Torque and kinematics conceptual example

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.970
- [Instructor] We are told
a student hangs blocks

00:00:02.970 --> 00:00:06.850
with different masses
from a pulley of mass, M,

00:00:06.850 --> 00:00:08.650
and radius, r,

00:00:08.650 --> 00:00:11.420
and releases them from rest.

00:00:11.420 --> 00:00:15.230
The student measures
the time of the fall, t,

00:00:15.230 --> 00:00:20.180
and the magnitude of the
angular velocity, omega sub f,

00:00:20.180 --> 00:00:22.720
when the block reaches a distance, d,

00:00:22.720 --> 00:00:24.370
below the axle.

00:00:24.370 --> 00:00:28.700
There is considerable friction
between the axle and pulley.

00:00:28.700 --> 00:00:30.250
Which of the following plots

00:00:30.250 --> 00:00:32.110
could show the student's measurements

00:00:32.110 --> 00:00:36.380
as a function of the applied
torque on the pulley?

00:00:36.380 --> 00:00:37.340
So pause this video

00:00:37.340 --> 00:00:39.590
and see if you can figure
it out on your own.

00:00:40.620 --> 00:00:44.260
All right, so let's first
understand what these axes are,

00:00:44.260 --> 00:00:45.330
and then we'll talk a little bit

00:00:45.330 --> 00:00:48.460
about what's going on with
this mass pulley system.

00:00:48.460 --> 00:00:51.060
So, on our horizontal axis here,

00:00:51.060 --> 00:00:53.780
we have our applied torque.

00:00:53.780 --> 00:00:57.130
Now where is the applied
torque in this system?

00:00:57.130 --> 00:00:59.120
Well, you're going to have some mass here,

00:00:59.120 --> 00:01:00.930
and we assume we're on a planet

00:01:00.930 --> 00:01:02.470
or probably on Earth.

00:01:02.470 --> 00:01:05.790
So you're going to have
gravity pulling down on this,

00:01:05.790 --> 00:01:08.900
so you're gonna have the force
of the weight of this object

00:01:08.900 --> 00:01:10.030
pulling down,

00:01:10.030 --> 00:01:12.810
and we know that this has radius, r.

00:01:12.810 --> 00:01:14.810
So that force times that radius

00:01:14.810 --> 00:01:16.510
is going to be the applied torque.

00:01:16.510 --> 00:01:18.830
And of course, the more mass you have,

00:01:18.830 --> 00:01:20.600
the more weight you're going to have,

00:01:20.600 --> 00:01:22.260
and you're going to have more torque.

00:01:22.260 --> 00:01:23.400
So this is really showing

00:01:23.400 --> 00:01:27.440
that as the student applies
larger and larger masses,

00:01:27.440 --> 00:01:28.610
right over here,

00:01:28.610 --> 00:01:31.320
we're going to have more
and more and more torque.

00:01:31.320 --> 00:01:34.160
Now, what is in our vertical
axis right over here?

00:01:34.160 --> 00:01:36.460
Omega sub f divided by t.

00:01:36.460 --> 00:01:39.900
So this is our final angular velocity,

00:01:39.900 --> 00:01:42.560
once this thing has
gone down a distance, d,

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that's what they tell us,

00:01:43.850 --> 00:01:47.080
divided by the time it took to get there.

00:01:47.080 --> 00:01:50.410
Well, if we assume that it
started from a standstill,

00:01:50.410 --> 00:01:54.540
you could view this as
your angular acceleration.

00:01:54.540 --> 00:01:56.710
So this is the angular acceleration

00:01:56.710 --> 00:01:59.120
as a function of applied torque.

00:01:59.120 --> 00:02:02.180
So I could write angular
acceleration right over there,

00:02:02.180 --> 00:02:05.010
or you could say your
average angular acceleration

00:02:05.950 --> 00:02:06.930
over that time period.

00:02:06.930 --> 00:02:10.010
From when you start to time, t.

00:02:10.010 --> 00:02:11.880
So what do we think would happen?

00:02:11.880 --> 00:02:14.390
Well, if you have small masses here,

00:02:14.390 --> 00:02:16.580
it would create a small torque.

00:02:16.580 --> 00:02:18.520
But if you have considerable friction,

00:02:18.520 --> 00:02:19.910
which they talk about right over here.

00:02:19.910 --> 00:02:22.950
There is considerable friction
between the axle and pulley.

00:02:22.950 --> 00:02:25.010
So for small masses right over here,

00:02:25.010 --> 00:02:28.080
the torque will likely
not be strong enough

00:02:28.080 --> 00:02:29.900
to overcome the friction.

00:02:29.900 --> 00:02:31.250
And in that situation,

00:02:31.250 --> 00:02:33.950
you would not have any
angular acceleration.

00:02:33.950 --> 00:02:35.980
Everything would just stay still.

00:02:35.980 --> 00:02:37.000
But then at some point,

00:02:37.000 --> 00:02:39.980
once your mass gets large
enough, right over here,

00:02:39.980 --> 00:02:41.710
then your torque would be large enough

00:02:41.710 --> 00:02:46.270
to overcome the friction in
between the axle and the pulley.

00:02:46.270 --> 00:02:49.260
So which of these graphs describe that?

00:02:49.260 --> 00:02:50.683
Or which of these plots?

00:02:51.860 --> 00:02:53.520
Well, this first one,

00:02:53.520 --> 00:02:56.790
describes a situation when
you have a low applied torque,

00:02:56.790 --> 00:02:58.560
you have a high angular acceleration.

00:02:58.560 --> 00:02:59.730
That does not make sense,

00:02:59.730 --> 00:03:01.480
so we should rule that out.

00:03:01.480 --> 00:03:04.340
This one kinda shows angular acceleration

00:03:04.340 --> 00:03:06.450
is indifferent to applied torque.

00:03:06.450 --> 00:03:07.680
And that doesn't make sense.

00:03:07.680 --> 00:03:09.340
The more applied torque you have,

00:03:09.340 --> 00:03:11.000
the larger masses here,

00:03:11.000 --> 00:03:12.990
well then you should have
more angular acceleration.

00:03:12.990 --> 00:03:14.700
So that doesn't make sense.

00:03:14.700 --> 00:03:17.550
Now, both of these show an upward trend

00:03:17.550 --> 00:03:20.010
where as you have more applied torque,

00:03:20.010 --> 00:03:22.130
you have more angular acceleration.

00:03:22.130 --> 00:03:23.780
So which of these would you pick?

00:03:24.730 --> 00:03:27.360
Well, this one here, on the right,

00:03:27.360 --> 00:03:29.160
this kind of paints a picture

00:03:29.160 --> 00:03:31.370
that as soon as you
have any applied torque,

00:03:31.370 --> 00:03:33.610
you start having angular acceleration.

00:03:33.610 --> 00:03:35.530
And if this was a frictionless system

00:03:35.530 --> 00:03:37.320
or if there was very little friction here,

00:03:37.320 --> 00:03:39.020
maybe this would make sense.

00:03:39.020 --> 00:03:40.040
But as we talked about,

00:03:40.040 --> 00:03:41.280
for low masses,

00:03:41.280 --> 00:03:43.040
which should have a low weight,

00:03:43.040 --> 00:03:45.410
so it would generate low applied torque,

00:03:45.410 --> 00:03:47.530
it wouldn't be able to overcome friction.

00:03:47.530 --> 00:03:49.390
And so you should not have

00:03:49.390 --> 00:03:52.470
some angular acceleration for low torques,

00:03:52.470 --> 00:03:54.170
but then as the torque gets larger,

00:03:54.170 --> 00:03:55.410
you will overcome the friction

00:03:55.410 --> 00:03:58.640
and then start to accelerate the system.

00:03:58.640 --> 00:04:00.793
And so I would pick B.


Bank balance sheets and fractional reserve banking

WEBVTT
Kind: captions
Language: en

00:00:00.170 --> 00:00:01.730
- [Narrator] In this
video we're gonna talk

00:00:01.730 --> 00:00:05.200
about balance sheets, and in
particular, balance sheets

00:00:05.200 --> 00:00:09.400
for banks in a fractional
reserve lending system.

00:00:09.400 --> 00:00:11.840
Now, it's not just banks
that have balance sheets.

00:00:11.840 --> 00:00:13.700
All corporations have a balance sheet.

00:00:13.700 --> 00:00:16.620
You can even have your own
individual balance sheet.

00:00:16.620 --> 00:00:19.880
That is a snap shot of,
what do you have of value

00:00:19.880 --> 00:00:22.690
and what do you owe to other people?

00:00:22.690 --> 00:00:25.120
Now the things that you have of value,

00:00:25.120 --> 00:00:26.180
if we're talking about a bank,

00:00:26.180 --> 00:00:28.680
the things that the bank has of value,

00:00:28.680 --> 00:00:31.320
those are called assets.

00:00:31.320 --> 00:00:34.050
And that could be cash that
the bank has in it's vaults.

00:00:34.050 --> 00:00:36.490
It might be property that the bank owns.

00:00:36.490 --> 00:00:38.540
And the things that the bank owes

00:00:38.540 --> 00:00:41.713
to other people are liabilities.

00:00:42.826 --> 00:00:45.310
Liabilities.

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And, this might be money that
the bank owes to someone else,

00:00:49.260 --> 00:00:52.040
some future obligation.

00:00:52.040 --> 00:00:54.733
Now there's this other notion of equity.

00:00:55.800 --> 00:00:57.670
And when we're talking
about a corporation,

00:00:57.670 --> 00:01:01.170
like a bank, equity is what's left over.

00:01:01.170 --> 00:01:04.180
If you take your assets and
you subtract your liabilities

00:01:04.180 --> 00:01:06.480
equity is, you could
view it as the net worth.

00:01:06.480 --> 00:01:09.530
How much net value is
owned by the shareholders.

00:01:09.530 --> 00:01:12.550
And to make this tangible
you can look at an analogy

00:01:12.550 --> 00:01:14.260
to your everyday life.

00:01:14.260 --> 00:01:16.450
If you're thinking about your
own personal balance sheet,

00:01:16.450 --> 00:01:20.090
let's say the only asset
you owned was a car

00:01:20.090 --> 00:01:22.477
that was worth $10,000.

00:01:22.477 --> 00:01:24.680
$10,000 car.

00:01:24.680 --> 00:01:26.340
So that is your asset.

00:01:26.340 --> 00:01:29.140
And let's say that you
only have one liability.

00:01:29.140 --> 00:01:31.610
You had to borrow $8,000 in order to buy

00:01:31.610 --> 00:01:33.380
that $10,000 car.

00:01:33.380 --> 00:01:36.440
So this is something that
you owed to other folks,

00:01:36.440 --> 00:01:39.110
so that is your liability.

00:01:39.110 --> 00:01:40.300
What would be your equity here?

00:01:40.300 --> 00:01:41.960
What would be your net worth?

00:01:41.960 --> 00:01:44.480
Well, if you own something worth $10,000,

00:01:44.480 --> 00:01:46.230
if all of your assets are 10 thousand,

00:01:46.230 --> 00:01:49.118
but you owe eight thousand,
what you have left over

00:01:49.118 --> 00:01:52.700
that's going to be $2,000.

00:01:52.700 --> 00:01:56.310
And so this would be your
equity, in every language,

00:01:56.310 --> 00:01:58.280
as often times the net worth.

00:01:58.280 --> 00:02:01.710
And so assets minus
liabilities is equal to equity.

00:02:01.710 --> 00:02:04.880
Or, if you add liabilities
to both sides of that

00:02:04.880 --> 00:02:07.740
you could say that assets are equal

00:02:07.740 --> 00:02:12.740
to liabilities plus, plus equity.

00:02:13.850 --> 00:02:16.100
And so when you see many balance sheets

00:02:16.100 --> 00:02:19.520
it's typical to see it in two columns.

00:02:19.520 --> 00:02:21.990
On the left hand side you have assets,

00:02:21.990 --> 00:02:26.600
and on the right hand side you
have liabilities plus equity.

00:02:26.600 --> 00:02:29.990
And these two things should
add up to the same amount.

