Central limit theorem

WEBVTT
Kind: captions
Language: en

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In this video, I want to
talk about what is easily

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one of the most fundamental and
profound concepts in statistics

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and maybe in all of mathematics.

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And that's the
central limit theorem.

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And what it tells us
is we can start off

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with any distribution that
has a well-defined mean and

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variance-- and if it has
a well-defined variance,

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it has a well-defined
standard deviation.

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And it could be a continuous
distribution or a discrete one.

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I'll draw a discrete one,
just because it's easier

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to imagine, at least for
the purposes of this video.

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So let's say I have a discrete
probability distribution

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function.

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And I want to be
very careful not

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to make it look anything close
to a normal distribution.

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Because I want to show you
the power of the central limit

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theorem.

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So let's say I have
a distribution.

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Let's say it could take
on values 1 through 6.

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1, 2, 3, 4, 5, 6.

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It's some kind of crazy dice.

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It's very likely to get a one.

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Let's say it's
impossible-- well,

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let me make that
a straight line.

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You have a very high
likelihood of getting a 1.

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Let's say it's
impossible to get a 2.

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Let's say it's an OK likelihood
of getting a 3 or a 4.

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Let's say it's
impossible to get a 5.

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And let's say it's very
likely to get a 6 like that.

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So that's my probability
distribution function.

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If I were to draw a
mean-- this the symmetric,

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so maybe the mean would
be something like that.

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The mean would be halfway.

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So that would be my
mean right there.

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The standard
deviation maybe would

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look-- it would be
that far and that

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far above and below the mean.

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But that's my discrete
probability distribution

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function.

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Now what I'm going to do
here, instead of just taking

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samples of this
random variable that's

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described by this probability
distribution function,

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I'm going to take samples of it.

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But I'm going to
average the samples

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and then look at
those samples and see

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the frequency of the
averages that I get.

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And when I say average,
I mean the mean.

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Let me define something.

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Let's say my sample size-- and
I could put any number here.

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But let's say first off we try a
sample size of n is equal to 4.

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And what that means is I'm going
to take four samples from this.

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So let's say the first
time I take four samples--

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so my sample sizes is
four-- let's say I get a 1.

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Let's say I get another 1.

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And let's say I get a 3.

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And I get a 6.

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So that right there is my
first sample of sample size 4.

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I know the terminology
can get confusing.

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Because this is the sample
that's made up of four samples.

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But then when we talk about the
sample mean and the sampling

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distribution of the
sample mean, which we're

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going to talk more and more
about over the next few videos,

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normally the sample refers
to the set of samples

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from your distribution.

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And the sample size tells
you how many you actually

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took from your distribution.

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But the terminology
can be very confusing,

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because you could easily view
one of these as a sample.

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But we're taking four
samples from here.

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We have a sample size of four.

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And what I'm going to do is
I'm going to average them.

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So let's say the mean-- I
want to be very careful when

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I say average.

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The mean of this first
sample of size 4 is what?

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1 plus 1 is 2.

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2 plus 3 is 5.

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5 plus 6 is 11.

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11 divided by 4 is 2.75.

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That is my first sample mean
for my first sample of size 4.

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Let me do another one.

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My second sample of size 4,
let's say that I get a 3, a 4.

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Let's say I get another 3.

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And let's say I get a 1.

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I just didn't happen
to get a 6 that time.

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And notice I can't
get a 2 or a 5.

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It's impossible for
this distribution.

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The chance of getting
a 2 or 5 is 0.

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So I can't have any
2s or 5s over here.

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So for the second
sample of sample size 4,

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my second sample mean is
going to be 3 plus 4 is 7.

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7 plus 3 is 10 plus 1 is 11.

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11 divided by 4,
once again, is 2.75.

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Let me do one more,
because I really

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want to make it clear
what we're doing here.

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So I do one more.

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Actually, we're going
to do a gazillion more.

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But let me just do
one more in detail.

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So let's say my third
sample of sample size 4--

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so I'm going to
literally take 4 samples.

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So my sample is
made up of 4 samples

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from this original
crazy distribution.

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Let's say I get a 1,
a 1, and a 6 and a 6.

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And so my third sample mean
is going to be 1 plus 1 is 2.

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2 plus 6 is 8.

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8 plus 6 is 14.

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14 divided by 4 is 3 and 1/2.

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And as I find each
of these sample

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means-- so for each of my
samples of sample size 4,

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I figure out a mean.

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And as I do each
of them, I'm going

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to plot it on a
frequency distribution.

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And this is all going to
amaze you in a few seconds.

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So I plot this all on a
frequency distribution.

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So I say, OK, on
my first sample,

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my first sample mean was 2.75.

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So I'm plotting the actual
frequency of the sample

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means I get for each sample.

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So 2.75, I got it one time.

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So I'll put a little plot there.

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So that's from that
one right there.

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And the next time,
I also got a 2.75.

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That's a 2.75 there.

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So I got it twice.

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So I'll plot the
frequency right there.

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Then I got a 3 and 1/2.

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So all the possible values,
I could have a three,

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I could have a 3.25, I
could have a 3 and 1/2.

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So then I have the 3 and 1/2,
so I'll plot it right there.

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And what I'm going
to do is I'm going

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to keep taking these samples.

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Maybe I'll take 10,000 of them.

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So I'm going to keep
taking these samples.

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So I go all the way to S 10,000.

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I just do a bunch of these.

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And what it's going to look like
over time is each of these--

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I'm going to make it
a dot, because I'm

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going to have to zoom out.

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So if I look at it like
this, over time-- it still

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has all the values that it
might be able to take on,

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2.75 might be here.

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So this first dot is
going to be-- this one

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right here is going
to be right there.

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And that second one is
going to be right there.

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Then that one at 3.5 is
going to look right there.

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But I'm going to
do it 10,000 times.

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Because I'm going
to have 10,000 dots.

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And let's say as I do it, I'm
going just keep plotting them.

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I'm just going to keep
plotting the frequencies.

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I'm just going to
keep plotting them

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over and over and over again.

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And what you're going
to see is, as I take

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many, many samples
of size 4, I'm

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going to have
something that's going

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to start kind of approximating
a normal distribution.

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So each of these dots represent
an incidence of a sample mean.

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So as I keep adding on
this column right here,

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that means I kept getting
the sample mean 2.75.

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So over time.

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I'm going to have
something that's

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starting to approximate
a normal distribution.

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And that is a neat thing about
the central limit theorem.

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So an orange, that's the
case for n is equal to 4.

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This was a sample size of 4.

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Now, if I did the same thing
with a sample size of maybe

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20-- so in this case, instead
of just taking 4 samples

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from my original crazy
distribution, every sample

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I take 20 instances
of my random variable,

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and I average those 20.

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And then I plot the
sample mean on here.

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So in that case,
I'm going to have

00:07:04.190 --> 00:07:06.810
a distribution that
looks like this.

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And we'll discuss
this in more videos.

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But it turns out if I were
to plot 10,000 of the sample

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means here, I'm going
to have something

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that, two things-- it's going
to even more closely approximate

00:07:17.610 --> 00:07:18.527
a normal distribution.

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And we're going to
see in future videos,

00:07:20.193 --> 00:07:22.250
it's actually going to
have a smaller-- well,

00:07:22.250 --> 00:07:23.240
let me be clear.

00:07:23.240 --> 00:07:25.809
It's going to have
the same mean.

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So that's the mean.

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This is going to
have the same mean.

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So it's going to have a
smaller standard deviation.

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Well, I should plot
these from the bottom

00:07:33.660 --> 00:07:34.590
because you kind of stack it.

00:07:34.590 --> 00:07:37.090
One you get one, then another
instance and another instance.

00:07:37.090 --> 00:07:38.980
But this is going to
more and more approach

00:07:38.980 --> 00:07:40.460
a normal distribution.

00:07:40.460 --> 00:07:44.930
So this is what's super
cool about the central limit

00:07:44.930 --> 00:07:45.540
theorem.

00:07:45.540 --> 00:07:53.080
As your sample size
becomes larger--

00:07:53.080 --> 00:07:55.080
or you could even say as
it approaches infinity.

00:07:55.080 --> 00:07:56.410
But you really don't
have to get that close

00:07:56.410 --> 00:07:58.790
to infinity to really get
close to a normal distribution.

00:07:58.790 --> 00:08:01.380
Even if you have a
sample size of 10 or 20,

00:08:01.380 --> 00:08:04.360
you're already getting very
close to a normal distribution,

00:08:04.360 --> 00:08:06.060
in fact about as
good an approximation

00:08:06.060 --> 00:08:07.850
as we see in our everyday life.

00:08:07.850 --> 00:08:11.740
But what's cool is we can start
with some crazy distribution.

00:08:11.740 --> 00:08:15.121
This has nothing to do
with a normal distribution.

00:08:15.121 --> 00:08:17.370
This was n equals 4, but if
we have a sample size of n

00:08:17.370 --> 00:08:19.910
equals 10 or n
equals 100, and we

00:08:19.910 --> 00:08:22.477
were to take 100 of these,
instead of four here,

00:08:22.477 --> 00:08:24.310
and average them and
then plot that average,

00:08:24.310 --> 00:08:27.060
the frequency of it, then we
take 100 again, average them,

00:08:27.060 --> 00:08:28.910
take the mean, plot
that again, and if we

00:08:28.910 --> 00:08:30.395
do that a bunch
of times, in fact,

00:08:30.395 --> 00:08:32.020
if we were to do that
an infinite time,

00:08:32.020 --> 00:08:33.394
we would find that
we, especially

00:08:33.394 --> 00:08:35.210
if we had an
infinite sample size,

00:08:35.210 --> 00:08:37.909
we would find a perfect
normal distribution.

00:08:37.909 --> 00:08:39.419
That's the crazy thing.

00:08:39.419 --> 00:08:42.409
And it doesn't apply just
to taking the sample mean.

00:08:42.409 --> 00:08:44.285
Here we took the
sample mean every time.

00:08:44.285 --> 00:08:46.160
But you could have also
taken the sample sum.

00:08:46.160 --> 00:08:48.890
The central limit theorem
would have still applied.

00:08:48.890 --> 00:08:51.100
But that's what's so
super useful about it.

00:08:51.100 --> 00:08:54.300
Because in life, there's all
sorts of processes out there,

00:08:54.300 --> 00:08:57.010
proteins bumping into
each other, people doing

00:08:57.010 --> 00:09:00.822
crazy things, humans
interacting in weird ways.

00:09:00.822 --> 00:09:02.780
And you don't know the
probability distribution

00:09:02.780 --> 00:09:04.364
functions for any
of those things.

00:09:04.364 --> 00:09:05.780
But what the central
limit theorem

00:09:05.780 --> 00:09:08.640
tells us is if we add a
bunch of those actions

00:09:08.640 --> 00:09:11.400
together, assuming that they
all have the same distribution,

00:09:11.400 --> 00:09:13.650
or if we were to take the
mean of all of those actions

00:09:13.650 --> 00:09:16.930
together, and if we were to plot
the frequency of those means,

00:09:16.930 --> 00:09:18.890
we do get a normal distribution.

