:17.290 --> 00:00:19.490
So let's think
about a population.

00:00:19.490 --> 00:00:22.890
So let's say this is the
population right over here.

00:00:22.890 --> 00:00:26.310
And it is of size
capital N. And we also

00:00:26.310 --> 00:00:28.900
have a sample of
that population,

00:00:28.900 --> 00:00:30.710
so a sample of that population.

00:00:30.710 --> 00:00:34.470
And in its size, we have
lowercase n data points.

00:00:34.470 --> 00:00:38.280
So let's think about all of
the parameters and statistics

00:00:38.280 --> 00:00:40.820
that we know about so far.

00:00:40.820 --> 00:00:46.320
So the first is the idea
of the mean, of the mean.

00:00:46.320 --> 00:00:50.270
So if we're trying to calculate
the mean for the population,

00:00:50.270 --> 00:00:54.574
is that going to be a
parameter or a statistic?

00:00:54.574 --> 00:00:56.990
Well, when we're trying to
calculate it on the population,

00:00:56.990 --> 00:00:59.120
we are calculating a parameter.

00:00:59.120 --> 00:01:01.350
We are calculating a parameter.

00:01:01.350 --> 00:01:03.240
So let me write this down.

00:01:03.240 --> 00:01:08.370
So this is going to be--
so for the population we

00:01:08.370 --> 00:01:11.430
are calculating a parameter.

00:01:11.430 --> 00:01:13.260
It is a parameter.

00:01:13.260 --> 00:01:16.130
And when we calculate, when we
attempt to calculate something

00:01:16.130 --> 00:01:23.520
for a sample we would call
that a statistic-- statistic.

00:01:23.520 --> 00:01:27.200
So how do we think about
the mean for a population?

00:01:27.200 --> 00:01:31.510
Well, first of all, we denote
it with the Greek letter mu.

00:01:31.510 --> 00:01:35.630
And we essentially take every
data point in our population.

00:01:35.630 --> 00:01:37.722
So we take the sum
of every data point.

00:01:37.722 --> 00:01:39.180
So we start at the
first data point

00:01:39.180 --> 00:01:41.430
and we go all the way to
the capital Nth data point.

00:01:41.430 --> 00:01:44.290
So every data point we add up.

00:01:44.290 --> 00:01:46.340
So this is the i-th
data point, so x

00:01:46.340 --> 00:01:49.610
sub 1 plus x sub 2 all the
way to x sub capital N.

00:01:49.610 --> 00:01:54.020
And then we divide by the total
number of data points we have.

00:01:54.020 --> 00:01:57.500
Well, how do we calculate
the sample mean?

00:01:57.500 --> 00:02:00.240
Well, the sample mean--
we do a very similar thing

00:02:00.240 --> 00:02:01.240
with the sample.

00:02:01.240 --> 00:02:05.040
And we denote it with
a x with a bar over it.

00:02:05.040 --> 00:02:10.160
And that's going to be taking
every data point in the sample,

00:02:10.160 --> 00:02:12.824
so going up to a lower
case n, adding them up

00:02:12.824 --> 00:02:15.365
--so these are the sum of all
the data points in our sample--

00:02:15.365 --> 00:02:17.600
and then dividing by
the number of data

00:02:17.600 --> 00:02:19.890
points that we actually had.

00:02:19.890 --> 00:02:23.650
Now, the other thing
that we're trying

00:02:23.650 --> 00:02:26.020
to calculate for the population,
which was a parameter,

00:02:26.020 --> 00:02:29.130
and then we'll also try to
calculate it for the sample

00:02:29.130 --> 00:02:31.010
and estimate it
for the population,

00:02:31.010 --> 00:02:34.620
was the variance, which was
a measure of how dispersed

00:02:34.620 --> 00:02:39.180
or how much of the data
points vary from the mean.

00:02:39.180 --> 00:02:43.810
So let's write variance
right over here.

00:02:43.810 --> 00:02:47.630
And how do we denote
any calculate variance

00:02:47.630 --> 00:02:50.150
for a population?