00:02:29.990 --> 00:02:32.720
So whatever assets are,
liabilities plus equity

00:02:32.720 --> 00:02:34.710
should add up to that same amount.

00:02:34.710 --> 00:02:39.670
So now let's use this framework
to start ourselves a bank.

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And so, let's say we immediately
go and buy a building

00:02:42.180 --> 00:02:44.200
and some equipment
worth a million dollars.

00:02:44.200 --> 00:02:46.160
Just to even have a place to run the bank.

00:02:46.160 --> 00:02:50.260
And so, we immediately
have assets of building,

00:02:50.260 --> 00:02:55.260
building plus equipment, plus
equipment, of $1,000,000.

00:02:59.460 --> 00:03:02.120
Now I just said, whatever
our total assets are,

00:03:02.120 --> 00:03:04.560
that would be our
liabilities plus our equity.

00:03:04.560 --> 00:03:09.160
So in this situation what
are our liabilities so far?

00:03:09.160 --> 00:03:11.110
Remember, the balance
sheet gives us a snap shot

00:03:11.110 --> 00:03:12.750
at any moment in time.

00:03:12.750 --> 00:03:14.740
Well, so far I don't
owe anything to anyone.

00:03:14.740 --> 00:03:16.470
I'll just assume that I
had that million dollars,

00:03:16.470 --> 00:03:19.310
I didn't have to borrow from
anyone to get that million.

00:03:19.310 --> 00:03:22.430
So I have zero dollars in liabilities.

00:03:22.430 --> 00:03:23.960
And so what would the equity be?

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Pause the video and think about that.

00:03:27.120 --> 00:03:30.190
Well, liabilities plus equity
needs to be $1,000,000.

00:03:30.190 --> 00:03:34.530
If liabilities is zero, then
our equity is $1,000,000.

00:03:34.530 --> 00:03:36.450
So if I'm the owner of
the bank, this tell me

00:03:36.450 --> 00:03:39.817
that the value of what
I own is $1,000,000.

00:03:40.783 --> 00:03:43.270
But as we know, banks don't
exist just to be a building.

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They take deposits from people

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and then they make loans to people.

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So let's say someone's
walkin' down the street

00:03:48.930 --> 00:03:51.060
and they see our bank and say,
Hey, that looks like a good

00:03:51.060 --> 00:03:52.440
place to deposit their money.

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It looks like a safe place,

00:03:53.680 --> 00:03:55.150
maybe they'll get some interest on it.

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And so they come and they give

00:03:57.550 --> 00:03:59.620
a million dollar cash deposit.

00:03:59.620 --> 00:04:02.400
They have a suitcase with a
million dollars of cash in it.

00:04:02.400 --> 00:04:05.483
So how would that be reflected
on this balance sheet?

00:04:06.360 --> 00:04:10.670
Well, it would actually get
categorized as reserves.

00:04:10.670 --> 00:04:13.730
Reserves you can view as
the federal reserve notes,

00:04:13.730 --> 00:04:16.590
the cash, that it has on hand.

00:04:16.590 --> 00:04:18.430
It could be money that's in it's vault.

00:04:18.430 --> 00:04:21.390
It could be an account that
it has with the central bank,

00:04:21.390 --> 00:04:23.110
although that gets a little
bit more sophisticated.

00:04:23.110 --> 00:04:26.480
But you can visualize it as it's cash.

00:04:26.480 --> 00:04:28.320
One way to simplify it.

00:04:28.320 --> 00:04:31.500
And so in this situation your reserves

00:04:31.500 --> 00:04:34.440
are now going to be $1,000,000.

00:04:34.440 --> 00:04:35.290
Where did that come from?

00:04:35.290 --> 00:04:38.350
It came from that suitcase of
cash that that person gave.

00:04:38.350 --> 00:04:40.670
Now, what happens on the right hand side

00:04:40.670 --> 00:04:41.980
of this balance sheet?

00:04:41.980 --> 00:04:43.500
They didn't just give us the money.

00:04:43.500 --> 00:04:46.340
At some future point in
time they might withdraw

00:04:46.340 --> 00:04:48.410
some or all of that money.

00:04:48.410 --> 00:04:49.960
Our liabilities now,

00:04:49.960 --> 00:04:52.910
so we now have a demand deposit.

00:04:52.910 --> 00:04:54.350
Let me write it this way.

00:04:54.350 --> 00:04:59.350
Demand deposit for $1,000,000.

00:05:00.770 --> 00:05:02.610
And this is a good time
to pause this video

00:05:02.610 --> 00:05:03.940
and really understand this

00:05:03.940 --> 00:05:06.840
because this is essential
for understanding banks,

00:05:06.840 --> 00:05:09.180
is that yes, we got that cash,

00:05:09.180 --> 00:05:10.820
but it's offset by a liability.

00:05:10.820 --> 00:05:12.320
Because at some point in the future

00:05:12.320 --> 00:05:14.810
we have to give that million dollars back

00:05:14.810 --> 00:05:16.830
to that person who made that deposit.

00:05:16.830 --> 00:05:19.630
And they can come on demand.

00:05:19.630 --> 00:05:21.257
Now you might be saying,
"All right this is all nice,

00:05:21.257 --> 00:05:23.760
"but how am I, as a bank,
going to make money?"

00:05:23.760 --> 00:05:25.840
And the main way that banks make money

00:05:25.840 --> 00:05:28.140
is by making loans.

00:05:28.140 --> 00:05:30.240
But how do they loan out
the money if all of this

00:05:30.240 --> 00:05:32.380
is on demand deposit?

00:05:32.380 --> 00:05:34.400
Well, in a fractional reserve system

00:05:34.400 --> 00:05:36.820
you don't have to keep all
of your demand deposits

00:05:36.820 --> 00:05:38.420
on hand as reserves.

00:05:38.420 --> 00:05:40.750
You can actually lend
out a good chunk of it.

00:05:40.750 --> 00:05:44.820
And it's dictated by what the
required reserve ratios are.

00:05:44.820 --> 00:05:46.360
So let's say in the country we're in,

00:05:46.360 --> 00:05:50.200
or the jurisdiction we're in,
the required reserve ratio,

00:05:50.200 --> 00:05:55.200
so required reserve ratio,
is 10% of demand deposits.

00:06:02.010 --> 00:06:05.510
That means that we can, whatever
our demand deposits are,

00:06:05.510 --> 00:06:07.690
we have to keep 10% of that in reserves,

00:06:07.690 --> 00:06:11.110
and then the excess
reserves we can loan out.

00:06:11.110 --> 00:06:12.941
And so I can now group my reserves.

00:06:12.941 --> 00:06:15.770
Instead of saying a million
dollars of just total reserves,

00:06:15.770 --> 00:06:19.760
I could sub-categorize
it as required reserves,

00:06:19.760 --> 00:06:24.760
required reserves, and excess reserves.

00:06:27.190 --> 00:06:29.800
Now, what do you think are
going to be the required

00:06:29.800 --> 00:06:32.180
and excess reserves in this scenario?

00:06:32.180 --> 00:06:34.440
Pause this video and figure it out.

00:06:34.440 --> 00:06:36.470
Well, required and access
are going to add up

00:06:36.470 --> 00:06:38.780
to my total million dollars of reserves.

00:06:38.780 --> 00:06:42.230
The required reserves are
10% of my demand deposits.

00:06:42.230 --> 00:06:44.550
So I have to keep 10%
of this million dollars,

00:06:44.550 --> 00:06:47.840
which is $100,000, and then the rest

00:06:47.840 --> 00:06:51.450
is excess reserves of $900,000.

00:06:51.450 --> 00:06:55.010
And so this model is based on
the idea that statistically,

00:06:55.010 --> 00:06:56.550
especially if these demand deposits

00:06:56.550 --> 00:06:58.180
are coming from many different people,

00:06:58.180 --> 00:07:01.950
it's unlikely that more
than 10% will be withdrawn

00:07:01.950 --> 00:07:03.010
at any moment in time.

00:07:03.010 --> 00:07:04.430
And we talk about runs on banks

00:07:04.430 --> 00:07:06.010
where this tends to break down.

00:07:06.010 --> 00:07:08.800
And so banks will then
lend out this other 90%

00:07:08.800 --> 00:07:12.160
of the demand deposits, or up
to 90% of the demand deposits,

00:07:12.160 --> 00:07:14.250
and they lend them out
to people they think

00:07:14.250 --> 00:07:16.280
are likely to pay the money with interest.

00:07:16.280 --> 00:07:18.290
And that interest is how
banks make the money.

00:07:18.290 --> 00:07:20.700
And so this bank could say,
Hey I have 900 thousand

00:07:20.700 --> 00:07:23.880
of excess reserves, which
I can use to make loans

00:07:23.880 --> 00:07:24.840
to other people.

00:07:24.840 --> 00:07:26.690
And so let's say a
bunch of people come by,

00:07:26.690 --> 00:07:28.550
and they have some good ideas,
and we think that they're

00:07:28.550 --> 00:07:29.860
going to pay us back.

00:07:29.860 --> 00:07:32.710
So what we do is, we can
take those excess reserves,

00:07:32.710 --> 00:07:36.370
up to $900,000 of them,
and instead of having

00:07:36.370 --> 00:07:39.290
this excess reserves we
can put 'em out as loans.

00:07:39.290 --> 00:07:44.250
So we can make loans of $900,000.

00:07:44.250 --> 00:07:45.960
Now one thing that might
be counterintuitive

00:07:45.960 --> 00:07:48.570
to some of you, is say,
wait I'm used to a loan

00:07:48.570 --> 00:07:50.040
being something like a liability.

00:07:50.040 --> 00:07:51.687
Here we just said we owed people money,

00:07:51.687 --> 00:07:53.620
and so that was an
obligation to other people.

00:07:53.620 --> 00:07:55.390
Why is it an asset here?

00:07:55.390 --> 00:07:57.530
Well, it depends if you are the lender

00:07:57.530 --> 00:07:58.980
or your the person borrowing the money.

00:07:58.980 --> 00:08:01.400
Here, as the bank, we're the lender.

00:08:01.400 --> 00:08:03.910
The loan is an asset
because someone is going

00:08:03.910 --> 00:08:06.300
to pay us money back in the future.

00:08:06.300 --> 00:08:08.010
This has value.

00:08:08.010 --> 00:08:10.960
If we owed money to someone else,

00:08:10.960 --> 00:08:13.490
well then that would be a liability.

00:08:13.490 --> 00:08:15.420
And so, now notice here, the whole time

00:08:15.420 --> 00:08:18.160
that we were doing this,
the assets were equal

00:08:18.160 --> 00:08:20.150
to the liabilities plus equity.

00:08:20.150 --> 00:08:22.700
Assets right now are $2,000,000.

00:08:22.700 --> 00:08:25.200
One million, plus 100
thousand, plus 900 thousand.

00:08:25.200 --> 00:08:28.090
And our liabilities plus
equity are $2,000,000.

00:08:28.090 --> 00:08:31.973
One million in liabilities,
one million in equity.


Bank balance sheet free response question

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:01.881
- [Instructor] The following is

00:00:01.881 --> 00:00:04.410
the balance sheet of First Superior Bank.

00:00:04.410 --> 00:00:06.420
And so it's here on the asset side,

00:00:06.420 --> 00:00:11.330
it has $200 of reserves
and $1,800 of loans

00:00:11.330 --> 00:00:14.390
so its total assets are $2,000.

00:00:14.390 --> 00:00:15.790
And then that should be the same

00:00:15.790 --> 00:00:17.590
as its liabilities and equity.

00:00:17.590 --> 00:00:21.530
And we see here that it has
$2,000 in demand deposits,

00:00:21.530 --> 00:00:23.950
that's a liability 'cause
people can come to the bank

00:00:23.950 --> 00:00:25.510
and say, "Hey, I want that money."

00:00:25.510 --> 00:00:27.110
These are checkable deposits so they

00:00:27.110 --> 00:00:29.460
could even try to withdraw that money.

00:00:29.460 --> 00:00:32.390
And since all of that
$2,000 is on the liability

00:00:32.390 --> 00:00:34.580
side there's no equity right over here.

00:00:34.580 --> 00:00:37.020
These two should sum up to $2,000,

00:00:37.020 --> 00:00:39.363
the same as you have on your assets side.