00:09:18.890 --> 00:09:22.120
And that's frankly why the
normal distribution shows up

00:09:22.120 --> 00:09:25.790
so much in statistics
and why, frankly, it's

00:09:25.790 --> 00:09:28.330
a very good
approximation for the sum

00:09:28.330 --> 00:09:31.460
or the means of a
lot of processes.

00:09:31.460 --> 00:09:33.931
Normal distribution.

00:09:33.931 --> 00:09:36.430
What I'm going to show you in
the next video is I'm actually

00:09:36.430 --> 00:09:39.013
going to show you that this is
a reality, that as you increase

00:09:39.013 --> 00:09:41.360
your sample size, as
you increase your n,

00:09:41.360 --> 00:09:43.180
and as you take a
lot of sample means,

00:09:43.180 --> 00:09:45.980
you're going to have a frequency
plot that looks very, very

00:09:45.980 --> 00:09:48.490
close to a normal distribution.


Standard error of the mean

WEBVTT
Kind: captions
Language: en

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We've seen in the
last several videos,

00:00:02.310 --> 00:00:05.324
you start off with any
crazy distribution.

00:00:05.324 --> 00:00:06.490
It doesn't have to be crazy.

00:00:06.490 --> 00:00:08.156
It could be a nice,
normal distribution.

00:00:08.156 --> 00:00:09.890
But to really make
the point that you

00:00:09.890 --> 00:00:11.200
don't have to have a
normal distribution,

00:00:11.200 --> 00:00:12.340
I like to use crazy ones.

00:00:12.340 --> 00:00:14.720
So let's say you have some
kind of crazy distribution

00:00:14.720 --> 00:00:16.011
that looks something like that.

00:00:16.011 --> 00:00:17.230
It could look like anything.

00:00:17.230 --> 00:00:18.604
So we've seen
multiple times, you

00:00:18.604 --> 00:00:20.970
take samples from this
crazy distribution.

00:00:20.970 --> 00:00:28.030
So let's say you were to take
samples of n is equal to 10.

00:00:28.030 --> 00:00:32.240
So we take 10 instances
of this random variable,

00:00:32.240 --> 00:00:36.300
average them out, and
then plot our average.

00:00:36.300 --> 00:00:37.630
We get one instance there.

00:00:37.630 --> 00:00:38.692
We keep doing that.

00:00:38.692 --> 00:00:39.400
We do that again.

00:00:39.400 --> 00:00:42.280
We take 10 samples from
this random variable,

00:00:42.280 --> 00:00:44.490
average them, plot them again.

00:00:44.490 --> 00:00:47.420
Eventually, you do this
a gazillion times--

00:00:47.420 --> 00:00:49.110
in theory, infinite
number of times--

00:00:49.110 --> 00:00:52.030
and you're going to
approach the sampling

00:00:52.030 --> 00:00:53.679
distribution of the sample mean.

00:00:53.679 --> 00:00:55.220
And n equals 10,
it's not going to be

00:00:55.220 --> 00:00:58.810
a perfect normal distribution,
but it's going to be close.

00:00:58.810 --> 00:01:00.750
It would be perfect
only if n was infinity.

00:01:00.750 --> 00:01:05.044
But let's say we eventually--
all of our samples,

00:01:05.044 --> 00:01:06.710
we get a lot of
averages that are there.

00:01:06.710 --> 00:01:07.585
That stacks up there.

00:01:07.585 --> 00:01:08.459
That stacks up there.

00:01:08.459 --> 00:01:10.220
And eventually, we'll
approach something

00:01:10.220 --> 00:01:13.284
that looks something like that.

00:01:13.284 --> 00:01:14.700
And we've seen
from the last video

00:01:14.700 --> 00:01:17.440
that, one, if-- let's say
we were to do it again.

00:01:17.440 --> 00:01:21.200
And this time, let's say
that n is equal to 20.

00:01:21.200 --> 00:01:25.126
One, the distribution that we
get is going to be more normal.

00:01:25.126 --> 00:01:26.500
And maybe in future
videos, we'll

00:01:26.500 --> 00:01:29.456
delve even deeper into things
like kurtosis and skew.

00:01:29.456 --> 00:01:30.830
But it's going to
be more normal.

00:01:30.830 --> 00:01:33.620
But even more important here,
or I guess even more obviously

00:01:33.620 --> 00:01:35.570
to us than we saw,
then, in the experiment,

00:01:35.570 --> 00:01:37.620
it's going to have a
lower standard deviation.

00:01:37.620 --> 00:01:39.411
So they're all going
to have the same mean.

00:01:39.411 --> 00:01:43.450
Let's say the mean here is 5.

00:01:43.450 --> 00:01:45.420
Then the mean here is
also going to be 5.

00:01:45.420 --> 00:01:47.860
The mean of our sampling
distribution of the sample mean

00:01:47.860 --> 00:01:48.977
is going to be 5.

00:01:48.977 --> 00:01:50.310
It doesn't matter what our n is.

00:01:50.310 --> 00:01:52.162
If our n is 20, it's
still going to be 5.

00:01:52.162 --> 00:01:53.620
But our standard
deviation is going

00:01:53.620 --> 00:01:55.817
to be less in either
of these scenarios.

00:01:55.817 --> 00:01:57.400
And we saw that just
by experimenting.

00:01:57.400 --> 00:01:58.100
It might look like this.

00:01:58.100 --> 00:01:59.683
It's going to be
more normal, but it's

00:01:59.683 --> 00:02:01.490
going to have a tighter
standard deviation.

00:02:01.490 --> 00:02:03.360
So maybe it'll look like that.

00:02:03.360 --> 00:02:06.720
And if we did it with an
even larger sample size--

00:02:06.720 --> 00:02:08.560
let me do that in
a different color.

00:02:08.560 --> 00:02:10.509
If we do that with an
even larger sample size,

00:02:10.509 --> 00:02:13.110
n is equal to 100,
what we're going to get

00:02:13.110 --> 00:02:16.160
is something that fits the
normal distribution even

00:02:16.160 --> 00:02:16.940
better.

00:02:16.940 --> 00:02:19.880
We take 100 instances
of this random variable,

00:02:19.880 --> 00:02:21.207
average them, plot it.

00:02:21.207 --> 00:02:22.790
100 instances of
this random variable,

00:02:22.790 --> 00:02:23.780
average them, plot it.

00:02:23.780 --> 00:02:25.082
We just keep doing that.

00:02:25.082 --> 00:02:27.040
If we keep doing that,
what we're going to have

00:02:27.040 --> 00:02:30.050
is something that's even more
normal than either of these.

00:02:30.050 --> 00:02:31.500
So it's going to
be a much closer

00:02:31.500 --> 00:02:33.610
fit to a true
normal distribution,

00:02:33.610 --> 00:02:35.710
but even more obvious
to the human eye,

00:02:35.710 --> 00:02:37.760
it's going to be even tighter.

00:02:37.760 --> 00:02:40.325
So it's going to be a very
low standard deviation.

00:02:40.325 --> 00:02:41.950
It's going to look
something like that.

00:02:41.950 --> 00:02:47.500
I'll show you that on the
simulation app probably later

00:02:47.500 --> 00:02:48.860
in this video.

00:02:48.860 --> 00:02:49.750
So two things happen.

00:02:49.750 --> 00:02:51.716
As you increase your
sample size for every time

00:02:51.716 --> 00:02:53.590
you do the average, two
things are happening.

00:02:53.590 --> 00:02:57.200
You're becoming more normal,
and your standard deviation

00:02:57.200 --> 00:02:58.030
is getting smaller.

00:02:58.030 --> 00:03:00.950
So the question might arise,
well, is there a formula?

00:03:00.950 --> 00:03:04.200
So if I know the
standard deviation--

00:03:04.200 --> 00:03:07.670
so this is my standard deviation
of just my original probability

00:03:07.670 --> 00:03:08.410
density function.

00:03:08.410 --> 00:03:11.970
This is the mean of my original
probability density function.

00:03:11.970 --> 00:03:14.040
So if I know the
standard deviation,

00:03:14.040 --> 00:03:16.240
and I know n is going
to change depending

00:03:16.240 --> 00:03:19.880
on how many samples I'm taking
every time I do a sample mean.

00:03:19.880 --> 00:03:23.910
If I know my standard deviation,
or maybe if I know my variance.

00:03:23.910 --> 00:03:26.269
The variance is just the
standard deviation squared.

00:03:26.269 --> 00:03:27.810
If you don't remember
that, you might

00:03:27.810 --> 00:03:29.770
want to review those videos.

00:03:29.770 --> 00:03:34.100
But if I know the variance
of my original distribution,

00:03:34.100 --> 00:03:38.219
and if I know what my n
is, how many samples I'm

00:03:38.219 --> 00:03:40.510
going to take every time
before I average them in order

00:03:40.510 --> 00:03:46.160
to plot one thing in my sampling
distribution of my sample mean,

00:03:46.160 --> 00:03:50.480
is there a way to predict what
the mean of these distributions

00:03:50.480 --> 00:03:52.000
are?

00:03:52.000 --> 00:03:54.160
The standard deviation
of these distributions.

00:03:54.160 --> 00:03:56.350
And to make it so you don't get
confused between that and that,

00:03:56.350 --> 00:03:57.370
let me say the variance.

00:03:57.370 --> 00:03:58.520
If you know the variance,
you can figure out

00:03:58.520 --> 00:04:00.070
the standard deviation
because one is just

00:04:00.070 --> 00:04:01.400
the square root of the other.

00:04:01.400 --> 00:04:06.310
So this is the variance of
our original distribution.

00:04:06.310 --> 00:04:09.292
Now, to show that this is
the variance of our sampling

00:04:09.292 --> 00:04:11.750
distribution of our sample
mean, we'll write it right here.

00:04:11.750 --> 00:04:16.480
This is the variance
of our sample mean.

00:04:16.480 --> 00:04:20.140
Remember, our true mean is
this, that the Greek letter

00:04:20.140 --> 00:04:22.140
mu is our true mean.

00:04:22.140 --> 00:04:25.610
This is equal to the mean.

00:04:25.610 --> 00:04:28.430
While an x with a line
over it means sample mean.

00:04:31.224 --> 00:04:32.640
So here, what we're
saying is this

00:04:32.640 --> 00:04:35.136
is the variance of
our sample means.

00:04:35.136 --> 00:04:37.010
Now, this is going to
be a true distribution.

00:04:37.010 --> 00:04:39.200
This isn't an estimate.

00:04:39.200 --> 00:04:42.620
If we magically knew
the distribution,

00:04:42.620 --> 00:04:45.080
there's some true variance here.

00:04:45.080 --> 00:04:48.030
And of course, the mean--
so this has a mean.

00:04:48.030 --> 00:04:50.450
This, right here-- if we can
just get our notation right--

00:04:50.450 --> 00:04:55.340
this is the mean of the sampling
distribution of the sampling

00:04:55.340 --> 00:04:55.840
mean.

00:04:55.840 --> 00:04:57.920
So this is the
mean of our means.

00:04:57.920 --> 00:05:00.020
It just happens to
be the same thing.

00:05:00.020 --> 00:05:02.914
This is the mean of
our sample means.