00:02:50.150 --> 00:02:52.490
Well, for population, we'd
say that the variance --we

00:02:52.490 --> 00:02:56.190
use a Greek letter sigma
squared-- is equal to--

00:02:56.190 --> 00:02:59.530
and you can view it as the
mean of the squared distances

00:02:59.530 --> 00:03:02.050
from the population mean.

00:03:02.050 --> 00:03:05.310
But what we do is we
take, for each data point,

00:03:05.310 --> 00:03:07.790
so i equal 1 all
the way to n, we

00:03:07.790 --> 00:03:11.406
take that data point, subtract
from it the population mean.

00:03:11.406 --> 00:03:12.780
So if you want to
calculate this,

00:03:12.780 --> 00:03:14.220
you'd want to figure this out.

00:03:14.220 --> 00:03:15.150
Well, that's one way to do it.

00:03:15.150 --> 00:03:16.733
We'll see there's
other ways to do it,

00:03:16.733 --> 00:03:18.750
where you can calculate
them at the same time.

00:03:18.750 --> 00:03:21.150
But the easiest or
the most intuitive

00:03:21.150 --> 00:03:24.000
is to calculate this first,
then for each of the data points

00:03:24.000 --> 00:03:26.160
take the data point and
subtract it from that,

00:03:26.160 --> 00:03:29.850
subtract the mean
from that, square it,

00:03:29.850 --> 00:03:34.300
and then divide by the total
number of data points you have.

00:03:34.300 --> 00:03:38.520
Now, we get to the interesting
part-- sample variance.

00:03:38.520 --> 00:03:42.040
There's are several ways-- where
when people talk about sample

00:03:42.040 --> 00:03:45.020
variance, there's several
tools in their toolkits

00:03:45.020 --> 00:03:47.290
or there's several
ways to calculate it.

00:03:47.290 --> 00:03:51.492
One way is the biased
sample variance,

00:03:51.492 --> 00:03:55.360
the non unbiased estimator
of the population variance.

00:03:55.360 --> 00:03:58.250
And that's denoted,
usually denoted, by s

00:03:58.250 --> 00:04:00.510
with a subscript n.

00:04:00.510 --> 00:04:05.109
And what is the biased
estimator, how we calculate it?

00:04:05.109 --> 00:04:07.650
Well, we would calculate it very
similar to how we calculated

00:04:07.650 --> 00:04:08.858
the variance right over here.

00:04:08.858 --> 00:04:12.620
But what we would do it for
our sample, not our population.

00:04:12.620 --> 00:04:18.040
So for every data point in our
sample --so we have n of them--

00:04:18.040 --> 00:04:20.470
we take that data point.

00:04:20.470 --> 00:04:23.570
And from it, we subtract
our sample mean.

00:04:23.570 --> 00:04:25.860
We subtract our sample
mean, square it,

00:04:25.860 --> 00:04:28.889
and then divide by the number
of data points that we have.

00:04:28.889 --> 00:04:30.930
But we already talked
about it in the last video.

00:04:30.930 --> 00:04:34.390
How would we find-- what is
our best unbiased estimate

00:04:34.390 --> 00:04:35.850
of the population variance?

00:04:35.850 --> 00:04:37.910
This is usually what
we're trying to get at.

00:04:37.910 --> 00:04:40.440
We're trying to find an unbiased
estimate of the population

00:04:40.440 --> 00:04:41.810
variance.

00:04:41.810 --> 00:04:44.326
Well, in the last video,
we talked about that,

00:04:44.326 --> 00:04:45.950
if we want to have
an unbiased estimate

00:04:45.950 --> 00:04:47.620
--and here, in
this video, I want

00:04:47.620 --> 00:04:50.370
to give you a sense
of the intuition why.

00:04:50.370 --> 00:04:52.530
We would take the sum.

00:04:52.530 --> 00:04:55.471
So we're going to go through
every data point in our sample.

00:04:55.471 --> 00:04:56.970
We're going to take
that data point,

00:04:56.970 --> 00:05:00.430
subtract from it the
sample mean, square that.

00:05:00.430 --> 00:05:04.940
But instead of dividing by
n, we divide by n minus 1.

00:05:04.940 --> 00:05:09.140
We're dividing by
a smaller number.

00:05:09.140 --> 00:05:11.230
We're dividing by
a smaller number.