00:00:39.363 --> 00:00:44.363
Assume that the required
reserve ratio is 10%.

00:00:44.540 --> 00:00:46.930
And just as a review, that's
the percent of deposits

00:00:46.930 --> 00:00:49.286
that the bank needs to keep as reserves

00:00:49.286 --> 00:00:52.766
and we can see that it's at
that reserve ratio right now.

00:00:52.766 --> 00:00:57.546
It has $2,000 in deposits
and so it needs to keep 10%

00:00:57.546 --> 00:01:01.720
as reserves, 10% of $2,000 is $200,

00:01:01.720 --> 00:01:04.845
so it's at its minimum reserves already.

00:01:04.845 --> 00:01:09.300
Part A says what is the
dollar value of new loans

00:01:09.300 --> 00:01:13.610
that First Superior
Bank can make, explain.

00:01:13.610 --> 00:01:14.710
Pause this video and see if you

00:01:14.710 --> 00:01:16.483
can figure that out yourself.

00:01:17.470 --> 00:01:18.960
Well as I just mentioned,

00:01:18.960 --> 00:01:22.410
this bank is already at
its minimum reserves.

00:01:22.410 --> 00:01:25.220
It's already loaned out
as much as it could.

00:01:25.220 --> 00:01:29.120
If you get $2,000 in deposits
and you have a 10% reserve

00:01:29.120 --> 00:01:32.330
ratio, that means you can
loan out 90% of that $2,000

00:01:32.330 --> 00:01:35.604
and it has already loaned
out 90% of the $2,000.

00:01:35.604 --> 00:01:38.420
So what is the dollar value of new loans

00:01:38.420 --> 00:01:40.290
that First Superior Bank can make?

00:01:40.290 --> 00:01:41.820
Well it's zero dollars.

00:01:41.820 --> 00:01:46.357
Zero dollars because
already at minimum reserves.

00:01:50.650 --> 00:01:54.173
Minimum reserves.

00:01:55.760 --> 00:01:58.120
Alright, let's do part B.

00:01:58.120 --> 00:02:02.770
Mr. Smith deposits $100 of
cash in a demand deposit

00:02:02.770 --> 00:02:04.881
account in First Superior Bank.

00:02:04.881 --> 00:02:08.217
Calculate the maximum amount of new loans

00:02:08.217 --> 00:02:11.710
that First Superior Bank can now make.

00:02:11.710 --> 00:02:12.720
So pause this video again

00:02:12.720 --> 00:02:14.890
and see if you can figure that out.

00:02:14.890 --> 00:02:17.190
Well there's a couple of ways
you could think about it.

00:02:17.190 --> 00:02:20.080
Before Mr. Smith makes
that deposit we already saw

00:02:20.080 --> 00:02:23.229
in part A that First Superior
Bank can't make any new loans

00:02:23.229 --> 00:02:27.743
and so now if it gets $100
of cash in a demand deposit

00:02:27.743 --> 00:02:30.379
account because the reserve ratio is 10%,

00:02:30.379 --> 00:02:33.811
the bank needs to keep 10%
of that deposit as reserves

00:02:33.811 --> 00:02:37.080
and it can loan out the other 90%.

00:02:37.080 --> 00:02:40.250
So you could just say well it
could loan out 90% of that new

00:02:40.250 --> 00:02:44.737
deposit and so it would be 90% times $100

00:02:45.998 --> 00:02:50.998
which is equal to $90 of new loans.

00:02:51.105 --> 00:02:53.370
Another way that you
could think about it is

00:02:53.370 --> 00:02:57.040
after Mr. Smith's deposit,
the demand deposits right over

00:02:57.040 --> 00:03:01.790
here goes to $2,100 and
that increase in liabilities

00:03:01.790 --> 00:03:03.510
is offset by an increase in assets.

00:03:03.510 --> 00:03:06.460
It just got $100 of cash so the reserves

00:03:06.460 --> 00:03:09.998
go from $200 to $300.

00:03:09.998 --> 00:03:13.939
Now at this point, First
Superior Bank is clearly above

00:03:13.939 --> 00:03:16.660
its minimum reserve requirements.

00:03:16.660 --> 00:03:20.740
300 over 2,100 is more than 10%.

00:03:20.740 --> 00:03:22.760
So it can loan out some money.

00:03:22.760 --> 00:03:24.270
How much could it loan out?

00:03:24.270 --> 00:03:27.878
Well it has to keep 10%
of this $2,100 as reserves

00:03:27.878 --> 00:03:32.740
so it needs to keep $210 right over here

00:03:32.740 --> 00:03:35.480
and so that other $90 it could loan out.

00:03:35.480 --> 00:03:36.830
And so this could be $1,890

00:03:39.000 --> 00:03:42.193
and so it can make $90 in new loans.

00:03:43.380 --> 00:03:46.480
As a result of Mr.
Smith's $100 cash deposit,

00:03:46.480 --> 00:03:50.142
calculate the maximum change over time

00:03:50.142 --> 00:03:53.556
in each of the following
in the banking system.

00:03:53.556 --> 00:03:57.400
So part one is loans, so
what's the maximum change

00:03:57.400 --> 00:04:00.040
over time in the banking system of loans?

00:04:00.040 --> 00:04:01.990
Pause this video, try to figure it out.

00:04:02.830 --> 00:04:06.529
Well as we just said Mr.
Smith by depositing $100.

00:04:06.529 --> 00:04:10.155
First Superior Bank can make $90 of loans

00:04:10.155 --> 00:04:12.842
so that would be $90 of
new loans but then whoever

00:04:12.842 --> 00:04:15.850
they loan that money to,
they could then deposit

00:04:15.850 --> 00:04:18.960
that in a bank and then
that bank could loan out

00:04:18.960 --> 00:04:23.960
90% of that, so then it
would be plus 0.9 times 90.

00:04:25.130 --> 00:04:30.130
Now this bank that got 0.9
times $90 which is $81,

00:04:30.670 --> 00:04:33.390
it can then loan out 90% of that.

00:04:33.390 --> 00:04:37.203
So it's gonna be plus 0.9 times this,

00:04:37.203 --> 00:04:40.657
so 0.9 squared times 90
and you just keep going

00:04:40.657 --> 00:04:42.845
like this and then this
gives us essentially

00:04:42.845 --> 00:04:45.100
the equation for the multiplier

00:04:45.100 --> 00:04:46.670
which you might have memorized.

00:04:46.670 --> 00:04:47.740
This is just going to be equal

00:04:47.740 --> 00:04:52.740
to 90 times one over one
minus you could say this 0.9

00:04:56.215 --> 00:04:58.208
or you could say this is the same thing as

00:04:58.208 --> 00:05:03.130
90 times one over the
reserve ratio which is 0.1

00:05:03.130 --> 00:05:06.990
and one over .1 is 10 so this
is going to be equal to $900.

00:05:10.180 --> 00:05:11.670
Now what is going to be the maximum

00:05:11.670 --> 00:05:14.440
change over time in demand deposits?

00:05:14.440 --> 00:05:16.673
Pause this video again and
try to figure that out.

00:05:17.640 --> 00:05:21.010
When Mr. Smith first
deposits that $100 in cash

00:05:21.010 --> 00:05:25.050
at First Superior Bank,
that creates a $100 increase

00:05:25.050 --> 00:05:28.880
demand deposit here on First
Superior Bank's balance sheet.

00:05:28.880 --> 00:05:31.110
So that is, let me just write it this way.

00:05:31.110 --> 00:05:34.700
So you have 100 but then we
already said that First Superior

00:05:34.700 --> 00:05:38.690
Bank could loan out as much as
90% of that and whoever they

00:05:38.690 --> 00:05:41.410
loan that to, they could
put all of that into a bank

00:05:41.410 --> 00:05:45.400
as a demand deposit and so
that's gonna be plus 0.9

00:05:45.400 --> 00:05:49.684
times 100 and then that bank
could then loan out .9 times

00:05:49.684 --> 00:05:51.903
this and then whoever gets that loan,

00:05:51.903 --> 00:05:55.020
that could be, they could
deposit that in a bank

00:05:55.020 --> 00:05:59.110
creating a demand deposit
so plus 0.9 squared

00:05:59.110 --> 00:06:02.540
times 100 and it keeps
going on and on and on.

00:06:02.540 --> 00:06:05.310
And you could just view this
as this is going to be equal

00:06:05.310 --> 00:06:10.310
to 100 times one over one
minus 0.9 which is the same

00:06:11.527 --> 00:06:16.049
thing as 100 times one
over the reserve ratio

00:06:16.049 --> 00:06:18.658
and so this is just gonna be 100 times 10

00:06:18.658 --> 00:06:23.482
which is going to be $1,000.

00:06:23.482 --> 00:06:28.244
Part D, as a result of Mr.
Smith's $100 cash deposit,

00:06:28.244 --> 00:06:33.244
calculate the maximum change
over time in the money supply.

00:06:34.070 --> 00:06:36.670
So pause this video again
and try to think about it.

00:06:37.530 --> 00:06:39.277
So you might be tempted to say,

00:06:39.277 --> 00:06:41.367
"Hey, maybe that's just
going to be the same thing

00:06:41.367 --> 00:06:45.000
"as the total maximum
change in demand deposits."

00:06:45.000 --> 00:06:46.957
But we have to be very careful.

00:06:46.957 --> 00:06:51.957
That first $100 already existed
as cash in the money supply,

00:06:52.486 --> 00:06:55.820
the maximum change is everything else.

00:06:55.820 --> 00:06:57.955
It's this part right over here.

00:06:57.955 --> 00:07:01.364
Mr. Smith's deposits $100 in
cash, that's not new money

00:07:01.364 --> 00:07:04.787
but then the bank can
loan out 90% of that.

00:07:04.787 --> 00:07:09.040
So that's $90 in new money
that someone could use

00:07:09.040 --> 00:07:11.237
to then deposit in another
bank and so that other

00:07:11.237 --> 00:07:14.646
bank could then loan out 90% of that 90

00:07:14.646 --> 00:07:16.760
and we've seen this drill before,

00:07:16.760 --> 00:07:18.340
that was the exact same calculation

00:07:18.340 --> 00:07:19.970
for part one right over here.

00:07:19.970 --> 00:07:22.930
This is just going to be
equal to 90 times one over

00:07:22.930 --> 00:07:26.590
the reserve ratio which is equal to $900.

00:07:28.890 --> 00:07:32.820
Part E, provide one reason
why the actual change

00:07:32.820 --> 00:07:35.844
in money supply can be
smaller than the maximum

00:07:35.844 --> 00:07:39.060
change you identified in part D.

00:07:39.060 --> 00:07:42.120
Pause this video and see
if you can figure that out.

00:07:42.120 --> 00:07:44.000
Well there's actually two good reasons why

00:07:44.000 --> 00:07:46.190
the actual change in money supply might

00:07:46.190 --> 00:07:48.470
be smaller than what we just calculated.

00:07:48.470 --> 00:07:50.206
One reason is if banks decide to keep

00:07:50.206 --> 00:07:52.379
more than the minimum reserves.

00:07:52.379 --> 00:07:57.379
So banks, banks might
keep more than minimum,

00:08:04.290 --> 00:08:08.270
minimum reserves.

00:08:08.270 --> 00:08:11.150
So you could imagine even
though by law according

00:08:11.150 --> 00:08:13.724
to this world, they have to
keep a reserve ratio of 10%,

00:08:13.724 --> 00:08:16.869
if on average the banks
decide to keep a reserve ratio

00:08:16.869 --> 00:08:21.140
of 20% then every time
they would loan out 80%

00:08:21.140 --> 00:08:22.930
of whatever they get in deposits

00:08:22.930 --> 00:08:25.520
and so you would have a lower number here.

00:08:25.520 --> 00:08:28.510
There's another world where
the people who get the loans

00:08:28.510 --> 00:08:31.770
don't deposit all of that
into the banking system.

00:08:31.770 --> 00:08:33.890
So if you wanted a second
reason, we've already answered

00:08:33.890 --> 00:08:36.150
their question, we've given
one reason but a second

00:08:36.150 --> 00:08:38.420
reason is people might not deposit all

00:08:38.420 --> 00:08:40.866
of their funds in the banking system.