00:05:02.914 --> 00:05:04.580
It's going to be the
same thing as that,

00:05:04.580 --> 00:05:07.171
especially if we do the
trial over and over again.

00:05:07.171 --> 00:05:08.670
But anyway, the
point of this video,

00:05:08.670 --> 00:05:12.530
is there any way to figure
out this variance given

00:05:12.530 --> 00:05:15.640
the variance of the original
distribution and your n?

00:05:15.640 --> 00:05:16.770
And it turns out, there is.

00:05:16.770 --> 00:05:18.420
And I'm not going
to do a proof here.

00:05:18.420 --> 00:05:19.920
I really want to give
you the intuition of it.

00:05:19.920 --> 00:05:21.628
And I think you already
do have the sense

00:05:21.628 --> 00:05:25.600
that every trial you take, if
you take 100, you're much more

00:05:25.600 --> 00:05:28.400
likely, when you average those
out, to get close to the true

00:05:28.400 --> 00:05:31.070
mean than if you took
an n of 2 or an n of 5.

00:05:31.070 --> 00:05:34.150
You're just very unlikely
to be far away if you

00:05:34.150 --> 00:05:36.720
took 100 trials as
opposed to taking five.

00:05:36.720 --> 00:05:38.350
So I think you know
that, in some way,

00:05:38.350 --> 00:05:40.740
it should be inversely
proportional to n.

00:05:40.740 --> 00:05:43.412
The larger your n, the
smaller a standard deviation.

00:05:43.412 --> 00:05:45.870
And it actually turns out it's
about as simple as possible.

00:05:45.870 --> 00:05:47.994
It's one of those magical
things about mathematics.

00:05:47.994 --> 00:05:50.180
And I'll prove it
to you one day.

00:05:50.180 --> 00:05:52.150
I want to give you a
working knowledge first.

00:05:52.150 --> 00:05:53.780
With statistics, I'm
always struggling

00:05:53.780 --> 00:05:56.155
whether I should be formal in
giving you rigorous proofs,

00:05:56.155 --> 00:05:57.864
but I've come to the
conclusion that it's

00:05:57.864 --> 00:05:59.904
more important to get the
working knowledge first

00:05:59.904 --> 00:06:02.740
in statistics, and then, later,
once you've gotten all of that

00:06:02.740 --> 00:06:05.330
down, we can get into
the real deep math of it

00:06:05.330 --> 00:06:06.310
and prove it to you.

00:06:06.310 --> 00:06:09.856
But I think experimental proofs
are all you need for right now,

00:06:09.856 --> 00:06:12.230
using those simulations to
show that they're really true.

00:06:12.230 --> 00:06:14.900
So it turns out that the
variance of your sampling

00:06:14.900 --> 00:06:16.910
distribution of
your sample mean is

00:06:16.910 --> 00:06:20.450
equal to the variance of
your original distribution--

00:06:20.450 --> 00:06:23.350
that guy right
there-- divided by n.

00:06:23.350 --> 00:06:24.490
That's all it is.

00:06:24.490 --> 00:06:29.410
So if this up here has a
variance of-- let's say

00:06:29.410 --> 00:06:32.020
this up here has
a variance of 20.

00:06:32.020 --> 00:06:34.730
I'm just making that number up.

00:06:34.730 --> 00:06:36.440
And then let's say your n is 20.

00:06:36.440 --> 00:06:38.720
Then the variance of your
sampling distribution

00:06:38.720 --> 00:06:40.850
of your sample mean
for an n of 20-- well,

00:06:40.850 --> 00:06:43.065
you're just going to take
the variance up here--

00:06:43.065 --> 00:06:46.020
your variance is 20--
divided by your n, 20.

00:06:46.020 --> 00:06:48.880
So here, your variance
is going to be

00:06:48.880 --> 00:06:51.239
20 divided by 20,
which is equal to 1.

00:06:51.239 --> 00:06:53.280
This is the variance of
your original probability

00:06:53.280 --> 00:06:54.350
distribution.

00:06:54.350 --> 00:06:55.709
And this is your n.

00:06:55.709 --> 00:06:57.500
What's your standard
deviation going to be?

00:06:57.500 --> 00:06:59.560
What's going to be the
square root of that?

00:06:59.560 --> 00:07:00.970
Standard deviation is going
to be the square root of 1.

00:07:00.970 --> 00:07:02.490
Well, that's also going to be 1.

00:07:02.490 --> 00:07:04.140
So we could also write this.

00:07:04.140 --> 00:07:12.260
We could take the square root
of both sides of this and say,

00:07:12.260 --> 00:07:14.520
the standard deviation of
the sampling distribution

00:07:14.520 --> 00:07:17.770
of the sample mean is often
called the standard deviation

00:07:17.770 --> 00:07:19.547
of the mean, and
it's also called--

00:07:19.547 --> 00:07:22.130
I'm going to write this down--
the standard error of the mean.

00:07:27.640 --> 00:07:30.370
All of these things I just
mentioned, these all just mean

00:07:30.370 --> 00:07:32.400
the standard deviation
of the sampling

00:07:32.400 --> 00:07:33.920
distribution of the sample mean.

00:07:33.920 --> 00:07:35.400
That's why this is confusing.

00:07:35.400 --> 00:07:37.397
Because you use the
word "mean" and "sample"

00:07:37.397 --> 00:07:38.230
over and over again.

00:07:38.230 --> 00:07:39.920
And if it confuses
you, let me know.

00:07:39.920 --> 00:07:42.170
I'll do another video or
pause and repeat or whatever.

00:07:42.170 --> 00:07:44.253
But if we just take the
square root of both sides,

00:07:44.253 --> 00:07:46.640
the standard error of the
mean, or the standard deviation

00:07:46.640 --> 00:07:48.960
of the sampling distribution
of the sample mean,

00:07:48.960 --> 00:07:51.440
is equal to the
standard deviation

00:07:51.440 --> 00:07:55.350
of your original function,
of your original probability

00:07:55.350 --> 00:07:57.870
density function, which
could be very non-normal,

00:07:57.870 --> 00:07:59.920
divided by the square root of n.

00:07:59.920 --> 00:08:03.540
I just took the square root of
both sides of this equation.

00:08:03.540 --> 00:08:06.020
Personally, I like
to remember this,

00:08:06.020 --> 00:08:08.900
that the variance is just
inversely proportional to n,

00:08:08.900 --> 00:08:10.380
and then I like to
go back to this,

00:08:10.380 --> 00:08:12.004
because this is very
simple in my head.

00:08:12.004 --> 00:08:13.760
You just take the
variance divided by n.

00:08:13.760 --> 00:08:15.480
Oh, and if I want the
standard deviation,

00:08:15.480 --> 00:08:17.271
I just take the square
roots of both sides,

00:08:17.271 --> 00:08:18.610
and I get this formula.

00:08:18.610 --> 00:08:23.980
So here, when n is 20,
the standard deviation

00:08:23.980 --> 00:08:26.350
of the sampling distribution
of the sample mean

00:08:26.350 --> 00:08:27.440
is going to be 1.

00:08:27.440 --> 00:08:32.520
Here, when n is
100, our variance--

00:08:32.520 --> 00:08:35.240
so our variance of the sampling
mean of the sample distribution

00:08:35.240 --> 00:08:38.419
or our variance of the mean, of
the sample mean, we could say,

00:08:38.419 --> 00:08:43.360
is going to be equal to 20, this
guy's variance, divided by n.

00:08:43.360 --> 00:08:46.990
So it equals-- n is 100--
so it equals one fifth.

00:08:46.990 --> 00:08:50.260
Now, this guy's
standard deviation

00:08:50.260 --> 00:08:53.201
or the standard deviation
of the sampling distribution

00:08:53.201 --> 00:08:55.450
of the sample mean, or the
standard error of the mean,

00:08:55.450 --> 00:08:56.950
is going to the
square root of that.

00:08:56.950 --> 00:08:58.820
So 1 over the square root of 5.

00:08:58.820 --> 00:09:02.400
And so this guy will
have to be a little bit

00:09:02.400 --> 00:09:04.055
under one half the
standard deviation,

00:09:04.055 --> 00:09:05.930
while this guy had a
standard deviation of 1.

00:09:05.930 --> 00:09:07.500
So you see it's
definitely thinner.

00:09:07.500 --> 00:09:08.280
Now, I know what you're saying.

00:09:08.280 --> 00:09:09.738
Well, Sal, you just
gave a formula.

00:09:09.738 --> 00:09:11.260
I don't necessarily believe you.

00:09:11.260 --> 00:09:13.130
Well, let's see
if we can prove it

00:09:13.130 --> 00:09:16.110
to ourselves using
the simulation.

00:09:16.110 --> 00:09:20.870
So just for fun, I'll just
mess with this distribution

00:09:20.870 --> 00:09:21.840
a little bit.

00:09:21.840 --> 00:09:23.464
So that's my new distribution.

00:09:23.464 --> 00:09:24.880
And let me take
an n-- let me take

00:09:24.880 --> 00:09:26.540
two things it's easy to
take the square root of,

00:09:26.540 --> 00:09:28.470
because we're looking
at standard deviations.

00:09:28.470 --> 00:09:33.340
So let's say we take
an n of 16 and n of 25.

00:09:33.340 --> 00:09:35.552
And let's do 10,000 trials.

00:09:35.552 --> 00:09:37.260
So in this case, every
one of the trials,

00:09:37.260 --> 00:09:38.968
we're going to take
16 samples from here,

00:09:38.968 --> 00:09:41.850
average them, plot it here,
and then do a frequency plot.

00:09:41.850 --> 00:09:44.850
Here, we're going to do a 25 at
a time and then average them.

00:09:44.850 --> 00:09:46.960
I'll do it once animated
just to remember.

00:09:46.960 --> 00:09:50.670
So I'm taking 16
samples, plot it there.

00:09:50.670 --> 00:09:53.540
I take 16 samples, as described
by this probability density

00:09:53.540 --> 00:09:55.240
function, or 25 now.

00:09:55.240 --> 00:09:57.120
Plot it down here.

00:09:57.120 --> 00:10:02.230
Now, if I do that 10,000
times, what do I get?

00:10:02.230 --> 00:10:03.520
What do I get?

00:10:03.520 --> 00:10:04.040
All right.

00:10:04.040 --> 00:10:07.326
So here, just visually, you can
tell just when n was larger,

00:10:07.326 --> 00:10:08.950
the standard deviation
here is smaller.

00:10:08.950 --> 00:10:10.450
This is more squeezed together.

00:10:10.450 --> 00:10:12.260
But actually, let's
write this stuff down.

00:10:12.260 --> 00:10:13.760
Let's see if I can
remember it here.

00:10:13.760 --> 00:10:14.450
Here, n is 6.

00:10:14.450 --> 00:10:17.840
So in this random distribution
I made, my standard deviation

00:10:17.840 --> 00:10:19.180
was 9.3.

00:10:19.180 --> 00:10:20.520
I'm going to remember these.

00:10:20.520 --> 00:10:24.610
Our standard deviation for
the original thing was 9.3.

00:10:24.610 --> 00:10:27.650
And so standard
deviation here was 2.3,

00:10:27.650 --> 00:10:29.620
and the standard
deviation here is 1.87.