00:05:11.230 --> 00:05:13.180
And when you divide
by a smaller number,

00:05:13.180 --> 00:05:16.870
you're going to
get a larger value.

00:05:16.870 --> 00:05:20.620
So this is going to be larger.

00:05:20.620 --> 00:05:23.470
This is going to be smaller.

00:05:23.470 --> 00:05:25.955
And this one, we refer
to the unbiased estimate.

00:05:28.940 --> 00:05:33.480
And this one, we refer
to the biased estimate.

00:05:33.480 --> 00:05:36.750
If people just
write this, they're

00:05:36.750 --> 00:05:38.255
talking about the
sample variance.

00:05:38.255 --> 00:05:40.130
It's a good idea to
clarify which one they're

00:05:40.130 --> 00:05:40.759
talking about.

00:05:40.759 --> 00:05:42.550
But if you had to guess
and people give you

00:05:42.550 --> 00:05:44.970
no further information,
they're probably

00:05:44.970 --> 00:05:48.530
talking about the unbiased
estimate of the variance.

00:05:48.530 --> 00:05:51.050
So you'd probably
divide by n minus 1.

00:05:51.050 --> 00:05:56.390
But let's think about why
this estimate would be biased

00:05:56.390 --> 00:05:59.017
and why we might want to have
an estimate like that is larger.

00:05:59.017 --> 00:06:01.600
And then maybe in the future,
we could have a computer program

00:06:01.600 --> 00:06:03.960
or something that really
makes us feel better,

00:06:03.960 --> 00:06:05.710
that dividing by
n minus 1 gives us

00:06:05.710 --> 00:06:09.640
a better estimate of the
true population variance.

00:06:09.640 --> 00:06:13.320
So let's imagine all the
data in a population.

00:06:13.320 --> 00:06:16.530
And I'm just going to plot
them on number a line.

00:06:16.530 --> 00:06:19.020
So this is my number line.

00:06:19.020 --> 00:06:20.700
This is my number line.

00:06:20.700 --> 00:06:23.960
And let me plot all the data
points in my population.

00:06:23.960 --> 00:06:25.780
So this is some data.

00:06:25.780 --> 00:06:27.620
This is some data.

00:06:27.620 --> 00:06:29.710
Here's some data.

00:06:29.710 --> 00:06:32.260
And here is some data here.

00:06:32.260 --> 00:06:34.400
And I can just do as
many points as I want.

00:06:34.400 --> 00:06:36.750
So these are just points
on the number line.

00:06:36.750 --> 00:06:39.160
Now, let's say I take
a sample of this.

00:06:39.160 --> 00:06:40.924
So this is my entire population.

00:06:40.924 --> 00:06:41.840
So let's see how many.

00:06:41.840 --> 00:06:47.640
I have 1 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14.

00:06:47.640 --> 00:06:49.940
So in this case, what
would be my big N?

00:06:49.940 --> 00:06:52.090
My big N would be 14.

00:06:52.090 --> 00:06:54.480
Big N would be 14.

00:06:54.480 --> 00:06:57.750
Now, let's say I take a sample,
a lowercase n of-- let's say

00:06:57.750 --> 00:07:01.400
my sample size is 3.

00:07:01.400 --> 00:07:04.450
I could take-- well, before
I even think about that,

00:07:04.450 --> 00:07:08.900
let's think about roughly where
the mean of this population

00:07:08.900 --> 00:07:09.430
would sit.

00:07:09.430 --> 00:07:10.846
So the way I drew
it --and I'm not

00:07:10.846 --> 00:07:12.490
going to calculate
exactly-- it looks

00:07:12.490 --> 00:07:17.030
like the mean might sit some
place roughly right over here.

00:07:17.030 --> 00:07:19.180
So the mean, the
true population mean,

00:07:19.180 --> 00:07:22.200
the parameter's going
to sit right over here.

00:07:22.200 --> 00:07:24.440
Now, let's think about what
happens when we sample.

00:07:24.440 --> 00:07:26.900
And I'm going to do just a
very small sample size just

00:07:26.900 --> 00:07:28.316
to give us the
intuition, but this

00:07:28.316 --> 00:07:29.800
is true of any sample size.