00:08:40.866 --> 00:08:42.973
So I will leave you there.


Introducing Khan Academy Kids

WEBVTT
Kind: captions
Language: en

00:00:00.230 --> 00:00:01.063
- Hi everyone,

00:00:01.063 --> 00:00:03.600
Sal here with my three-year-old son Azad,

00:00:03.600 --> 00:00:05.420
and we're excited to announce the launch of

00:00:05.420 --> 00:00:07.060
Khan Academy Kids!

00:00:07.060 --> 00:00:09.020
Which is designed to take students like Azad,

00:00:09.020 --> 00:00:11.880
ages two to five, to become lifelong learners.

00:00:11.980 --> 00:00:14.200
- [Reya] Hi friend, welcome to my room!

00:00:14.200 --> 00:00:15.880
- [Sal] Kids love this program because it's really fun.

00:00:15.880 --> 00:00:16.580
- [Peck] Awesome!

00:00:16.580 --> 00:00:19.340
- [Sal] Friendly characters like our narrator Kodi Bear –

00:00:19.340 --> 00:00:21.360
- [Kodi] Welcome to Khan Academy Kids!

00:00:21.370 --> 00:00:23.850
– interact with them throughout the learning experience.

00:00:23.850 --> 00:00:26.530
And we've combined tasks like math and reading

00:00:26.530 --> 00:00:28.860
with activities like drawing and storytelling.

00:00:28.860 --> 00:00:30.240
We really think parents and teachers

00:00:30.240 --> 00:00:31.900
are going to love Khan Academy Kids

00:00:31.900 --> 00:00:33.810
because it's aligned with important standards

00:00:33.810 --> 00:00:34.940
like the Common Core

00:00:34.940 --> 00:00:37.360
and the Head Start Learning Outcomes Framework.

00:00:37.370 --> 00:00:38.920
There are thousands of activities

00:00:38.920 --> 00:00:40.530
and original books in the program.

00:00:40.530 --> 00:00:42.160
Khan Academy Kids also promotes

00:00:42.160 --> 00:00:43.630
social emotional development

00:00:43.630 --> 00:00:45.800
through topics like empathy and sharing.

00:00:45.800 --> 00:00:47.640
And what's more, we've partnered with

00:00:47.640 --> 00:00:48.920
Super Simple Songs,

00:00:48.920 --> 00:00:50.420
Bellwether Media,

00:00:50.420 --> 00:00:53.100
and National Geographic Young Explorer Magazine

00:00:53.100 --> 00:00:55.960
to provide learning materials kids will love.

00:00:55.970 --> 00:00:58.010
Khan Academy Kids is a free mobile app

00:00:58.010 --> 00:00:59.510
available in the App Store,

00:00:59.510 --> 00:01:01.840
and in beta for Google Play and Amazon.

00:01:01.840 --> 00:01:04.180
Download today and let the joyful learning begin!

00:01:06.220 --> 00:01:07.400
[Sal] Hi everyone, Sal here.

00:01:07.400 --> 00:01:08.160
I'm here to announce–

00:01:08.160 --> 00:01:09.940
[Azad] Blaaaaaah. [Giggles]

00:01:09.940 --> 00:01:11.660
[Sal] Shh.

00:01:11.980 --> 00:01:14.140
Hi everyone, I'm here with my son Azad,

00:01:14.140 --> 00:01:16.320
who's three years old, to announce the launch of

00:01:16.320 --> 00:01:18.573
Khan Academy Kids, which is designed

00:01:18.573 --> 00:01:21.730
to get students aged two to five

00:01:21.730 --> 00:01:23.640
ready to become learners for life.

00:01:24.460 --> 00:01:26.200
- [Azad] Download today.

00:01:26.200 --> 00:01:27.960
- [Sal] Yeah, say it loud and funny!

00:01:27.960 --> 00:01:29.820
- [Azad] Download today!

00:01:30.020 --> 00:01:32.880
- [Sal] Download today and let the joyful learning begin!

00:01:39.880 --> 00:01:40.920
How do they do this in movies?


Standard deviation of residuals or root mean square deviation (RMSD)

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.810
- [Tutor] So we are interested in studying

00:00:01.810 --> 00:00:04.040
the relationship between the amount

00:00:04.040 --> 00:00:07.021
that folks study for a test
and their score on a test,

00:00:07.021 --> 00:00:10.560
where the score is between zero and six

00:00:10.560 --> 00:00:13.370
and so what we're going to
do is go look at the people

00:00:13.370 --> 00:00:16.700
who took the test, we're
going to plot for each person

00:00:16.700 --> 00:00:18.760
the amount that they
studied and their score,

00:00:18.760 --> 00:00:20.560
so for example, this data point is someone

00:00:20.560 --> 00:00:23.380
who studied an hour and
they got a one on the test

00:00:23.380 --> 00:00:25.470
and then we're going to
fit a regression line

00:00:25.470 --> 00:00:29.500
and this blue regression line
is the actual regression line

00:00:29.500 --> 00:00:31.240
for these four data points

00:00:31.240 --> 00:00:34.184
and here is the equation
for that regression line.

00:00:34.184 --> 00:00:37.067
Now there's a couple of
things to keep in mind,

00:00:37.067 --> 00:00:39.149
normally when you're doing
this type of analysis,

00:00:39.149 --> 00:00:42.480
you would do it with far
more than four data points,

00:00:42.480 --> 00:00:44.950
the reason why I kept this to four

00:00:44.950 --> 00:00:46.980
is because we are actually
going to calculate

00:00:46.980 --> 00:00:51.680
how good a fit this
regression line is by hand

00:00:51.680 --> 00:00:53.870
and typically you would not do it by hand,

00:00:53.870 --> 00:00:55.331
we have computers for that.

00:00:55.331 --> 00:00:57.480
Now the way that we're going to measure

00:00:57.480 --> 00:00:59.910
how good a fit this regression line is

00:00:59.910 --> 00:01:03.420
to the data has several names,

00:01:03.420 --> 00:01:08.420
one name is the standard
deviation of the residuals,

00:01:08.640 --> 00:01:11.960
another name is the root
mean square deviation,

00:01:11.960 --> 00:01:14.086
sometimes abbreviated RMSD,

00:01:14.086 --> 00:01:17.154
sometimes it's called
root mean square error,

00:01:17.154 --> 00:01:19.858
so what we're going to
do is is for every point,

00:01:19.858 --> 00:01:22.553
we're going to calculate the residual

00:01:22.553 --> 00:01:23.950
and then we're going to square it

00:01:23.950 --> 00:01:25.330
and then we're gonna add up

00:01:25.330 --> 00:01:28.271
the sum of those squared residuals,

00:01:28.271 --> 00:01:30.950
so we're gonna take the sum

00:01:30.950 --> 00:01:33.260
of the residuals,

00:01:33.260 --> 00:01:36.260
residuals squared

00:01:36.260 --> 00:01:38.600
and then we're going to divide that

00:01:38.600 --> 00:01:42.060
by the number of data
points we have minus two

00:01:42.060 --> 00:01:44.100
and we can talk in future videos

00:01:44.100 --> 00:01:45.670
or a more advanced statistics class

00:01:45.670 --> 00:01:48.852
of why you divide by two,
but it's related to the idea

00:01:48.852 --> 00:01:51.324
that what we're calculating
here is a statistic

00:01:51.324 --> 00:01:54.090
and we're trying to
estimate a true parameter

00:01:54.090 --> 00:01:56.380
as best as possible and n minus two

00:01:56.380 --> 00:01:58.110
actually does the trick for us.

00:01:58.110 --> 00:01:59.851
But to calculate the root
mean square deviation,

00:01:59.851 --> 00:02:02.960
we would then take a square root of this

00:02:02.960 --> 00:02:04.750
and some of you might recognize

00:02:04.750 --> 00:02:08.440
strong parallels between
this and how we calculated

00:02:08.440 --> 00:02:11.663
sample standard deviation
early in our statistics career

00:02:11.663 --> 00:02:13.520
and I encourage you to think about it.

00:02:13.520 --> 00:02:15.529
But let's actually calculate it by hand,

00:02:15.529 --> 00:02:17.730
as I mentioned earlier in this video,

00:02:17.730 --> 00:02:20.260
to see how things actually play out.

00:02:20.260 --> 00:02:24.240
So to do that, I'm going to
ourselves a little table here,

00:02:24.240 --> 00:02:28.300
so let's say that is our
x value in that column,

00:02:28.300 --> 00:02:31.233
let's make this our y value,

00:02:32.150 --> 00:02:33.610
let's make this

00:02:33.610 --> 00:02:35.590
y hat, which is going to be equal

00:02:35.590 --> 00:02:39.380
to 2.5x minus two

00:02:39.380 --> 00:02:42.750
and then let's make this
the residual squared,

00:02:42.750 --> 00:02:47.420
which is going to be our y
value minus our y hat value,

00:02:47.420 --> 00:02:51.134
our actual minus our estimate
for that given x squared

00:02:51.134 --> 00:02:53.680
and then we're going to sum them all up,

00:02:53.680 --> 00:02:56.830
divide by n minus two
and take the square root.

00:02:56.830 --> 00:02:58.416
So first let's do this data point,

00:02:58.416 --> 00:03:01.360
so that's the point 1,1,

00:03:01.360 --> 00:03:03.000
1,1,

00:03:03.000 --> 00:03:05.370
now what is the estimate
from our regression line?

00:03:05.370 --> 00:03:07.359
Well, for that x value,
when x is equal to one,

00:03:07.359 --> 00:03:10.013
it's gonna be 2.5 times one minus two,

00:03:10.013 --> 00:03:12.570
so it's gonna be 2.5

00:03:12.570 --> 00:03:14.520
times one minus two,

00:03:14.520 --> 00:03:18.340
which is equal to 0.5

00:03:18.340 --> 00:03:22.530
and so our residual squared
is going to be one minus 0.5,

00:03:22.530 --> 00:03:26.490
one minus 0.5 squared,

00:03:26.490 --> 00:03:29.330
which is equal to, that's
gonna be .5 squared,

00:03:29.330 --> 00:03:32.512
which is gonna be 0.25.

00:03:32.512 --> 00:03:34.740
Alright, let's do the next data point,

00:03:34.740 --> 00:03:36.560
we have this one right over here,

00:03:36.560 --> 00:03:39.430
it is 2,2,

00:03:39.430 --> 00:03:41.810
now our estimate from the regression line

00:03:41.810 --> 00:03:45.760
when x equals two is going to be equal

00:03:45.760 --> 00:03:48.990
to 2.5 times our x value,

00:03:48.990 --> 00:03:51.430
times two minus two,

00:03:51.430 --> 00:03:55.090
which is going to be equal to three

00:03:55.090 --> 00:04:00.090
and so our residual squared is
going to be two minus three,

00:04:00.770 --> 00:04:04.252
two minus three squared,
which is negative one squared,

00:04:04.252 --> 00:04:07.170
which is going to be equal to one,

00:04:07.170 --> 00:04:09.820
then we can go to this point,

00:04:09.820 --> 00:04:12.423
so that's the point 2,3, 2,3,

00:04:13.580 --> 00:04:16.420
now our estimate from our regression line

00:04:16.420 --> 00:04:21.380
is going to be 2.5 times
our x value, times two

00:04:21.380 --> 00:04:26.380
minus two, which is going
to be equal to three

00:04:27.320 --> 00:04:29.101
and so our residual
here is going to be zero

00:04:29.101 --> 00:04:31.994
and you can see that that point
sits on the regression line,

00:04:31.994 --> 00:04:34.640
so it's going to be three minus three,

00:04:34.640 --> 00:04:38.200
three minus three squared,
which is equal to zero

00:04:38.200 --> 00:04:39.750
and then last but not least,

00:04:39.750 --> 00:04:44.250
we have this point right
over here, when x is three,

00:04:44.250 --> 00:04:47.410
our y value, this person
studied three hours

00:04:47.410 --> 00:04:51.560
and they got a six on the
test, so y is equal to six,

00:04:51.560 --> 00:04:54.126
and so our estimate from
the regression line,

00:04:54.126 --> 00:04:56.160
you can see what you would
have expected to get,

00:04:56.160 --> 00:04:59.910
based on the regression line is 2.5

00:04:59.910 --> 00:05:03.580
times our x value, times three minus two

00:05:03.580 --> 00:05:06.750
is equal to 5.5

00:05:06.750 --> 00:05:11.735
and so our residual squared
is six minus 5.5 squared,

00:05:11.735 --> 00:05:13.235
it is 5.5 squared,

00:05:14.261 --> 00:05:16.253
so it's .5 squared, which is 0.25.