00:10:29.620 --> 00:10:32.982
Let's see if it
conforms to our formula.

00:10:32.982 --> 00:10:35.065
So I'm going to take this
off screen for a second,

00:10:35.065 --> 00:10:39.060
and I'm going to go back
and do some mathematics.

00:10:39.060 --> 00:10:40.890
So I have this on
my other screen

00:10:40.890 --> 00:10:42.960
so I can remember those numbers.

00:10:42.960 --> 00:10:46.980
So, in the trial we just
did, my wacky distribution

00:10:46.980 --> 00:10:56.510
had a standard deviation of 9.3.

00:10:56.510 --> 00:11:01.092
When n was equal to 16--
just doing the experiment,

00:11:01.092 --> 00:11:03.050
doing a bunch of trials
and averaging and doing

00:11:03.050 --> 00:11:06.650
all the thing-- we got
the standard deviation

00:11:06.650 --> 00:11:08.650
of the sampling distribution
of the sample mean,

00:11:08.650 --> 00:11:10.080
or the standard
error of the mean.

00:11:10.080 --> 00:11:15.590
We experimentally
determined it to be 2.33.

00:11:15.590 --> 00:11:20.180
And then when n
is equal to 25, we

00:11:20.180 --> 00:11:24.930
got the standard error of
the mean being equal to 1.87.

00:11:24.930 --> 00:11:28.460
Let's see if it conforms
to our formulas.

00:11:28.460 --> 00:11:32.510
So we know that the
variance-- or we could almost

00:11:32.510 --> 00:11:35.270
say the variance of the
mean or the standard error--

00:11:35.270 --> 00:11:38.549
the variance of the sampling
distribution of the sample mean

00:11:38.549 --> 00:11:41.090
is equal to the variance of our
original distribution divided

00:11:41.090 --> 00:11:41.930
by n.

00:11:41.930 --> 00:11:44.150
Take the square
roots of both sides.

00:11:44.150 --> 00:11:46.690
Then you get standard
error of the mean

00:11:46.690 --> 00:11:49.860
is equal to standard deviation
of your original distribution,

00:11:49.860 --> 00:11:51.850
divided by the square root of n.

00:11:51.850 --> 00:11:54.490
So let's see if this works
out for these two things.

00:11:54.490 --> 00:11:58.830
So if I were to take 9.3--
so let me do this case.

00:11:58.830 --> 00:12:04.100
So 9.3 divided by the
square root of 16-- n

00:12:04.100 --> 00:12:08.110
is 16-- so divided by the
square root of 16, which is 4.

00:12:08.110 --> 00:12:09.440
What do I get?

00:12:09.440 --> 00:12:11.710
So 9.3 divided by 4.

00:12:11.710 --> 00:12:14.980
Let me get a little
calculator out here.

00:12:14.980 --> 00:12:17.970
Let's see.

00:12:17.970 --> 00:12:21.070
We want to divide
9.3 divided by 4.

00:12:21.070 --> 00:12:25.070
9.3 divided by our square
root of n-- n was 16,

00:12:25.070 --> 00:12:32.110
so divided by 4--
is equal to 2.32.

00:12:32.110 --> 00:12:41.590
So this is equal to 2.32, which
is pretty darn close to 2.33.

00:12:41.590 --> 00:12:43.190
This was after 10,000 trials.

00:12:43.190 --> 00:12:44.690
Maybe right after
this I'll see what

00:12:44.690 --> 00:12:46.950
happens if we did 20,000
or 30,000 trials where

00:12:46.950 --> 00:12:49.130
we take samples of
16 and average them.

00:12:49.130 --> 00:12:50.550
Now let's look at this.

00:12:50.550 --> 00:12:53.760
Here, we would take 9.3.

00:12:53.760 --> 00:12:55.440
So let me draw a
little line here.

00:12:55.440 --> 00:12:56.190
Maybe scroll over.

00:12:56.190 --> 00:12:57.470
That might be better.

00:12:57.470 --> 00:13:00.210
So we take our
standard deviation

00:13:00.210 --> 00:13:02.770
of our original
distribution-- so just

00:13:02.770 --> 00:13:04.960
that formula that we've
derived right here

00:13:04.960 --> 00:13:08.110
would tell us that our
standard error should

00:13:08.110 --> 00:13:09.720
be equal to the
standard deviation

00:13:09.720 --> 00:13:12.940
of our original
distribution, 9.3,

00:13:12.940 --> 00:13:16.460
divided by the square root of
n, divided by square root of 25.

00:13:16.460 --> 00:13:18.560
4 was just the
square root of 16.

00:13:18.560 --> 00:13:22.220
So this is equal to
9.3 divided by 5.

00:13:22.220 --> 00:13:24.340
And let's see if it's 1.87.

00:13:24.340 --> 00:13:28.490
So let me get my
calculator back.

00:13:28.490 --> 00:13:36.430
So if I take 9.3 divided
by 5, what do I get?

00:13:36.430 --> 00:13:41.870
1.86, which is
very close to 1.87.

00:13:41.870 --> 00:13:49.690
So we got in this case 1.86.

00:13:49.690 --> 00:13:52.310
So as you can see, what
we got experimentally

00:13:52.310 --> 00:13:55.659
was almost exactly-- and
this is after 10,000 trials--

00:13:55.659 --> 00:13:56.700
of what you would expect.

00:13:56.700 --> 00:13:58.830
Let's do another 10,000.

00:13:58.830 --> 00:14:00.250
So you got another
10,000 trials.

00:14:00.250 --> 00:14:01.600
Well, we're still
in the ballpark.

00:14:01.600 --> 00:14:03.016
We're not going
to-- maybe I can't

00:14:03.016 --> 00:14:07.520
hope to get the exact
number rounded or whatever.

00:14:07.520 --> 00:14:09.890
But, as you can see,
hopefully that'll be pretty

00:14:09.890 --> 00:14:13.600
satisfying to you, that the
variance of the sampling

00:14:13.600 --> 00:14:19.120
distribution of the
sample mean is just

00:14:19.120 --> 00:14:21.254
going to be equal
to the variance

00:14:21.254 --> 00:14:22.920
of your original
distribution, no matter

00:14:22.920 --> 00:14:25.160
how wacky that
distribution might be,

00:14:25.160 --> 00:14:29.720
divided by your sample size,
by the number of samples you

00:14:29.720 --> 00:14:33.281
take for every basket
that you average,

00:14:33.281 --> 00:14:35.030
I guess is the best
way to think about it.

00:14:35.030 --> 00:14:36.590
And sometimes this
can get confusing,

00:14:36.590 --> 00:14:40.600
because you are taking samples
of averages based on samples.

00:14:40.600 --> 00:14:42.730
So when someone says
sample size, you're like,

00:14:42.730 --> 00:14:45.020
is sample size the
number of times

00:14:45.020 --> 00:14:46.990
I took averages or
the number of things

00:14:46.990 --> 00:14:48.880
I'm taking averages
of each time?

00:14:48.880 --> 00:14:51.180
And it doesn't hurt
to clarify that.

00:14:51.180 --> 00:14:52.930
Normally when they
talk about sample size,

00:14:52.930 --> 00:14:54.260
they're talking about n.

00:14:54.260 --> 00:14:57.370
And, at least in my head,
when I think of the trials

00:14:57.370 --> 00:14:59.260
as you take a sample
of size of 16,

00:14:59.260 --> 00:15:01.120
you average it,
that's one trial.

00:15:01.120 --> 00:15:01.802
And you plot it.

00:15:01.802 --> 00:15:03.760
Then you do it again,
and you do another trial.

00:15:03.760 --> 00:15:05.110
And you do it over
and over again.

00:15:05.110 --> 00:15:07.193
But anyway, hopefully this
makes everything clear.

00:15:07.193 --> 00:15:09.050
And then you now
also understand how

00:15:09.050 --> 00:15:14.040
to get to the standard
error of the mean.


Sampling distribution of the sample mean 2

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:03.810
We hopefully now have a
respectable working knowledge

00:00:03.810 --> 00:00:06.850
of the sampling distribution
of the sample mean.

00:00:06.850 --> 00:00:08.870
And what I want to do in this
video is explore a little bit

00:00:08.870 --> 00:00:12.640
more on how that distribution
changes as we change

00:00:12.640 --> 00:00:14.920
our sample size n.

00:00:14.920 --> 00:00:16.130
I'll write n down right here.

00:00:16.130 --> 00:00:17.290
Our sample size n.

00:00:17.290 --> 00:00:20.090
So just as a bit of review, we
saw before we can just start

00:00:20.090 --> 00:00:22.480
off with any crazy
distribution, maybe it looks

00:00:22.480 --> 00:00:23.815
something like this.

00:00:23.815 --> 00:00:26.230
I'll do a discrete
distribution.

00:00:26.230 --> 00:00:27.930
Really to model anything
at some point you have

00:00:27.930 --> 00:00:29.000
to make it discreet.

00:00:29.000 --> 00:00:30.885
It could be a very granular
discrete distribution.

00:00:30.885 --> 00:00:33.380
But let's say it's something
crazy that looks like this.

00:00:33.380 --> 00:00:36.670
This is clearly not a
normal distribution.

00:00:36.670 --> 00:00:39.110
But we saw in the first video
if you take, let's say,

00:00:39.110 --> 00:00:40.870
sample sizes of 4.

00:00:40.870 --> 00:00:43.880
So if you took 4 numbers from
this distribution, 4 random

00:00:43.880 --> 00:00:47.140
numbers where let's say this
is the probably of a 1,

00:00:47.140 --> 00:00:51.880
2, 3, 4, 5, 6, 7, 8, 9.

00:00:51.880 --> 00:00:55.200
If you took 4 numbers at a time
and averaged them-- let me do

00:00:55.200 --> 00:00:59.580
that here-- if you took 4
numbers at a time, let's say we

00:00:59.580 --> 00:01:01.760
used this distribution to
generate 4 random

00:01:01.760 --> 00:01:02.500
numbers, right?

00:01:02.500 --> 00:01:03.930
We're very likely to get a 9.

00:01:03.930 --> 00:01:05.680
We're definitely not going
to get any 7's or 8's.

00:01:05.680 --> 00:01:07.480
We're definitely not
going to get a 4.

00:01:07.480 --> 00:01:08.470
We might get a 1 or 2.

00:01:08.470 --> 00:01:09.750
3 is also very likely.

00:01:09.750 --> 00:01:10.780
Five is very likely.

00:01:10.780 --> 00:01:13.690
So we use this function
to essentially generate

00:01:13.690 --> 00:01:14.810
random numbers for us.

00:01:14.810 --> 00:01:17.890
And we take samples of 4 and
then we average them up.

00:01:17.890 --> 00:01:20.680
So let's say our first average
is, I don't know, let's say

00:01:20.680 --> 00:01:25.985
it's a 9, it's a 5, it's
another 9, and then it's a 1.

00:01:25.985 --> 00:01:26.640
So what is that?

00:01:26.640 --> 00:01:28.010
That's 14 plus 10.

00:01:28.010 --> 00:01:29.460
24 divided by 4.

00:01:29.460 --> 00:01:33.380
The average for this first
trial, for this first sample of

00:01:33.380 --> 00:01:35.400
4, is going to be 6, right?