00:07:29.800 --> 00:07:34.310
So let's say we have
sample size of 3.

00:07:34.310 --> 00:07:36.170
So there is some
possibility, when

00:07:36.170 --> 00:07:39.410
we take our sample size of 3,
that we happen to sample it

00:07:39.410 --> 00:07:43.630
in a way that our sample mean is
pretty close to our population

00:07:43.630 --> 00:07:44.130
mean.

00:07:44.130 --> 00:07:47.090
So for example, if we sampled
to that point, that point,

00:07:47.090 --> 00:07:50.290
and that point, I could imagine
in our sample mean might

00:07:50.290 --> 00:07:53.130
actually said pretty
close, pretty close

00:07:53.130 --> 00:07:56.440
to our population mean.

00:07:56.440 --> 00:07:58.810
But there's a
distinct possibility,

00:07:58.810 --> 00:08:01.490
there's a distinct
possibility, that maybe when

00:08:01.490 --> 00:08:04.660
I take a sample, I
sample that and that.

00:08:04.660 --> 00:08:07.960
And the key idea here is
when you take a sample,

00:08:07.960 --> 00:08:12.170
your sample mean is always
going to sit within your sample.

00:08:12.170 --> 00:08:15.670
And so there is a possibility
that when you take your sample,

00:08:15.670 --> 00:08:18.282
your mean could even be
outside of the sample.

00:08:18.282 --> 00:08:19.740
And so in this
situation-- and this

00:08:19.740 --> 00:08:20.860
is just to give
you an intuition.

00:08:20.860 --> 00:08:22.276
So here, your
sample mean is going

00:08:22.276 --> 00:08:26.980
to be sitting
someplace in there.

00:08:26.980 --> 00:08:29.120
And so if you were to just
calculate the distance

00:08:29.120 --> 00:08:33.110
from each of this points to the
sample mean --so this distance,

00:08:33.110 --> 00:08:34.510
that distance,
and you square it,

00:08:34.510 --> 00:08:36.801
and you were to divide by
the number of data points you

00:08:36.801 --> 00:08:38.330
have-- this is
going to be a much

00:08:38.330 --> 00:08:41.600
lower estimate than the true
variance the true variance,

00:08:41.600 --> 00:08:45.520
from the actual population mean,
where these things are much,

00:08:45.520 --> 00:08:46.979
much, much further.

00:08:46.979 --> 00:08:49.520
Now, you're always not going to
have the true population mean

00:08:49.520 --> 00:08:50.570
outside of your sample.

00:08:50.570 --> 00:08:52.490
But it's possible that you do.

00:08:52.490 --> 00:08:56.710
So in general, when you
just take your points,

00:08:56.710 --> 00:08:59.490
find the squared distance
to your sample mean, which

00:08:59.490 --> 00:09:01.730
is always going to sit
inside of your data

00:09:01.730 --> 00:09:05.230
even though the true population
mean could be outside of it,

00:09:05.230 --> 00:09:07.081
or it could be at
one end of your data,

00:09:07.081 --> 00:09:08.830
however, you might
want to think about it,

00:09:08.830 --> 00:09:11.820
you are likely to
be underestimating,

00:09:11.820 --> 00:09:13.400
you're likely to
be underestimating

00:09:13.400 --> 00:09:15.510
the true population variance.

00:09:15.510 --> 00:09:18.665
So this right over here is an
underestimate-- underestimate.

00:09:18.665 --> 00:09:21.530
And it does turn out that
if you just-- instead

00:09:21.530 --> 00:09:23.940
of dividing by n, you
divide by n minus 1,

00:09:23.940 --> 00:09:26.510
you'll get a slightly
larger sample variance.

00:09:26.510 --> 00:09:29.937
And this is an
unbiased estimate.

00:09:29.937 --> 00:09:32.520
In the next video --and I might
not to get to it immediately--

00:09:32.520 --> 00:09:35.040
I would like to generate some
type of a computer program

00:09:35.040 --> 00:09:38.730
that is more convincing that
this is a better estimate,

00:09:38.730 --> 00:09:42.080
this is a better estimate
of the population variance

00:09:42.080 --> 00:09:44.230
than this is.