00:05:17.790 --> 00:05:19.090
So now the next step,

00:05:19.090 --> 00:05:22.790
let me take the sum of all
of these squared residuals,

00:05:22.790 --> 00:05:26.820
so this is, let me just write
it this way, do it like this,

00:05:26.820 --> 00:05:31.340
so the sum of the residuals,

00:05:31.340 --> 00:05:33.500
residuals squared

00:05:33.500 --> 00:05:35.870
is equal to,

00:05:35.870 --> 00:05:37.413
if I just sum all of this up,

00:05:38.348 --> 00:05:41.847
it's going to be 1.5,

00:05:41.847 --> 00:05:43.405
1.5

00:05:43.405 --> 00:05:46.800
and then if I divide that by n minus two,

00:05:46.800 --> 00:05:48.477
so if I divide by n minus two,

00:05:48.477 --> 00:05:51.150
that's going to be equal
to, I have four data points,

00:05:51.150 --> 00:05:52.469
so I'm gonna divide by four minus two,

00:05:52.469 --> 00:05:54.730
so I'm gonna divide by two

00:05:54.730 --> 00:05:57.540
and then I'm gonna wanna
take the square root of that,

00:05:57.540 --> 00:05:59.730
then I'm gonna take
the square root of that

00:05:59.730 --> 00:06:02.720
and so this is going to get us

00:06:02.720 --> 00:06:05.250
1.5 over two is the same
thing as three-fourths,

00:06:05.250 --> 00:06:06.820
so it's the square root of three-fourths

00:06:06.820 --> 00:06:09.448
or the square root of three over two

00:06:09.448 --> 00:06:12.610
and you could use a calculator
to figure what that is,

00:06:12.610 --> 00:06:14.676
to figure out what that is as a decimal,

00:06:14.676 --> 00:06:16.040
but this gives us a sense

00:06:16.040 --> 00:06:18.700
of how good a fit this regression line is,

00:06:18.700 --> 00:06:20.200
the closer this is to zero,

00:06:20.200 --> 00:06:22.900
the better the fit of the regression line,

00:06:22.900 --> 00:06:24.965
the further away from zero, the worst fit

00:06:24.965 --> 00:06:26.720
and what would be the units

00:06:26.720 --> 00:06:29.070
for the root mean square deviation?

00:06:29.070 --> 00:06:31.300
Well, it would be in terms of

00:06:31.300 --> 00:06:34.340
whatever your units are for your y axis,

00:06:34.340 --> 00:06:36.900
in this case, it would
be the score on the test

00:06:36.900 --> 00:06:40.094
and that's one of the other
values of this calculation,

00:06:40.094 --> 00:06:42.520
of taking the square root of the sum

00:06:42.520 --> 00:06:46.810
of the squares of the residuals
dividing by n minus two.

00:06:46.810 --> 00:06:50.410
So big picture, this square
root of three over two

00:06:50.410 --> 00:06:53.060
can be viewed as the approximate size

00:06:53.060 --> 00:06:55.810
of a typical or average prediction error

00:06:55.810 --> 00:06:57.360
between these points

00:06:57.360 --> 00:07:00.013
and what the regression
line would have predicted

00:07:00.013 --> 00:07:03.320
or you could view it
as the approximate size

00:07:03.320 --> 00:07:05.753
of a typical or average residual.


Give the gift of life-changing education

WEBVTT
Kind: captions
Language: en

00:00:00.340 --> 00:00:02.310
(soft music)

00:00:02.310 --> 00:00:04.920
I belong to a very poor family.

00:00:04.920 --> 00:00:08.570
At home - it's just a room - we live there.

00:00:08.570 --> 00:00:13.010
My father is a car mechanic
and my mother is a housewife.

00:00:13.010 --> 00:00:17.300
As a child, I used to fear mathematics.

00:00:17.300 --> 00:00:20.410
Whenever I used to have
one question or one doubt,

00:00:20.410 --> 00:00:23.438
I used to feel very shy asking teachers

00:00:23.438 --> 00:00:27.993
because I heard that my friends
will laugh at me and stuff.

00:00:28.960 --> 00:00:32.530
But then, Khan Academy videos are best

00:00:32.530 --> 00:00:35.680
because I can play whenever I want,

00:00:35.680 --> 00:00:37.273
how many times I wanted -

00:00:38.157 --> 00:00:42.490
- so because of videos, my
concepts were crystal clear.

00:00:42.490 --> 00:00:46.660
Now, I am able to teach
my fellow students.

00:00:46.660 --> 00:00:51.560
I am in love with mathematics
just because of Khan Academy

00:00:51.560 --> 00:00:53.850
and of course Khan Academy is best

00:00:53.850 --> 00:00:55.713
because it is free of cost.

00:00:56.580 --> 00:01:00.730
I also want to prove the world
and of course my village,

00:01:00.730 --> 00:01:04.440
that having a girl child is not a problem.

00:01:04.440 --> 00:01:06.330
Basically, I want to change the world

00:01:06.330 --> 00:01:09.400
and education is the best way to do it.

00:01:09.400 --> 00:01:11.240
So, I started with myself,

00:01:11.240 --> 00:01:13.893
then of course my village
and then the whole world.


Root mean square deviation (RMSD)

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:01.810
- [Instructor] So we are
interested in studying

00:00:01.810 --> 00:00:05.030
the relationship between
the amount that folks study

00:00:05.030 --> 00:00:07.370
for a test and their score on a test,

00:00:07.370 --> 00:00:10.560
where the score is between zero and six.

00:00:10.560 --> 00:00:12.860
And so what we're going to do is go look

00:00:12.860 --> 00:00:14.600
at the people who took the test,

00:00:14.600 --> 00:00:16.700
we're going to plot for each person

00:00:16.700 --> 00:00:18.760
the amount that they
studied and their score,

00:00:18.760 --> 00:00:20.560
so for example, this data point is someone

00:00:20.560 --> 00:00:23.380
who studied an hour and
they got a one on the test,

00:00:23.380 --> 00:00:25.470
and then we're going to
fit a regression line,

00:00:25.470 --> 00:00:29.500
and this blue regression line
is the actual regression line

00:00:29.500 --> 00:00:31.240
for these four data points.

00:00:31.240 --> 00:00:34.670
And here is the equation
for that regression line.

00:00:34.670 --> 00:00:37.400
Now there's a couple of
things to keep in mind.

00:00:37.400 --> 00:00:39.670
Normally when you're doing
this type of analysis,

00:00:39.670 --> 00:00:42.820
you would do it with far
more than four data points.

00:00:42.820 --> 00:00:44.950
The reason why I kept this to four

00:00:44.950 --> 00:00:46.980
is because we are actually
going to calculate

00:00:46.980 --> 00:00:51.680
how good a fit this
regression line is by hand.

00:00:51.680 --> 00:00:53.870
And typically, you
would not do it by hand.

00:00:53.870 --> 00:00:55.890
We have computers for that.

00:00:55.890 --> 00:00:57.480
Now, the way that we're going to measure

00:00:57.480 --> 00:01:01.270
how good a fit this
regression line is to the data

00:01:01.270 --> 00:01:03.420
has several names.

00:01:03.420 --> 00:01:08.420
One name is the standard
deviation of the residuals.

00:01:08.640 --> 00:01:11.960
Another name is the root
mean square deviation,

00:01:11.960 --> 00:01:14.550
sometimes abbreviated RMSD.

00:01:14.550 --> 00:01:17.790
Sometimes it's called
root mean square error.

00:01:17.790 --> 00:01:20.400
So what we're going to do
is, is for every point,

00:01:20.400 --> 00:01:22.920
we're going to calculate the residual,

00:01:22.920 --> 00:01:24.380
and then we're going to square it,

00:01:24.380 --> 00:01:25.330
and then we're gonna add up

00:01:25.330 --> 00:01:28.730
the sum of those squared residuals.

00:01:28.730 --> 00:01:32.757
So we're gonna take the
sum of the residuals,

00:01:32.757 --> 00:01:37.757
residuals squared, and then
we're going to divide that

00:01:38.600 --> 00:01:42.060
by the number of data
points we have minus two.

00:01:42.060 --> 00:01:44.100
And we can talk in future videos,

00:01:44.100 --> 00:01:45.670
or a more advanced statistics class,

00:01:45.670 --> 00:01:48.280
of why you divide by two.

00:01:48.280 --> 00:01:50.850
It's related to the idea that
what we're calculating here

00:01:50.850 --> 00:01:53.090
is a statistic, and
we're trying to estimate

00:01:53.090 --> 00:01:55.110
a true parameter as best as possible,

00:01:55.110 --> 00:01:58.110
and n minus two actually
does the trick for us.

00:01:58.110 --> 00:02:00.200
But to calculate the root
mean squared deviation,

00:02:00.200 --> 00:02:02.960
we would then take a square root of this.

00:02:02.960 --> 00:02:05.920
And some of you might
recognize strong parallels

00:02:05.920 --> 00:02:10.460
between this and how we calculated
sample standard deviation

00:02:10.460 --> 00:02:12.140
early in our statistics career,

00:02:12.140 --> 00:02:13.950
and I encourage you to think about it.

00:02:13.950 --> 00:02:15.910
But let's actually calculate it by hand,

00:02:15.910 --> 00:02:17.730
as I mentioned earlier in this video,

00:02:17.730 --> 00:02:20.260
to see how things actually play out.

00:02:20.260 --> 00:02:22.890
So to do that, I'm going to give ourselves

00:02:22.890 --> 00:02:26.780
a little table here, so let's
say that is our x value,

00:02:26.780 --> 00:02:31.534
in that column, let's
make this our y value,

00:02:31.534 --> 00:02:35.590
let's make this y-hat
which is going to be equal

00:02:35.590 --> 00:02:40.590
to 2.5x minus two, and
then let's make this

00:02:41.280 --> 00:02:44.730
the residual squared, which is going to be

00:02:44.730 --> 00:02:47.420
our y value minus our y-hat value,

00:02:47.420 --> 00:02:51.740
our actual minus our estimate
for that given x, squared,

00:02:51.740 --> 00:02:53.680
and then we're going to sum them all up,

00:02:53.680 --> 00:02:56.830
divide by n minus two,
and take the square root.

00:02:56.830 --> 00:02:58.890
So first, let's do this data point.

00:02:58.890 --> 00:03:03.000
So that's the point, one
comma one, one comma one.

00:03:03.000 --> 00:03:05.670
Now what is the estimate
from our regression line?

00:03:05.670 --> 00:03:07.770
Well, for that x value,
when x is equal to one,

00:03:07.770 --> 00:03:10.410
it's gonna be 2.5 times one minus two.

00:03:10.410 --> 00:03:14.520
So it's gonna be 2.5 times one minus two,

00:03:14.520 --> 00:03:19.520
which is equal to 0.5, and
so our residual squared

00:03:20.610 --> 00:03:25.610
is going to be one minus
0.5, one minus 0.5, squared,

00:03:26.490 --> 00:03:29.860
which is equal to, that's
gonna be .5 squared,

00:03:29.860 --> 00:03:32.830
which is going to be 0.25.

00:03:32.830 --> 00:03:34.740
Alright, let's do the next data point.

00:03:34.740 --> 00:03:36.560
We have this one right over here.

00:03:36.560 --> 00:03:39.430
It is two comma two.

00:03:39.430 --> 00:03:41.810
Now our estimate from the regression line,

00:03:41.810 --> 00:03:46.793
when x equals two, is
going to be equal to 2.5

00:03:48.140 --> 00:03:51.430
times our x value times two, minus two,

00:03:51.430 --> 00:03:55.090
which is going to be equal to three.