00:01:35.400 --> 00:01:37.140
They add up to 24 divided by 4.

00:01:37.140 --> 00:01:38.450
So we would plot it right here.

00:01:38.450 --> 00:01:40.890
Our average was 6 that time.

00:01:40.890 --> 00:01:41.356
Just like that.

00:01:41.356 --> 00:01:42.820
And we'll just keep doing it.

00:01:42.820 --> 00:01:45.900
And we've seen in the past that
if you just keep doing this,

00:01:45.900 --> 00:01:47.860
this is going to start
looking something like

00:01:47.860 --> 00:01:48.910
a normal distribution.

00:01:48.910 --> 00:01:51.160
So maybe we'd do it again,
the average is 6 again.

00:01:51.160 --> 00:01:53.220
Maybe we do it again,
the average is 5.

00:01:53.220 --> 00:01:54.660
We do it again,
the average is 7.

00:01:54.660 --> 00:01:56.500
We do it again,
the average is 6.

00:01:56.500 --> 00:01:59.720
And then if you just do this a
ton, a ton of times, your

00:01:59.720 --> 00:02:01.900
distribution might look
something that looks very much

00:02:01.900 --> 00:02:02.930
like a normal distribution.

00:02:02.930 --> 00:02:04.180
So these boxes are
really small.

00:02:04.180 --> 00:02:06.760
So we just do a bunch of these
trials, at some point it

00:02:06.760 --> 00:02:09.240
might look a lot like
a normal distribution.

00:02:09.240 --> 00:02:11.170
Obviously there are
some average values.

00:02:11.170 --> 00:02:13.360
It won't be a perfect normal
distribution because you can

00:02:13.360 --> 00:02:17.110
never get anything less than
a 0, or anything less than

00:02:17.110 --> 00:02:18.590
a 1, really as an average.

00:02:18.590 --> 00:02:20.010
You can't get 0 as an average.

00:02:20.010 --> 00:02:21.440
And you can't get
anything more than 9.

00:02:21.440 --> 00:02:24.170
So it's not going to have
infinitely long tails but at

00:02:24.170 --> 00:02:26.480
least for the middle part of it
a normal distribution might

00:02:26.480 --> 00:02:28.090
be a good approximation.

00:02:28.090 --> 00:02:30.340
In this video what I want
to think about is what

00:02:30.340 --> 00:02:34.410
happens as we change n.

00:02:34.410 --> 00:02:37.030
So in this case n was 4.

00:02:37.030 --> 00:02:38.950
n is our sample size.

00:02:38.950 --> 00:02:41.290
Every time we do a trial we
took 4 and we took their

00:02:41.290 --> 00:02:42.700
average and we plotted it.

00:02:42.700 --> 00:02:44.470
We could have had n equal 10.

00:02:44.470 --> 00:02:48.915
We could have taken 10 samples
from this from this population,

00:02:48.915 --> 00:02:52.540
you could say, or from this
random variable, average them,

00:02:52.540 --> 00:02:54.000
and then plotted them here.

00:02:54.000 --> 00:02:55.910
And in the last video
we ran the simulation.

00:02:55.910 --> 00:02:58.090
I'm going to go back to
that simulation a second.

00:02:58.090 --> 00:02:59.800
We saw a couple of things.

00:02:59.800 --> 00:03:01.700
And I'll show it to you
at a little bit more

00:03:01.700 --> 00:03:03.420
depth this time.

00:03:03.420 --> 00:03:06.360
When n is pretty small, it
doesn't approach a normal

00:03:06.360 --> 00:03:07.950
distribution that well.

00:03:07.950 --> 00:03:11.170
So when n is small-- I mean,
let's take the extreme case.

00:03:11.170 --> 00:03:14.060
What happens when
n is equal to 1?

00:03:14.060 --> 00:03:17.800
That literally just means I
take 1 instance of this random

00:03:17.800 --> 00:03:18.630
variable and average it.

00:03:18.630 --> 00:03:20.410
Well it's just going
to be that thing.

00:03:20.410 --> 00:03:22.520
So if I just take a bunch of
trials from the thing and

00:03:22.520 --> 00:03:24.540
plot it over time, what's
it going to look like?

00:03:24.540 --> 00:03:25.710
Well it's definitely not
going to look like a

00:03:25.710 --> 00:03:26.460
normal distribution.

00:03:26.460 --> 00:03:28.050
It's going to look-- you're
going to have a couple

00:03:28.050 --> 00:03:29.970
of 1's, you're going to
have a couple of 2's.

00:03:29.970 --> 00:03:32.490
You're going to have
more 3's like that.

00:03:32.490 --> 00:03:33.750
You're going to have no 4's.

00:03:33.750 --> 00:03:35.630
You're going to have
a bunch of 5's.

00:03:35.630 --> 00:03:37.330
You're going to have some
6's that'll look like that.

00:03:37.330 --> 00:03:39.380
And you're going to
have a bunch of 9's.

00:03:39.380 --> 00:03:41.730
So there your sampling
distribution of the sample mean

00:03:41.730 --> 00:03:43.890
for an n of 1 is going to
look-- I don't care how many

00:03:43.890 --> 00:03:46.300
trials you do, it's not going
to look like a normal

00:03:46.300 --> 00:03:46.950
distribution.

00:03:46.950 --> 00:03:49.740
So the central limit theorem,
although I said you do a bunch

00:03:49.740 --> 00:03:50.900
of trials, it'll look like a
normal distribution,

00:03:50.900 --> 00:03:52.900
it definitely doesn't
work for n equal 1.

00:03:52.900 --> 00:03:55.460
As n gets larger though
it starts to make sense.

00:03:55.460 --> 00:03:59.030
That let's see if we've got n
equals 2--- and I'm all just

00:03:59.030 --> 00:04:00.700
doing this in my head, I
don't know what the actual

00:04:00.700 --> 00:04:03.520
distributions would look like--
but then, it's still would be

00:04:03.520 --> 00:04:06.370
difficult for it to become an
exact normal distribution.

00:04:06.370 --> 00:04:08.280
But then you can get more
instance-- that you could get

00:04:08.280 --> 00:04:10.610
more-- you know, you might get
things from all of the above.

00:04:10.610 --> 00:04:12.670
But you can only get two
in each of your baskets

00:04:12.670 --> 00:04:13.490
that your averaging.

00:04:13.490 --> 00:04:15.140
You're only going to
get 2 numbers, right?

00:04:15.140 --> 00:04:15.250
So?

00:04:15.250 --> 00:04:17.800
You're never going to for
example, you're never going to

00:04:17.800 --> 00:04:22.640
get 7.5 in your sampling
distribution of the sample mean

00:04:22.640 --> 00:04:25.170
for n is equal to 2 because
it's impossible to get a 7 and

00:04:25.170 --> 00:04:26.660
it's impossible to get an 8.

00:04:26.660 --> 00:04:30.680
So you're never going to get
7.5 as-- so maybe when you plot

00:04:30.680 --> 00:04:35.125
it, maybe it looks like this.

00:04:35.125 --> 00:04:38.020
But there will be a gap at 7.5
because that's impossible

00:04:38.020 --> 00:04:40.780
and maybe it looks
something like that.

00:04:40.780 --> 00:04:42.750
So it's still won't be
a normal distribution

00:04:42.750 --> 00:04:44.230
when n is equal to 2.

00:04:44.230 --> 00:04:46.480
So there's a couple of
interesting things here.

00:04:46.480 --> 00:04:48.550
So one thing-- and I didn't
mention this the first time

00:04:48.550 --> 00:04:50.160
because I really wanted you to
get the gut sense what the

00:04:50.160 --> 00:04:54.170
central limit theorem is-- the
central limit theorem says as n

00:04:54.170 --> 00:04:57.320
approaches-- really as it
approaches infinity then is

00:04:57.320 --> 00:04:59.075
when you get the real
normal distribution.

00:05:02.200 --> 00:05:04.870
But in kind of every day
practice, you don't have to get

00:05:04.870 --> 00:05:06.320
that much beyond n equals two.

00:05:06.320 --> 00:05:09.220
If you get to n equals 10 or n
equals 15, you're getting very

00:05:09.220 --> 00:05:10.400
close to a normal distribution.

00:05:10.400 --> 00:05:15.240
So this converges to a normal
distribution very quickly.

00:05:15.240 --> 00:05:16.880
Now the other thing is
you obviously wants

00:05:16.880 --> 00:05:17.840
many, many trials.

00:05:17.840 --> 00:05:20.220
So this is your sample size.

00:05:20.220 --> 00:05:21.360
That is your sample size.

00:05:21.360 --> 00:05:23.570
That's the size of
each of your baskets.

00:05:23.570 --> 00:05:25.700
In the very first video
I did on this, I took

00:05:25.700 --> 00:05:27.180
a sample size of 4.

00:05:27.180 --> 00:05:30.310
And in the simulation I did
in the last video, we did

00:05:30.310 --> 00:05:32.620
sample sizes of 4 and
10 and whatever else.

00:05:32.620 --> 00:05:33.572
This is a sample size of one.

00:05:33.572 --> 00:05:34.900
So that's our sample size.

00:05:34.900 --> 00:05:39.250
So as that approaches infinity
your actual sampling

00:05:39.250 --> 00:05:44.140
distribution of the sample of
the sample mean will approach a

00:05:44.140 --> 00:05:46.080
normal distribution.

00:05:46.080 --> 00:05:49.150
Now in order to actually see
that normal distribution and

00:05:49.150 --> 00:05:51.520
actually to prove it to
yourself, you would have to do

00:05:51.520 --> 00:05:54.590
this many, many-- remember the
normal distribution happens,

00:05:54.590 --> 00:05:57.560
this is essentially the
population or this is

00:05:57.560 --> 00:05:58.440
the random variable.

00:05:58.440 --> 00:06:00.350
That tells you all of
the possibilities.

00:06:00.350 --> 00:06:02.770
In real life, we seldom know
all the possibilities.

00:06:02.770 --> 00:06:07.060
In fact in real life, we seldom
know the pure probability

00:06:07.060 --> 00:06:07.820
generating function.

00:06:07.820 --> 00:06:09.320
Only if we're writing
it or if we're writing

00:06:09.320 --> 00:06:10.290
a computer program.

00:06:10.290 --> 00:06:12.530
Normally we're doing
samples and we're

00:06:12.530 --> 00:06:13.660
trying estimate things.

00:06:13.660 --> 00:06:17.490
So normally there's some random
variable and then maybe we'll

00:06:17.490 --> 00:06:20.880
do a bunch of-- we'd take it a
bunch of samples, we'd take

00:06:20.880 --> 00:06:22.460
their means and we'd plot them
and we're going to get some

00:06:22.460 --> 00:06:23.740
type of normal distribution.

00:06:23.740 --> 00:06:26.210
Let's say we take samples of
100 and we average them.

00:06:26.210 --> 00:06:28.160
We're going to get some
normal distribution.

00:06:28.160 --> 00:06:32.640
And in theory, as we take those
averages hundreds or thousands

00:06:32.640 --> 00:06:37.230
of times, our data set it's
going to more closely

00:06:37.230 --> 00:06:44.260
approximate that pure
sampling distribution

00:06:44.260 --> 00:06:49.100
of the sample mean.

00:06:49.100 --> 00:06:51.410
This thing is a
real distribution.

00:06:51.410 --> 00:06:54.770
It's a real distribution
with a real mean.

00:06:54.770 --> 00:06:56.415
It has a pure mean.

00:06:56.415 --> 00:07:04.410
So the mean of the sampling
distribution of the sample

00:07:04.410 --> 00:07:06.210
mean, we'll write it like that.

00:07:06.210 --> 00:07:10.200
Notice I didn't write it is
just the x with-- what this is,

00:07:10.200 --> 00:07:14.540
this is actually saying that
this is a real population mean,

00:07:14.540 --> 00:07:16.960
this is a real random
variable mean.

00:07:16.960 --> 00:07:19.640
If you look at every
possibility of all of the

00:07:19.640 --> 00:07:22.410
samples that you can take from
your original distribution,

00:07:22.410 --> 00:07:25.980
from some other random original
distribution, and you took all

00:07:25.980 --> 00:07:28.940
of the possibilities of
let's see sample size.

00:07:28.940 --> 00:07:29.970
Let's see were dealing
with the world where

00:07:29.970 --> 00:07:31.550
a sample size is 10.

00:07:31.550 --> 00:07:34.160
If you took all of the
combinations of 10 samples from

00:07:34.160 --> 00:07:38.320
some original distribution and
you averaged them out, this

00:07:38.320 --> 00:07:39.700
would describe that function.

00:07:39.700 --> 00:07:42.140
Of course in reality, if you
don't know the original

00:07:42.140 --> 00:07:45.150
distribution, you can't take an
infinite samples from it so you

00:07:45.150 --> 00:07:46.340
won't know every combination.

00:07:46.340 --> 00:07:52.840
But if you did it with 1,000--
if you did the trial 1,000

00:07:52.840 --> 00:07:55.320
times-- so 1,000 times you took
10 samples from some

00:07:55.320 --> 00:07:57.960
distribution and took 1,000
averages and then plotted

00:07:57.960 --> 00:08:01.680
them, you're going
to get pretty close.

00:08:01.680 --> 00:08:04.600
Now the next thing I want to
touch on is what happens as

00:08:04.600 --> 00:08:07.830
n-- we know as n approaches
infinity it becomes more of a

00:08:07.830 --> 00:08:11.170
normal distribution, but as I
said already, n equals 10 is

00:08:11.170 --> 00:08:13.295
pretty good and n equals
20 is even better.

00:08:13.295 --> 00:08:15.860
But we saw something in the
last video that at least I

00:08:15.860 --> 00:08:17.660
find pretty interesting.

00:08:17.660 --> 00:08:20.390
Let's say we start with this
crazy distribution up here.

00:08:20.390 --> 00:08:21.710
It really doesn't matter
what distribution

00:08:21.710 --> 00:08:22.770
we're starting with.

00:08:22.770 --> 00:08:29.440
We saw in the simulation that
when n is equal to 5, our graph

00:08:29.440 --> 00:08:32.905
after we try-- we take samples
of 5, average them and we do

00:08:32.905 --> 00:08:36.550
it 10,000 times-- our graph
look something like this.

00:08:36.550 --> 00:08:38.270
It's kind of wide like that.

00:08:38.270 --> 00:08:42.820
And then when we did n is equal
to 10 our graph looked a little

00:08:42.820 --> 00:08:45.040
bit-- it was actually a
little bit squeezed in like

00:08:45.040 --> 00:08:46.610
that a little bit more.

00:08:46.610 --> 00:08:49.980
So not only was it more
normal-- that's what the

00:08:49.980 --> 00:08:52.310
central limit theorem tells us
because we're taking larger

00:08:52.310 --> 00:08:56.340
sample sizes-- but it had a
smaller standard deviation or a

00:08:56.340 --> 00:08:57.650
smaller variance, right?

00:08:57.650 --> 00:09:00.420
The mean is going to be the
same either case but when our

00:09:00.420 --> 00:09:03.780
sample size was larger our
standard deviation

00:09:03.780 --> 00:09:04.710
became smaller.

00:09:04.710 --> 00:09:07.120
In fact, our standard deviation
became smaller than our

00:09:07.120 --> 00:09:11.590
original population
distribution-- or original

00:09:11.590 --> 00:09:12.720
probability density function.

00:09:12.720 --> 00:09:14.505
Let me show you that
with a simulation.

00:09:14.505 --> 00:09:16.580
So let me clear everything.

00:09:16.580 --> 00:09:18.730
And this simulation is as good
as any, so the first thing I

00:09:18.730 --> 00:09:20.750
want to show-- or this
distribution is as good as

00:09:20.750 --> 00:09:23.190
any-- the first thing I want to
show you is that n of 2

00:09:23.190 --> 00:09:24.910
is really not that good.

00:09:24.910 --> 00:09:29.340
So let's compare an n of 2
to let's say an n of 16.

00:09:29.340 --> 00:09:32.810
So when you compare an
n of 2 to an n of 16,

00:09:32.810 --> 00:09:33.730
let's do it once.

00:09:33.730 --> 00:09:35.490
So you get 1, 2 trials,
you average them.

00:09:35.490 --> 00:09:38.320
And then it's going to do 16
and then it's going to plot it

00:09:38.320 --> 00:09:40.530
down here and average there.

00:09:40.530 --> 00:09:43.640
Let's do that 10,000 times.

00:09:43.640 --> 00:09:46.520
So notice, when you took an n
of 2, even though we did it

00:09:46.520 --> 00:09:49.990
10,000 times, this is not
approaching a normal

00:09:49.990 --> 00:09:50.630
distribution.

00:09:50.630 --> 00:09:53.230
You can actually see it in the
skew and kurtosis numbers.

00:09:53.230 --> 00:09:56.120
It has a rightward positive
skew which means it has a

00:09:56.120 --> 00:09:57.990
longer tail to the right
than to the left.

00:09:57.990 --> 00:10:00.860
And then it has a negative
kurtosis which means that it's

00:10:00.860 --> 00:10:06.000
a little bit-- it has shorter
tales and smaller peaks than a

00:10:06.000 --> 00:10:08.270
standard normal distribution.

00:10:08.270 --> 00:10:11.000
Now when n is equal to
16 you do the same.

00:10:11.000 --> 00:10:14.930
So every time we took 16
samples from this distribution

00:10:14.930 --> 00:10:17.650
function up here and averaged
them-- and each of these dots

00:10:17.650 --> 00:10:22.050
represent an average and we did
it 10,001 times-- and notice

00:10:22.050 --> 00:10:24.110
the mean is the same in both
places but here all of a

00:10:24.110 --> 00:10:27.630
sudden, our kurtosis is much
smaller and our skew

00:10:27.630 --> 00:10:28.230
is much smaller.

00:10:28.230 --> 00:10:32.080
So we are more normal in
this into situation.

00:10:32.080 --> 00:10:34.000
But even a more interesting
thing is our standard

00:10:34.000 --> 00:10:35.480
deviation is smaller, right?

00:10:35.480 --> 00:10:37.300
This is more squeezed
in than that is.

00:10:37.300 --> 00:10:39.980
And it's definitely more
squeezed in then our

00:10:39.980 --> 00:10:41.700
original distribution.

00:10:41.700 --> 00:10:45.400
Now let me do it with 2-- let
me clear everything again.

00:10:45.400 --> 00:10:49.270
I like this distribution
because it's a very

00:10:49.270 --> 00:10:50.780
non-normal distribution.

00:10:50.780 --> 00:10:53.180
It looks like a bimodal
distribution of some kind.

00:10:53.180 --> 00:10:56.080
And let's take a scenario
where I take an n of--

00:10:56.080 --> 00:10:56.990
let's take two good n's.

00:10:56.990 --> 00:10:59.895
Let's take an n of 16-- that's
a nice healthy n-- and let's

00:10:59.895 --> 00:11:02.180
take an n of 25 and let's
compare them a little bit.

00:11:04.760 --> 00:11:08.720
So if we-- I'll do one trial
animated just because

00:11:08.720 --> 00:11:09.630
it's always nice to see.

00:11:09.630 --> 00:11:12.010
So first it's going to do 16
of these trials and average

00:11:12.010 --> 00:11:13.180
them and there we go.

00:11:13.180 --> 00:11:16.680
And then it's going to do 25 of
these trials and then average

00:11:16.680 --> 00:11:19.270
them and then there we go.

00:11:19.270 --> 00:11:21.550
Now let's do that-- what I
just did animated-- let's

00:11:21.550 --> 00:11:24.590
do it 10,000 times.

00:11:24.590 --> 00:11:26.270
Miracles of computers.

00:11:26.270 --> 00:11:28.360
Now notice something:
this is 10,000 times.

00:11:28.360 --> 00:11:30.480
These are both pretty
good approximations of

00:11:30.480 --> 00:11:31.860
normal distributions.

00:11:31.860 --> 00:11:34.660
The n is equal to
25 is more normal.

00:11:34.660 --> 00:11:38.180
It has less skew-- slightly
less skew than n is equal 16.

00:11:38.180 --> 00:11:43.180
It has slightly kurtosis which
means it's closer to being

00:11:43.180 --> 00:11:45.610
a normal distribution
than n is equal to 16.

00:11:45.610 --> 00:11:47.230
But even more interesting,
it's more squeezed in.

00:11:47.230 --> 00:11:49.210
It has a lower
standard deviation.

00:11:49.210 --> 00:11:53.120
The standard deviation here
is 2.1 and the standard

00:11:53.120 --> 00:11:56.120
deviation here is 2.64.

00:11:56.120 --> 00:11:59.300
So that's another-- I mean I
kind of touched on that in

00:11:59.300 --> 00:12:01.630
the last video-- and it
kind of makes sense.

00:12:01.630 --> 00:12:04.350
For every sample you do for
your average, the more you

00:12:04.350 --> 00:12:07.030
put into that sample, the
less standard deviation.

00:12:07.030 --> 00:12:08.440
Think of the extreme case.

00:12:08.440 --> 00:12:11.270
If instead of taking 16 samples
from our distribution every

00:12:11.270 --> 00:12:14.440
time or instead of taking 25,
if I were to take 1,000,000

00:12:14.440 --> 00:12:21.720
samples from this distribution
every time that sample mean is

00:12:21.720 --> 00:12:24.180
always going to be pretty
darn close to my mean.

00:12:24.180 --> 00:12:27.010
If I take 1,000,000 samples of
everything, if I essentially

00:12:27.010 --> 00:12:30.130
try to estimate a mean by
taking 1,000,000 samples, I'm

00:12:30.130 --> 00:12:31.860
going to get a pretty good
estimate of that mean.

00:12:31.860 --> 00:12:35.260
The probability that a
million numbers are all

00:12:35.260 --> 00:12:36.770
out here is very low.

00:12:36.770 --> 00:12:41.760
So if n is 1,000,000 of course
all of my sample means when I

00:12:41.760 --> 00:12:44.650
average them are all going to
be really tightly focused

00:12:44.650 --> 00:12:47.130
around the mean itself.