00:03:55.090 --> 00:03:58.030
And so, our residual
squared is going to be

00:03:58.030 --> 00:04:03.030
two minus three, two minus three, squared,

00:04:03.480 --> 00:04:04.760
which is negative one squared,

00:04:04.760 --> 00:04:07.170
which is going to be equal to one.

00:04:07.170 --> 00:04:09.820
Then we can go to this point.

00:04:09.820 --> 00:04:13.580
So that's the point two
comma three, two comma three.

00:04:13.580 --> 00:04:16.420
Now, our estimate from our regression line

00:04:16.420 --> 00:04:21.380
is going to be 2.5 times
our x value, times two,

00:04:21.380 --> 00:04:26.380
minus two, which is going
to be equal to three,

00:04:27.320 --> 00:04:29.570
and so our residual here
is going to be zero,

00:04:29.570 --> 00:04:32.420
and you can see that that point
sits on the regression line.

00:04:32.420 --> 00:04:36.250
So it's going to be three
minus three, three minus three,

00:04:36.250 --> 00:04:38.540
squared, which is equal to zero,

00:04:38.540 --> 00:04:39.750
and then last but not least,

00:04:39.750 --> 00:04:41.800
we have this point right over here.

00:04:41.800 --> 00:04:45.500
When x is three, our y value,

00:04:45.500 --> 00:04:47.410
this person studied three hours,

00:04:47.410 --> 00:04:51.560
and they got a six on the
test, so y is equal to six,

00:04:51.560 --> 00:04:54.570
and so our estimate from
the regression line,

00:04:54.570 --> 00:04:56.160
you could say what you
would've expected to get

00:04:56.160 --> 00:05:00.990
based on the regression line,
is 2.5 times our x value,

00:05:00.990 --> 00:05:05.600
times three, minus two, is equal to 5.5.

00:05:06.750 --> 00:05:11.540
And so our residual squared
is six minus 5.5 squared,

00:05:11.540 --> 00:05:16.253
minus 5.5, squared, so it's
.5 squared, which is 0.25.

00:05:17.790 --> 00:05:20.460
So now, the next step, let me take the sum

00:05:20.460 --> 00:05:23.850
of all of these square
residuals, so this is,

00:05:23.850 --> 00:05:25.130
let me just write it this way.

00:05:25.130 --> 00:05:26.820
Should we just do it like this?

00:05:26.820 --> 00:05:31.820
So the sum of the residuals,
residuals squared,

00:05:33.500 --> 00:05:37.260
is equal to, if I just sum all of this up,

00:05:37.260 --> 00:05:41.500
it's going to be point,
it's going to be 1.5.

00:05:41.500 --> 00:05:46.500
1.5, and then if I divide
that by n minus two,

00:05:46.800 --> 00:05:49.840
so if I divide by n minus two,
that's going to be equal to,

00:05:49.840 --> 00:05:51.930
I have four data points,
so I'm gonna divide

00:05:51.930 --> 00:05:54.730
by four minus two, so
I'm gonna divide by two,

00:05:54.730 --> 00:05:57.930
and then I'm gonna want to
take the square root of that.

00:05:57.930 --> 00:05:59.730
Then I'm gonna want to take
the square root of that,

00:05:59.730 --> 00:06:03.660
and so this is going
to get us, 1.5 over two

00:06:03.660 --> 00:06:06.810
is the same thing as 3/4, so
it's the square root of 3/4,

00:06:06.810 --> 00:06:10.030
or the square root of three over two.

00:06:10.030 --> 00:06:12.620
And you could use a calculator
to figure what that is,

00:06:12.620 --> 00:06:15.000
to figure out what that is as a decimal,

00:06:15.000 --> 00:06:17.260
but this gives us a
sense of how good a fit

00:06:17.260 --> 00:06:18.700
this regression line is.

00:06:18.700 --> 00:06:21.520
The closer this is to
zero, the better the fit

00:06:21.520 --> 00:06:22.900
of the regression line.

00:06:22.900 --> 00:06:25.490
The further away from zero, the worse fit.

00:06:25.490 --> 00:06:26.710
And what would be the units

00:06:26.710 --> 00:06:29.080
for the root mean square deviation?

00:06:29.080 --> 00:06:32.730
Well, it would be in terms
of whatever your units are

00:06:32.730 --> 00:06:34.950
for your y-axis; in this case,

00:06:34.950 --> 00:06:36.910
it would be the score on the test.

00:06:36.910 --> 00:06:40.410
And that's one of the other
values of this calculation,

00:06:40.410 --> 00:06:43.200
of taking the square root
of the sum of the squares

00:06:43.200 --> 00:06:46.543
of the residuals dividing by n minus two.


Introduction to residuals and least-squares regression

WEBVTT
Kind: captions
Language: en

00:00:00.120 --> 00:00:01.830
- [Instructor] Let's say
we're trying to understand

00:00:01.830 --> 00:00:06.070
the relationship between
people's height and their weight.

00:00:06.070 --> 00:00:09.380
So what we do is we go
to 10 different people,

00:00:09.380 --> 00:00:10.980
and we measure each of their heights

00:00:10.980 --> 00:00:12.200
and each of their weights.

00:00:12.200 --> 00:00:14.510
And so on this scatter plot here,

00:00:14.510 --> 00:00:16.830
each dot represents a person.

00:00:16.830 --> 00:00:20.590
So, for example, this dot
over here represents a person

00:00:20.590 --> 00:00:24.350
whose height was 60
inches, or five feet tall.

00:00:24.350 --> 00:00:26.870
So that's the point 60 comma,

00:00:26.870 --> 00:00:28.280
and whose weight,

00:00:28.280 --> 00:00:33.280
which we have on the
y-axis, was 125 pounds.

00:00:33.660 --> 00:00:35.650
And so when you look at this scatter plot,

00:00:35.650 --> 00:00:38.430
your eyes naturally see
some type of a trend.

00:00:38.430 --> 00:00:41.380
It seems like, generally speaking,

00:00:41.380 --> 00:00:44.740
as height increases,
weight increases as well.

00:00:44.740 --> 00:00:46.200
But I said generally speaking.

00:00:46.200 --> 00:00:47.610
You definitely have circumstances

00:00:47.610 --> 00:00:50.970
where there are taller
people who might weigh less.

00:00:50.970 --> 00:00:55.690
But an interesting question
is can we try to fit a line

00:00:55.690 --> 00:00:57.220
to this data?

00:00:57.220 --> 00:00:59.820
And this idea of trying to fit a line

00:00:59.820 --> 00:01:03.070
as closely as possible to as
many of the points as possible

00:01:03.070 --> 00:01:05.700
is known as linear,

00:01:05.700 --> 00:01:07.493
linear regression.

00:01:08.600 --> 00:01:12.180
Now, the most common technique
is to try to fit a line

00:01:12.180 --> 00:01:16.590
that minimizes the squared
distance to each of those points,

00:01:16.590 --> 00:01:19.200
and we're gonna talk more
about that in future videos.

00:01:19.200 --> 00:01:21.930
But for now, we want to get
an intuitive feel for that.

00:01:21.930 --> 00:01:23.340
So if you were to just eyeball it

00:01:23.340 --> 00:01:24.900
and look at a line like that,

00:01:24.900 --> 00:01:27.680
you wouldn't think that it would
be a particularly good fit.

00:01:27.680 --> 00:01:31.090
It looks like most of the
data sits above the line.

00:01:31.090 --> 00:01:35.310
Similarly, something like this
also doesn't look that great.

00:01:35.310 --> 00:01:38.980
Here most of our data points
are sitting below the line.

00:01:38.980 --> 00:01:42.630
But something like this
actually looks very good.

00:01:42.630 --> 00:01:44.990
It looks like it's getting
as close as possible

00:01:44.990 --> 00:01:47.200
to as many of the points as possible.

00:01:47.200 --> 00:01:51.030
It seems like it's describing
this general trend.

00:01:51.030 --> 00:01:54.490
And so this is the actual regression line.

00:01:54.490 --> 00:01:58.100
And the equation here, we would write as,

00:01:58.100 --> 00:02:00.450
we'd write y with a little hat over it.

00:02:00.450 --> 00:02:03.740
And that means that we're
trying to estimate a y

00:02:03.740 --> 00:02:04.840
for a given x.

00:02:04.840 --> 00:02:07.730
It's not always going to be
the actual y for a given x.

00:02:07.730 --> 00:02:10.610
Because, as we see, sometimes the points

00:02:10.610 --> 00:02:12.070
aren't sitting on the line.

00:02:12.070 --> 00:02:14.440
But we say y hat is equal to,

00:02:14.440 --> 00:02:18.810
and our y-intercept, for this
particular regression line,

00:02:18.810 --> 00:02:21.450
it is negative 140

00:02:21.450 --> 00:02:24.710
plus the slope 14 over three

00:02:24.710 --> 00:02:26.440
times x.

00:02:26.440 --> 00:02:29.210
Now, as we can see, for
most of these points,

00:02:29.210 --> 00:02:31.780
given the x-value of those points,

00:02:31.780 --> 00:02:34.780
the estimate that our
regression line gives

00:02:34.780 --> 00:02:37.430
is different than the actual value.

00:02:37.430 --> 00:02:39.600
And that difference between the actual

00:02:39.600 --> 00:02:41.570
and the estimate from the regression line

00:02:41.570 --> 00:02:44.620
is known as the residual.

00:02:44.620 --> 00:02:45.453
So let me write that down.

00:02:45.453 --> 00:02:48.863
So, for example, the
residual at that point,

00:02:48.863 --> 00:02:50.500
residual

00:02:50.500 --> 00:02:54.240
at that point is going to
be equal to, for a given x,

00:02:54.240 --> 00:02:56.360
the actual y-value

00:02:56.360 --> 00:02:58.920
minus the estimated y-value

00:02:58.920 --> 00:03:01.800
from the regression line for that same x.

00:03:01.800 --> 00:03:03.370
Or another way to think about it is,

00:03:03.370 --> 00:03:05.690
for that x-value, when x is equal to 60,

00:03:05.690 --> 00:03:07.620
we're talking about the
residual just at that point,

00:03:07.620 --> 00:03:09.940
it's going to be the actual y-value

00:03:09.940 --> 00:03:12.940
minus our estimate of what the y-value is

00:03:12.940 --> 00:03:16.100
from this regression
line for that x-value.

00:03:16.100 --> 00:03:17.070
So pause this video,

00:03:17.070 --> 00:03:19.080
and see if you can
calculate this residual.

00:03:19.080 --> 00:03:22.620
And you could visually imagine it as being

00:03:22.620 --> 00:03:24.423
this right over here.

00:03:25.460 --> 00:03:28.290
Well, to actually calculate the residual,

00:03:28.290 --> 00:03:32.060
you would take our actual
value, which is 125,

00:03:32.060 --> 00:03:33.730
for that x-value.

00:03:33.730 --> 00:03:36.140
Remember, we're calculating
the residual for a point.

00:03:36.140 --> 00:03:38.410
So it's the actual y there minus,

00:03:38.410 --> 00:03:41.480
what would be the estimated
y there for that x-value?

00:03:41.480 --> 00:03:43.270
Well, we could just go to this equation

00:03:43.270 --> 00:03:47.660
and say what would y hat
be when x is equal to 60?

00:03:47.660 --> 00:03:49.250
Well, it's going to be equal to,

00:03:49.250 --> 00:03:54.250
let's see, we have negative 140
plus 14 over three times 60.

00:03:55.480 --> 00:03:59.120
Let's see, 60 divided by three is 20.

00:03:59.120 --> 00:04:00.880
20 times

00:04:00.880 --> 00:04:02.950
14 is

00:04:02.950 --> 00:04:04.770
280.

00:04:04.770 --> 00:04:07.603
And so all of this is going to be 140.

00:04:09.020 --> 00:04:11.600
And so our residual, for this point,

00:04:11.600 --> 00:04:16.600
is going to be 125 minus
140, which is negative 15.

00:04:16.660 --> 00:04:19.290
And residuals indeed can be negative.