00:12:47.130 --> 00:12:50.660
So hopefully that kind of
makes sense to you as well.

00:12:50.660 --> 00:12:53.930
If it doesn't just think about
it or even use this tool and

00:12:53.930 --> 00:12:56.430
experiment with it just so you
can trust that is

00:12:56.430 --> 00:12:57.315
really the case.

00:12:57.315 --> 00:13:00.380
And it actually turns out that
there's a very clean formula

00:13:00.380 --> 00:13:03.730
that relates to standard
deviation of the original

00:13:03.730 --> 00:13:07.060
probability distribution
function to the standard

00:13:07.060 --> 00:13:09.110
deviation of the sampling
distribution of

00:13:09.110 --> 00:13:10.140
the sample mean.

00:13:10.140 --> 00:13:13.680
And as you can imagine it is a
function of your sample size,

00:13:13.680 --> 00:13:16.230
of how many samples you
take out in every basket

00:13:16.230 --> 00:13:17.110
before you average them.

00:13:17.110 --> 00:13:18.300
And I'll go over that
in the next video.


Electrostatics (part 1): Introduction to Charge and Coulomb's Law

WEBVTT
Kind: captions
Language: en

00:00:00.216 --> 00:00:02.374
- [Voiceover] Let's talk a
little bit about one of the,

00:00:02.374 --> 00:00:03.893
what I find to be one of the more

00:00:03.893 --> 00:00:05.469
mysterious forces of the universe.

00:00:05.469 --> 00:00:06.947
Actually, I find all of the forces

00:00:06.947 --> 00:00:09.569
of the universe to be fairly mysterious.

00:00:09.569 --> 00:00:11.882
So, let's talk a little bit about Charge.

00:00:11.882 --> 00:00:14.631
And we've all heard of charge.

00:00:14.631 --> 00:00:16.089
Charge the battery.

00:00:16.089 --> 00:00:18.621
This particle has charge.

00:00:18.621 --> 00:00:19.607
But when you really think about it

00:00:19.607 --> 00:00:23.926
all charge means is that there's
this property called charge

00:00:23.926 --> 00:00:28.926
and we know that if something
contains a positive charge

00:00:29.944 --> 00:00:32.739
and calling it positive
is a little bit arbitrary.

00:00:32.739 --> 00:00:35.142
It's not like protons have a
little plus written on them.

00:00:35.142 --> 00:00:36.200
We can could've called them negative.

00:00:36.200 --> 00:00:38.342
But that when something
has a positive charge

00:00:38.342 --> 00:00:41.185
and when something else
has a positive charge

00:00:42.431 --> 00:00:45.297
that they repel each other.

00:00:45.297 --> 00:00:47.533
And we also know that
if I had something else,

00:00:47.533 --> 00:00:48.733
another particle that happened

00:00:48.733 --> 00:00:50.533
to havea negative charge

00:00:50.533 --> 00:00:52.773
and once again the word
negative being applied

00:00:52.773 --> 00:00:54.672
to this is completely arbitrary.

00:00:54.672 --> 00:00:57.973
They could've called it
blue charge and red charge

00:00:57.973 --> 00:00:59.932
but all we know is that
when another object

00:00:59.932 --> 00:01:03.122
has the other charge, in this
case we call it negative,

00:01:03.122 --> 00:01:05.835
that's going to be attracted
to a positive charge.

00:01:05.835 --> 00:01:07.644
All right, so, what do
we know about charge?

00:01:07.644 --> 00:01:10.284
Charge is a property that particles have

00:01:10.284 --> 00:01:12.023
and if you put enough particles together,

00:01:12.023 --> 00:01:14.558
I guess objects have
that property as well.

00:01:14.558 --> 00:01:16.825
So, it's just a property

00:01:16.825 --> 00:01:18.886
and that's a way of saying

00:01:18.886 --> 00:01:20.243
that I really don't know what it is

00:01:20.243 --> 00:01:23.181
and frankly, no one
fundamentally knows what it is.

00:01:23.181 --> 00:01:24.283
I actually didn't.

00:01:24.283 --> 00:01:26.651
No one really fundamentally
knows anything.

00:01:26.651 --> 00:01:28.598
But charge is a property of particles

00:01:28.598 --> 00:01:30.179
and objects just like mass.

00:01:30.179 --> 00:01:31.118
I mean, if you think about it,

00:01:31.118 --> 00:01:33.878
mass is just a property and to some degree

00:01:33.878 --> 00:01:35.776
it seems a little bit
more real than charge

00:01:35.776 --> 00:01:38.816
because in our brains are wired

00:01:38.816 --> 00:01:42.593
to in some way comprehend what mass is

00:01:42.593 --> 00:01:45.731
but we're probably more
comprehending weight and volume more

00:01:45.731 --> 00:01:48.330
than mass but we can think more
about that at another time.

00:01:48.330 --> 00:01:50.071
Charge is a little bit more abstract

00:01:50.071 --> 00:01:55.071
because before we started
rubbing amber into our hair

00:01:55.130 --> 00:01:58.091
we really didn't experience much charge

00:01:58.091 --> 00:02:00.561
unless we got struck by lightning.

00:02:00.561 --> 00:02:02.978
So, charge is a property that
particles or objects have

00:02:02.978 --> 00:02:05.335
and we know that there
are two types of charge,

00:02:05.335 --> 00:02:10.335
which we've arbitrarily
named positive and negative

00:02:12.184 --> 00:02:15.213
and we know that like charges...

00:02:16.413 --> 00:02:21.178
like charges repel

00:02:21.993 --> 00:02:24.513
and opposite charges...

00:02:24.513 --> 00:02:29.513
opposite charge attract
or unlike charges attract.

00:02:30.497 --> 00:02:31.217
All right.

00:02:31.217 --> 00:02:32.991
So, what can we do with this?

00:02:32.991 --> 00:02:37.173
Well, if we have this property
I think a useful thing

00:02:37.173 --> 00:02:39.673
to do would be to measure the property

00:02:39.673 --> 00:02:41.996
and so we came up with units

00:02:41.996 --> 00:02:46.130
and so the unity of charge
is called the Coulomb.

00:02:46.130 --> 00:02:47.769
It's named after a scientist

00:02:47.769 --> 00:02:50.862
in the late 1700s who played
around a lot with charge.

00:02:50.862 --> 00:02:53.541
You can look up more
about him on Wikipedia.

00:02:53.541 --> 00:02:56.813
But it's called the
Coulomb and the Coulomb,

00:02:56.813 --> 00:03:00.475
there's a bunch of definitions but I like

00:03:00.475 --> 00:03:03.701
to think of it in terms
of elementary particles.

00:03:03.701 --> 00:03:06.153
Just because to some
degree, well, unless you go

00:03:06.153 --> 00:03:07.532
and start going into quantum theory

00:03:07.532 --> 00:03:09.252
and start thinking about quirks and stuff.

00:03:09.252 --> 00:03:11.553
The elementary charge is the charge

00:03:11.553 --> 00:03:13.671
on a proton or a neutron.

00:03:13.671 --> 00:03:15.915
So, I'll go into more detail in the future

00:03:15.915 --> 00:03:18.614
on actually the structure
on atoms and whatever else,

00:03:18.614 --> 00:03:20.072
but let me just draw a little example.

00:03:20.072 --> 00:03:24.732
So, a atom tends to have
some neutrons in them

00:03:24.732 --> 00:03:27.136
which don't have this charge property.

00:03:27.136 --> 00:03:30.525
It'll have some protons in 'em

00:03:30.525 --> 00:03:32.992
which have a positive
charge and once again

00:03:32.992 --> 00:03:35.973
that's kind of arbitrarily
defined as positive.

00:03:35.973 --> 00:03:39.197
We could've called it it has a red charge.

00:03:39.197 --> 00:03:41.331
And then it has these
things floating around

00:03:41.331 --> 00:03:44.569
that are much lighter than,
the much, much, lighter

00:03:44.569 --> 00:03:48.431
than the, than the protons and
the neutrons in the nucleus.

00:03:48.431 --> 00:03:51.524
And these are called electrons.

00:03:51.524 --> 00:03:52.763
It's not even clear that they,

00:03:52.763 --> 00:03:54.202
that they're real objects.

00:03:54.202 --> 00:03:58.172
They're almost, like, energy but sometimes

00:03:58.172 --> 00:03:59.791
it's useful to view them as objects.

00:03:59.791 --> 00:04:02.316
Sometimes it's useful to view them as,

00:04:02.316 --> 00:04:03.669
well, not as objects and we'll go

00:04:03.669 --> 00:04:05.888
into all of that more later.

00:04:05.888 --> 00:04:08.105
But electrons have a negative charge.

00:04:08.105 --> 00:04:10.855
And the fundamental
unit of charge as far as

00:04:10.855 --> 00:04:12.557
we are concerned right now before we start

00:04:12.557 --> 00:04:14.036
talking about quarks and other

00:04:14.036 --> 00:04:17.086
potentially subatomic
particles is the charge

00:04:17.102 --> 00:04:20.932
in an electron or proton and
they have the exact same charge

00:04:20.947 --> 00:04:23.351
and that elementary
charge is denoted by E.

00:04:23.351 --> 00:04:26.391
And to be frank, I'm not
sure whether E stands

00:04:26.391 --> 00:04:28.834
for elementary or E stands for electron,

00:04:28.834 --> 00:04:32.191
but actually E is equal
to the charge of a proton,

00:04:32.191 --> 00:04:37.187
so it probably stands for
elementary charge of a proton

00:04:39.166 --> 00:04:43.487
and the charge of an electron
is the negative of this.

00:04:43.487 --> 00:04:47.601
So, negative E is the
charge of an electron.

00:04:47.601 --> 00:04:50.724
Charge of electron.

00:04:50.724 --> 00:04:53.495
But if we didn't care about size,

00:04:53.495 --> 00:04:56.741
about sign than the
magnitudes are the same.

00:04:56.741 --> 00:04:59.581
So, that's the fundamental as far as

00:04:59.581 --> 00:05:01.602
we know as so far in our physics.

00:05:01.602 --> 00:05:03.424
That's the fundamental charge.

00:05:03.424 --> 00:05:04.758
The fundamental unit of charge is just

00:05:04.758 --> 00:05:07.006
the charge in a proton or neutron.

00:05:07.006 --> 00:05:09.123
So, how does a Coulomb relate to that?

00:05:09.123 --> 00:05:12.483
Well, a Coulomb, which we'll denote by C

00:05:12.483 --> 00:05:16.151
is equal to, and this is a
bit of a arbitrary number,

00:05:16.151 --> 00:05:18.141
but when we start doing
things with electricity

00:05:18.141 --> 00:05:21.242
we'll see why the Coulomb
was defined the way it is.

00:05:21.242 --> 00:05:23.621
But a Coulomb is six point two four

00:05:23.621 --> 00:05:28.621
times ten to eighteenth Es.

00:05:29.673 --> 00:05:31.113
Or you could say six point two four

00:05:31.113 --> 00:05:35.653
times ten to the eighteenth
times the charge on an electron.

00:05:35.653 --> 00:05:37.394
Actually, times the charge on a proton.

00:05:37.394 --> 00:05:40.834
And then of course, in terms of magnitude.