00:04:19.290 --> 00:04:23.580
If your residual is negative,
it means, for that x-value,

00:04:23.580 --> 00:04:27.950
your data point, your actual
y-value, is below the estimate.

00:04:27.950 --> 00:04:29.960
If we were to calculate the residual here

00:04:29.960 --> 00:04:32.070
or if we were to calculate
the residual here,

00:04:32.070 --> 00:04:35.130
our actual for that x-value
is above our estimate,

00:04:35.130 --> 00:04:37.720
so we would get positive residuals.

00:04:37.720 --> 00:04:40.400
And as you will see later
in your statistics career,

00:04:40.400 --> 00:04:42.770
the way that we calculate
these regression lines

00:04:42.770 --> 00:04:47.770
is all about minimizing the
square of these residuals.


Mirror equation example problems

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:02.820
- [Instructor] Mirror equation
problems can be intimidating

00:00:02.820 --> 00:00:04.240
when you first deal with them.

00:00:04.240 --> 00:00:05.360
And that's not because

00:00:05.360 --> 00:00:07.180
the mirror equation's all that difficult.

00:00:07.180 --> 00:00:08.013
It's kinda easy.

00:00:08.013 --> 00:00:10.280
It's just a few fractions added together.

00:00:10.280 --> 00:00:12.270
The place where it gets tricky is deciding

00:00:12.270 --> 00:00:14.420
whether these should be
positive or negative.

00:00:14.420 --> 00:00:17.240
So there's a bunch of positive
and negative sign decisions

00:00:17.240 --> 00:00:18.370
that you have to make.

00:00:18.370 --> 00:00:20.820
And if you make even one
of those incorrectly,

00:00:20.820 --> 00:00:22.230
you can get the wrong answer.

00:00:22.230 --> 00:00:24.450
So let's do a few mirror
equation problems,

00:00:24.450 --> 00:00:26.460
and you can see how the signs work.

00:00:26.460 --> 00:00:27.640
Now I have to warn you,

00:00:27.640 --> 00:00:29.930
everyone has their own sign convention.

00:00:29.930 --> 00:00:32.500
There's a lot of
different sign conventions

00:00:32.500 --> 00:00:33.900
when you deal with optics.

00:00:33.900 --> 00:00:35.760
The one I'm gonna use is the one

00:00:35.760 --> 00:00:38.640
that I feel like most
textbooks are using these days,

00:00:38.640 --> 00:00:40.300
and it's the one where anything

00:00:40.300 --> 00:00:42.640
on the front side of the mirror,

00:00:42.640 --> 00:00:46.240
so, by front side, your eyes
should be over here somewhere.

00:00:46.240 --> 00:00:48.350
So let's say your eyes
are over here, right?

00:00:48.350 --> 00:00:50.970
You're looking at some object.

00:00:50.970 --> 00:00:53.890
Maybe it's an arrow or a crayon
right here, a blue crayon,

00:00:53.890 --> 00:00:56.390
you're holding in front
of this mirror right here.

00:00:56.390 --> 00:00:58.440
So this is the mirror right here.

00:00:58.440 --> 00:01:01.750
And the convention I'm using
is that anything on this side

00:01:01.750 --> 00:01:04.610
of the mirror is gonna
be counted as positive.

00:01:04.610 --> 00:01:07.870
So if it's a focal point
on that side, positive.

00:01:07.870 --> 00:01:11.440
If it's an object distance
on this side, it's positive.

00:01:11.440 --> 00:01:13.710
And if the image distance
comes out positive,

00:01:13.710 --> 00:01:16.540
you'll know that it's also on
this front side of the mirror.

00:01:16.540 --> 00:01:18.190
Anything that comes out negative,

00:01:18.190 --> 00:01:19.910
so if we get a negative image distance

00:01:19.910 --> 00:01:21.800
after we do our calculation,

00:01:21.800 --> 00:01:24.000
we'll know that that thing
is behind the mirror.

00:01:24.000 --> 00:01:25.070
That kinda makes sense,

00:01:25.070 --> 00:01:28.490
negative like behind,
positive like in front.

00:01:28.490 --> 00:01:32.700
So this side over here is in
front of the mirror, positive.

00:01:32.700 --> 00:01:35.350
And back here would be behind the mirror,

00:01:35.350 --> 00:01:36.580
and that would be negative.

00:01:36.580 --> 00:01:38.130
So I've got some numbers in here already.

00:01:38.130 --> 00:01:39.430
Let's just solve this one.

00:01:39.430 --> 00:01:40.420
So what do we do?

00:01:40.420 --> 00:01:42.250
We're gonna use this mirror equation.

00:01:42.250 --> 00:01:44.580
We're gonna say that one
over the focal length,

00:01:44.580 --> 00:01:46.230
and already we have to decide

00:01:46.230 --> 00:01:47.830
on a positive or negative sign.

00:01:47.830 --> 00:01:49.860
So this mirror, the way
it's shaped right here,

00:01:49.860 --> 00:01:52.750
based on how we're looking
at it, is a concave mirror.

00:01:52.750 --> 00:01:54.970
And with the sign
conventions we just discussed

00:01:54.970 --> 00:01:57.180
and the signs I'm using in this formula,

00:01:57.180 --> 00:02:00.700
concave mirrors always have
a positive focal length.

00:02:00.700 --> 00:02:02.510
So, in other words, since this focal point

00:02:02.510 --> 00:02:05.760
is four centimeters from
the center of the mirror,

00:02:05.760 --> 00:02:07.490
I'm gonna have to plug in the focal length

00:02:07.490 --> 00:02:10.110
as positive four centimeters.

00:02:10.110 --> 00:02:11.740
Notice I'm not converting.

00:02:11.740 --> 00:02:12.573
That's okay.

00:02:12.573 --> 00:02:13.990
If you leave everything in centimeters,

00:02:13.990 --> 00:02:16.040
you'll just get an answer in centimeters.

00:02:16.040 --> 00:02:17.360
So it's okay, you don't have to convert

00:02:17.360 --> 00:02:19.370
as long as everything's in the same units.

00:02:19.370 --> 00:02:22.920
We'll set that equal to one
over the object distance.

00:02:22.920 --> 00:02:26.160
Just one other warning,
sometimes instead of d o,

00:02:26.160 --> 00:02:29.320
you'll see this as s
o for object distance,

00:02:29.320 --> 00:02:32.603
or you might see d i as
s i for image distance.

00:02:32.603 --> 00:02:33.653
It's the same thing.

00:02:33.653 --> 00:02:35.460
It's just a different letter.

00:02:35.460 --> 00:02:36.910
So the object distance, again,

00:02:36.910 --> 00:02:39.720
is on this side in front of the mirror.

00:02:39.720 --> 00:02:42.640
So it's gonna be positive 12 centimeters

00:02:42.640 --> 00:02:44.370
since it's located 12 centimeters

00:02:44.370 --> 00:02:46.190
from the center of the mirror.

00:02:46.190 --> 00:02:48.620
So this is also gonna be
positive 12 centimeters.

00:02:48.620 --> 00:02:52.530
And now we're gonna add to that
one over the image distance.

00:02:52.530 --> 00:02:53.960
The image distance, we don't know.

00:02:53.960 --> 00:02:55.360
I haven't drawn the image on here.

00:02:55.360 --> 00:02:56.420
It's gonna be a surprise.

00:02:56.420 --> 00:02:57.530
We don't know what this is gonna be,

00:02:57.530 --> 00:02:58.890
but we can solve for it.

00:02:58.890 --> 00:03:00.420
So we can solve for image distance.

00:03:00.420 --> 00:03:03.170
I'll subtract one over 12 from both sides,

00:03:03.170 --> 00:03:05.350
which will give us one
over four centimeters

00:03:05.350 --> 00:03:07.910
minus one over 12 centimeters,

00:03:07.910 --> 00:03:11.700
and that's gonna have to equal
one over the image distance.

00:03:11.700 --> 00:03:15.810
So 1/4 you could rewrite as 3/12.

00:03:15.810 --> 00:03:20.680
So 3/12 minus 1/12 is just gonna be 2/12,

00:03:20.680 --> 00:03:23.240
and that's gonna equal one
over the image distance.

00:03:23.240 --> 00:03:25.083
But 2/12 is just 1/6,

00:03:26.000 --> 00:03:29.540
so one over the image
distance is just gonna be 1/6.

00:03:29.540 --> 00:03:31.620
But that's what one over
the image distance is.

00:03:31.620 --> 00:03:34.050
Sometimes people forget
to flip this at the end.

00:03:34.050 --> 00:03:36.780
We don't want one over the image distance.

00:03:36.780 --> 00:03:38.220
We want the image distance.

00:03:38.220 --> 00:03:40.920
So finally, you take one over each side.

00:03:40.920 --> 00:03:43.300
And we solve, and we get
that the image distance

00:03:43.300 --> 00:03:45.160
is gonna be six centimeters.

00:03:45.160 --> 00:03:46.250
And it came out positive.

00:03:46.250 --> 00:03:47.390
That's important.

00:03:47.390 --> 00:03:50.340
This came out to be
positive six centimeters.

00:03:50.340 --> 00:03:51.900
So where's our image gonna be?

00:03:51.900 --> 00:03:54.610
So since this image distance
came out to be positive,

00:03:54.610 --> 00:03:57.080
our image is gonna be
in front of the mirror.

00:03:57.080 --> 00:03:58.070
So it's gonna be over here.

00:03:58.070 --> 00:04:00.790
It's gonna be six
centimeters from the mirror,

00:04:00.790 --> 00:04:02.180
somewhere around here.

00:04:02.180 --> 00:04:05.400
So at this point right here is
where our image is gonna be.

00:04:05.400 --> 00:04:06.870
But we don't know how big it's gonna be

00:04:06.870 --> 00:04:09.250
or whether it's right-side
up or upside down.

00:04:09.250 --> 00:04:11.550
To figure that out, we have
to use a different equation,

00:04:11.550 --> 00:04:14.080
and that equation's called
the magnification equation.

00:04:14.080 --> 00:04:16.660
It says that the magnification is equal to

00:04:16.660 --> 00:04:19.760
the height of the image divided
by the height of the object.

00:04:19.760 --> 00:04:23.670
That's equal to negative
the image distance

00:04:23.670 --> 00:04:26.030
divided by the object distance.

00:04:26.030 --> 00:04:29.100
So it turns out this ratio
of negative image distance

00:04:29.100 --> 00:04:32.740
over object distance is
always equal to the ratio

00:04:32.740 --> 00:04:35.240
of the height of the image
over the height of the object.

00:04:35.240 --> 00:04:36.810
So what's the height
of our image gonna be?

00:04:36.810 --> 00:04:37.880
Let's just solve for it.

00:04:37.880 --> 00:04:40.100
If we solve for the height of our image,

00:04:40.100 --> 00:04:42.280
we get that the height
of the image is gonna be,

00:04:42.280 --> 00:04:44.020
multiply both sides by h o,

00:04:44.020 --> 00:04:46.130
we'll have the negative
sign's already here,

00:04:46.130 --> 00:04:49.370
so negative height of the object times

00:04:49.370 --> 00:04:53.550
this ratio of the image distance
over the object distance.

00:04:53.550 --> 00:04:55.070
And now we can just plug in numbers.

00:04:55.070 --> 00:04:56.000
The height of the object,

00:04:56.000 --> 00:04:58.240
it says it's three
centimeters tall right here.

00:04:58.240 --> 00:04:59.710
So it's three centimeters,

00:04:59.710 --> 00:05:03.940
so negative three centimeters
times the ratio of,

00:05:03.940 --> 00:05:07.840
the image distance was six,
the object distance was 12.

00:05:07.840 --> 00:05:09.010
And so if you solve this,

00:05:09.010 --> 00:05:11.070
you'll get one half of negative three,

00:05:11.070 --> 00:05:14.430
which is negative 1.5 centimeters.

00:05:14.430 --> 00:05:17.870
The negative sign means that
this image got inverted.

00:05:17.870 --> 00:05:18.820
So it got flipped over.

00:05:18.820 --> 00:05:21.530
It's gonna be upside down
compared to what it was before.

00:05:21.530 --> 00:05:23.800
And the 1.5 is how tall it's gonna be.

00:05:23.800 --> 00:05:25.770
So what we end up with is

00:05:25.770 --> 00:05:29.900
an image six centimeters from
the mirror, and it's gonna be,

00:05:29.900 --> 00:05:32.980
have a height half as
tall, so 1.5 centimeters.

00:05:32.980 --> 00:05:34.750
And it's gonna be upside down

00:05:34.750 --> 00:05:36.450
because of this negative sign.

00:05:36.450 --> 00:05:39.360
So it's gonna be 1.5 centimeters
tall and upside down.

00:05:39.360 --> 00:05:42.850
That's what you're gonna see
if you look into this mirror.

00:05:42.850 --> 00:05:43.870
It's like a funhouse mirror.

00:05:43.870 --> 00:05:45.520
It's a weird, curved mirror.

00:05:45.520 --> 00:05:48.080
You'd see an upside-down image right here.

00:05:48.080 --> 00:05:50.260
It might look like you
could reach out and grab it,

00:05:50.260 --> 00:05:51.630
but it's gonna be an optical illusion.

00:05:51.630 --> 00:05:52.630
There's gonna be no object there.

00:05:52.630 --> 00:05:55.410
It's just gonna be the
image of this object here.

00:05:55.410 --> 00:05:58.150
So that's an example
using a concave mirror.

00:05:58.150 --> 00:06:00.510
What would change, what if we did this?

00:06:00.510 --> 00:06:01.990
What if we took our object,

00:06:01.990 --> 00:06:03.820
so say we don't put it here anymore,

00:06:03.820 --> 00:06:05.840
we move it inside here.

00:06:05.840 --> 00:06:08.340
So instead of being at 12 centimeters,

00:06:08.340 --> 00:06:11.300
we moved it to like three centimeters.

00:06:11.300 --> 00:06:12.940
What would we do differently?

00:06:12.940 --> 00:06:14.090
Everything would be the same.

00:06:14.090 --> 00:06:17.190
We just would plug in, instead
of positive 12 down here,

00:06:17.190 --> 00:06:19.120
we'd plug in positive three.

00:06:19.120 --> 00:06:21.530
So the mirror equation
works exactly the same.

00:06:21.530 --> 00:06:24.800
You still plug in whatever
that object distance is.

00:06:24.800 --> 00:06:26.270
You solve for your image distance.

00:06:26.270 --> 00:06:28.640
We're, of course, gonna get
a different image distance.

00:06:28.640 --> 00:06:29.510
But whatever you get,

00:06:29.510 --> 00:06:31.130
that would tell you where the image is.

00:06:31.130 --> 00:06:33.080
And then you would take that, plug it into

00:06:33.080 --> 00:06:35.730
the magnification equation
if you wanted to decide

00:06:35.730 --> 00:06:37.460
how big the image is gonna be

00:06:37.460 --> 00:06:39.850
and whether it's gonna be
right-side up or upside down.

00:06:39.850 --> 00:06:41.220
So these numbers are gonna be different,

00:06:41.220 --> 00:06:44.200
but you would use this
equation the exact same way.

00:06:44.200 --> 00:06:47.610
Now what would happen if instead
of using a concave mirror,

00:06:47.610 --> 00:06:50.170
we used a convex mirror?

00:06:50.170 --> 00:06:52.290
Let's say we used a
mirror shaped like this.

00:06:52.290 --> 00:06:54.610
So imagine our eye, again, is over here,

00:06:54.610 --> 00:06:57.250
looking at this object
inside of the mirror,

00:06:57.250 --> 00:06:59.230
and we're gonna see an
image of the object.

00:06:59.230 --> 00:07:00.480
We're gonna see the object right here,

00:07:00.480 --> 00:07:03.390
but we're also gonna see
the image of the object.

00:07:03.390 --> 00:07:05.370
This mirror, this time instead of concave,

00:07:05.370 --> 00:07:07.350
this is a convex mirror.

00:07:07.350 --> 00:07:09.960
So its focal length is behind the mirror.

00:07:09.960 --> 00:07:12.310
So what do we do now to
figure out where the image is?

00:07:12.310 --> 00:07:14.590
We, again, use the mirror equation.

00:07:14.590 --> 00:07:16.200
We're gonna use the same equation.

00:07:16.200 --> 00:07:18.410
We're gonna have one
over the focal length.

00:07:18.410 --> 00:07:20.240
And, again, I immediately have

00:07:20.240 --> 00:07:22.200
to make a decision on the sign.

00:07:22.200 --> 00:07:23.680
With the convention that I'm using,

00:07:23.680 --> 00:07:25.740
this focal length is behind the mirror.

00:07:25.740 --> 00:07:27.760
So this focal length for a convex mirror

00:07:27.760 --> 00:07:28.690
is gonna be negative.

00:07:28.690 --> 00:07:30.820
So this would be negative
four centimeters.

00:07:30.820 --> 00:07:33.960
And that's gonna equal one
over the object distance.

00:07:33.960 --> 00:07:35.220
Well, again, the object is

00:07:35.220 --> 00:07:37.470
in front of the mirror 12 centimeters.

00:07:37.470 --> 00:07:41.320
So this object distance is gonna
be positive 12 centimeters.

00:07:41.320 --> 00:07:43.380
And then we add to that
the image distance,

00:07:43.380 --> 00:07:44.500
which we don't know.

00:07:44.500 --> 00:07:45.830
This is what we're gonna solve for.

00:07:45.830 --> 00:07:46.810
So this time if we solve,

00:07:46.810 --> 00:07:48.590
we're gonna have one over negative four.

00:07:48.590 --> 00:07:51.330
And, again, we have to
subtract one over 12,

00:07:51.330 --> 00:07:52.820
and that's all gonna have to be equal

00:07:52.820 --> 00:07:55.570
to one over the image distance.

00:07:55.570 --> 00:07:58.900
So now on the left-hand
side, we have negative 1/4,

00:07:58.900 --> 00:08:01.650
but that's the same as negative 3/12.

00:08:01.650 --> 00:08:06.650
So negative 3/12 minus 1/12
is the same as negative 4/12,

00:08:07.760 --> 00:08:10.720
and that's got to equal one
over the image distance.

00:08:10.720 --> 00:08:14.820
But negative 4/12 is the
same as negative 1/3.

00:08:14.820 --> 00:08:16.840
So one over the image distance has

00:08:16.840 --> 00:08:19.330
to be equal to negative a third.

00:08:19.330 --> 00:08:22.530
And if we flip that over, we
get that the image distance

00:08:22.530 --> 00:08:26.090
finally is gonna be
negative three centimeters.

00:08:26.090 --> 00:08:29.820
So, in other words, this
here is negative 1/3.

00:08:29.820 --> 00:08:32.330
So when you flip that over,
you get that the image distance

00:08:32.330 --> 00:08:33.960
is negative three centimeters.

00:08:33.960 --> 00:08:35.490
Where is this image gonna be?

00:08:35.490 --> 00:08:37.050
Well, since it came out negative,

00:08:37.050 --> 00:08:39.210
that means it's behind the mirror

00:08:39.210 --> 00:08:41.290
'cause that's the sign
convention we're using.

00:08:41.290 --> 00:08:43.780
So it's gonna be three
centimeters behind the mirror.

00:08:43.780 --> 00:08:45.890
So it's gonna be like over here about

00:08:45.890 --> 00:08:48.160
at this point right around here somewhere,

00:08:48.160 --> 00:08:50.620
three centimeters behind this mirror.

00:08:50.620 --> 00:08:53.380
And, again, if we want to figure
out how tall it's gonna be,

00:08:53.380 --> 00:08:55.080
whether it's right-side up or upside down,

00:08:55.080 --> 00:08:57.200
we're gonna have to use
the magnification equation.

00:08:57.200 --> 00:08:59.750
And that magnification
equation looked like this.

00:08:59.750 --> 00:09:01.500
It said that the height of the image

00:09:01.500 --> 00:09:03.960
over the height of the
object had to be negative

00:09:03.960 --> 00:09:05.870
image distance over object distance.

00:09:05.870 --> 00:09:07.930
So let's just see what this
right-hand side's gonna be.

00:09:07.930 --> 00:09:09.720
If we just set this equal to,

00:09:09.720 --> 00:09:11.270
it's gonna be negative of,

00:09:11.270 --> 00:09:13.220
negative three is the image distance,

00:09:13.220 --> 00:09:14.780
so I have to plug in the negative three.

00:09:14.780 --> 00:09:16.600
And you keep that negative in there.

00:09:16.600 --> 00:09:19.120
This negative out here comes along always,

00:09:19.120 --> 00:09:20.700
but now we have another negative

00:09:20.700 --> 00:09:23.310
inside of this image
distance of negative three.

00:09:23.310 --> 00:09:26.220
And the object distance,
again, was 12 centimeters.

00:09:26.220 --> 00:09:27.300
So what are we left with?

00:09:27.300 --> 00:09:29.290
We get negative of negative three,

00:09:29.290 --> 00:09:31.470
which is positive three over 12,

00:09:31.470 --> 00:09:33.710
which is positive 1/4.

00:09:33.710 --> 00:09:35.180
And the centimeters cancel.

00:09:35.180 --> 00:09:38.730
So what this ratio tells you,
which is the magnification,

00:09:38.730 --> 00:09:40.940
is that the image is
not gonna be inverted.

00:09:40.940 --> 00:09:43.220
This positive means it's
gonna be right-side up.

00:09:43.220 --> 00:09:47.360
And the 1/4 ratio means
it's gonna be 1/4 as large,

00:09:47.360 --> 00:09:49.740
the image, I should say,
is gonna be 1/4 as large

00:09:49.740 --> 00:09:51.380
as the object is gonna be.

00:09:51.380 --> 00:09:54.240
And the reason is this ratio
is equal to height of the image

00:09:54.240 --> 00:09:55.180
over the height of the object.

00:09:55.180 --> 00:09:56.830
So, in other words, if I can multiply,

00:09:56.830 --> 00:09:58.790
if I want to multiply both sides,

00:09:58.790 --> 00:10:02.420
I can say that height of the
image is equal to positive 1/4

00:10:02.420 --> 00:10:04.770
times the height of the object.

00:10:04.770 --> 00:10:06.240
Well, the height of our object was

00:10:06.240 --> 00:10:07.960
three centimeters in this case.

00:10:07.960 --> 00:10:09.570
But whatever your height of the object is,

00:10:09.570 --> 00:10:13.173
you multiply it by this ratio
of negative d r over d o,

00:10:13.173 --> 00:10:14.700
and you get what the height
of the image is gonna be.

00:10:14.700 --> 00:10:16.720
So we're gonna get 1/4 three.

00:10:16.720 --> 00:10:20.530
So we get positive 3/4 of a centimeter.

00:10:20.530 --> 00:10:21.850
So it's gonna be tiny.

00:10:21.850 --> 00:10:22.683
This is gonna be little,

00:10:22.683 --> 00:10:25.050
little image that's
gonna be right-side up.

00:10:25.050 --> 00:10:26.530
So it's gonna be right here,

00:10:26.530 --> 00:10:27.940
but it's gonna be teeny.

00:10:27.940 --> 00:10:32.040
It's only gonna be like this,
3/4 of a centimeter tall.

00:10:32.040 --> 00:10:34.090
That's what our image is gonna look like.

00:10:34.090 --> 00:10:37.170
So recapping, you can
use the mirror equation

00:10:37.170 --> 00:10:40.060
to figure out where the
images are gonna be located.

00:10:40.060 --> 00:10:41.970
The sign convention we're using is that

00:10:41.970 --> 00:10:43.830
objects, images, and focal lengths

00:10:43.830 --> 00:10:45.910
in front of the mirror
are gonna be positive.

00:10:45.910 --> 00:10:48.350
Anything behind the mirror
is gonna be negative.

00:10:48.350 --> 00:10:50.140
And you could use the
magnification equation

00:10:50.140 --> 00:10:52.120
to figure out how tall
the image is gonna be

00:10:52.120 --> 00:10:56.520
relative to the object by taking
negative the image distance

00:10:56.520 --> 00:10:58.083
over the object distance.