00:05:40.834 --> 00:05:42.215
Because if I just say Coulomb

00:05:42.215 --> 00:05:43.695
I'm not really giving it direction.

00:05:43.695 --> 00:05:45.192
So, if you look at the other way around

00:05:45.192 --> 00:05:47.851
you could say the elementary
charge, the elementary charge,

00:05:47.851 --> 00:05:50.985
is equal to, at least its magnitude,

00:05:50.985 --> 00:05:55.985
one point six times ten to
the minus nineteen Coulombs.

00:05:56.743 --> 00:05:58.039
So, fair enough.

00:05:58.039 --> 00:06:01.170
This might be a useful number to memorize

00:06:01.170 --> 00:06:04.140
but it will usually be
provided for you in some way.

00:06:04.140 --> 00:06:05.133
So, what can do?

00:06:05.133 --> 00:06:07.253
We've, we say that these objects

00:06:07.253 --> 00:06:08.674
have this property called Charge.

00:06:08.674 --> 00:06:11.769
Like charges repel,
unlike charges attract.

00:06:11.769 --> 00:06:16.191
If we have enough of
these protons together

00:06:16.191 --> 00:06:18.304
then the whole object has charge.

00:06:18.304 --> 00:06:21.366
If we have more, let's
say, protons than electrons

00:06:21.366 --> 00:06:22.407
and we have a positive charge.

00:06:22.407 --> 00:06:23.785
If we have more electrons than protons,

00:06:23.785 --> 00:06:25.873
we have a negative charge.

00:06:25.873 --> 00:06:28.345
And we know that we've
defined this unit of charge

00:06:28.345 --> 00:06:29.946
called the Coulomb which is a bunch

00:06:29.946 --> 00:06:31.982
of the fundamental charge.

00:06:31.982 --> 00:06:33.204
So, let's play around with this

00:06:33.204 --> 00:06:36.233
and see if we can measure charge.

00:06:38.708 --> 00:06:41.134
So, in part of the initial,

00:06:41.134 --> 00:06:43.716
I guess we can call it
definition on what charge is

00:06:43.716 --> 00:06:48.112
I said that like charges repel.

00:06:48.112 --> 00:06:50.124
Right, like charges
repel, so if those, both

00:06:50.124 --> 00:06:53.488
of these are positive they're
going to repel each other.

00:06:53.488 --> 00:06:56.858
And unlike charges, if this
is negative, this is positive,

00:06:56.858 --> 00:06:59.184
they're gonna attract each other, right?

00:06:59.184 --> 00:07:02.231
So, by definition, if they
are moving each other,

00:07:02.231 --> 00:07:04.226
they're, these two particles are going

00:07:04.226 --> 00:07:05.745
to accelerate away from each other.

00:07:05.745 --> 00:07:06.465
These two particles are going

00:07:06.465 --> 00:07:08.664
to accelerate towards each other.

00:07:08.664 --> 00:07:10.944
The charge between these particles

00:07:10.944 --> 00:07:12.561
or the charge in each of these particules

00:07:12.561 --> 00:07:15.662
must be generating some
type of force, right?

00:07:15.662 --> 00:07:16.697
If there was no force being generated

00:07:16.697 --> 00:07:20.169
then they wouldn't repel
or attract each other.

00:07:20.169 --> 00:07:21.956
And this is where we get to Coulomb's law

00:07:21.956 --> 00:07:26.034
and this is why we name
charges after Coulomb.

00:07:26.049 --> 00:07:29.189
Coulomb figured out that the force

00:07:29.189 --> 00:07:32.168
between two charges is equal to,

00:07:32.168 --> 00:07:33.748
and this is going to be a vector quantity,

00:07:33.748 --> 00:07:36.228
and in about thirty seconds
I'll tell you which,

00:07:36.228 --> 00:07:37.948
what happens with the direction,

00:07:37.948 --> 00:07:42.948
is equal to some constant
times the first charge

00:07:43.218 --> 00:07:46.118
times the second charge

00:07:46.118 --> 00:07:49.769
divided by the distance
between them squared.

00:07:49.769 --> 00:07:51.072
And this is pretty neat because

00:07:51.072 --> 00:07:53.494
this looks an awful lot like,

00:07:53.494 --> 00:07:57.093
so, if we call this the
force, the electric force,

00:07:57.093 --> 00:08:00.185
that looks a lot like the
gravitational force equation.

00:08:00.185 --> 00:08:03.098
Let me write that down.

00:08:03.098 --> 00:08:06.676
Force from gravity between two masses is

00:08:06.676 --> 00:08:08.385
equal to the gravitational constant

00:08:08.385 --> 00:08:13.126
times M one times M two divided

00:08:13.126 --> 00:08:15.781
by the distance between them squared.

00:08:15.781 --> 00:08:17.522
So, so far the two forces that we covered,

00:08:17.522 --> 00:08:20.422
Gravity and now we're
covering electric force,

00:08:20.422 --> 00:08:22.864
and we'll eventually expand
this to electromagnetic force,

00:08:22.864 --> 00:08:25.822
it seems like they kind of act at distance

00:08:25.822 --> 00:08:27.122
in a similar way and both of these

00:08:27.122 --> 00:08:29.843
forces into gravity, in a vacuum.

00:08:29.843 --> 00:08:31.322
So, it doesn't matter if you have no air,

00:08:31.322 --> 00:08:34.323
if you have no substances
between the two particles,

00:08:34.323 --> 00:08:37.004
somehow they're communicating
with each other,

00:08:37.004 --> 00:08:38.775
which I find kind of amazing, right?

00:08:38.775 --> 00:08:41.656
You can have nothing
between these two particles

00:08:41.656 --> 00:08:44.253
but somehow this particle knows
that this particle is there

00:08:44.253 --> 00:08:46.173
and that particle knows
that that particle is there,

00:08:46.173 --> 00:08:49.175
and they start moving without having any,

00:08:49.175 --> 00:08:50.868
It's not like they have a
wire connecting to each other

00:08:50.868 --> 00:08:52.544
and someone's telling the other particle,

00:08:52.544 --> 00:08:54.104
"Hey, there's a particle
there, start moving."

00:08:54.104 --> 00:08:55.944
So, I don't know if you
find that as amazing as I do

00:08:55.944 --> 00:08:57.119
but think about it and you might.

00:08:57.119 --> 00:08:58.279
And it's just like gravity.

00:08:58.279 --> 00:09:00.301
I mean, the two masses
are in no way connected.

00:09:00.301 --> 00:09:01.640
They could be sitting in a vacuum

00:09:01.640 --> 00:09:04.099
but somehow they know that
the other particle is there

00:09:04.099 --> 00:09:06.649
and when we start learning about

00:09:06.649 --> 00:09:07.820
special relativity and all of that

00:09:07.820 --> 00:09:10.241
we'll learn that, oh,
maybe there's nothing there

00:09:10.241 --> 00:09:12.838
but maybe the masses are
somehow shaping the universe,

00:09:12.838 --> 00:09:14.409
and maybe that's happening with,

00:09:14.409 --> 00:09:16.704
with the electric charges as well.

00:09:16.704 --> 00:09:18.064
But all we know at this point

00:09:18.064 --> 00:09:20.046
is that we have these charges

00:09:20.046 --> 00:09:22.951
and that they exert a force on each other

00:09:22.951 --> 00:09:24.798
that's proportional to the product

00:09:24.798 --> 00:09:27.298
of their respective charges divided

00:09:27.298 --> 00:09:30.138
by the square of the
distance between them.

00:09:30.138 --> 00:09:32.781
And this constant right here

00:09:32.781 --> 00:09:36.991
that is, I always forget it, what was it?

00:09:36.991 --> 00:09:39.590
I think it's six point...

00:09:39.590 --> 00:09:42.543
I always forget what that constant is.

00:09:42.543 --> 00:09:45.746
It is nine times ten to the ninth.

00:09:45.746 --> 00:09:47.898
So, K, that's, it's rounded of course.

00:09:47.898 --> 00:09:51.432
That would amazing if it was exactly nine.

00:09:51.432 --> 00:09:55.262
Nine times ten to the ninth

00:09:55.262 --> 00:10:00.212
and the units are newton meters
squared per Coulomb squared,

00:10:03.143 --> 00:10:04.278
and why are the...

00:10:04.278 --> 00:10:06.198
Why are those the units?

00:10:06.198 --> 00:10:09.746
Well, pretty much because
at the end we have,

00:10:09.746 --> 00:10:10.999
we have Coulomb, Coulombs.

00:10:10.999 --> 00:10:14.077
We're gonna have Coulomb squared
divided by meters squared

00:10:14.077 --> 00:10:16.176
and we wanna finish with newtons

00:10:16.176 --> 00:10:18.156
so we wanna cancel out the Coulomb square

00:10:18.156 --> 00:10:18.987
by putting in the denominator.

00:10:18.987 --> 00:10:20.408
We wanna cancel out the meters square

00:10:20.408 --> 00:10:21.953
by putting in the numerator

00:10:21.953 --> 00:10:23.977
and then we'll end up with
the newtons to get the force.

00:10:23.977 --> 00:10:25.878
So, that's just where the units come from.

00:10:25.878 --> 00:10:27.757
So, given that let's figure out

00:10:27.757 --> 00:10:30.856
the force between two particles.

00:10:30.856 --> 00:10:33.696
So, let's say I have, I have,

00:10:33.696 --> 00:10:35.078
and I've spent ten minutes

00:10:35.078 --> 00:10:36.780
with a pretty long winded explanation

00:10:36.780 --> 00:10:39.299
but the actual problems you'll
see in your physics class

00:10:39.299 --> 00:10:42.678
are pretty straightforward
when it comes to Coulomb's law.

00:10:42.678 --> 00:10:47.434
So, they'll say, "Hey,
we have a positive..."

00:10:47.434 --> 00:10:50.725
We have a particle here that
has a positive charge of,

00:10:50.725 --> 00:10:54.274
let's call it, I don't know, plus,

00:10:55.397 --> 00:10:58.417
I don't know, let me
think of a good number.

00:10:58.417 --> 00:11:02.561
I don't know, plus five times ten

00:11:02.561 --> 00:11:06.000
to the minus three Coulombs.

00:11:06.000 --> 00:11:07.161
So, that's a positive charge.

00:11:07.161 --> 00:11:09.908
And then we have a negative charge here.

00:11:09.908 --> 00:11:13.955
And let's say that, I don't
know, how far will I make them.

00:11:13.955 --> 00:11:17.035
Let's say that they're half a meter apart,

00:11:17.035 --> 00:11:18.617
point five meters apart,

00:11:18.617 --> 00:11:21.930
and then I have a negative
charge here that is,

00:11:21.930 --> 00:11:26.917
let's say that this is ten minus ten times

00:11:28.332 --> 00:11:32.524
ten to the, I don't
know, minus two Coulombs.

00:11:32.524 --> 00:11:34.617
So, what is the force
between these two particles?

00:11:34.617 --> 00:11:37.276
So, if we just plug
them in to Coulomb's law

00:11:37.276 --> 00:11:41.635
we get the electric, the
force due to electricity,

00:11:41.635 --> 00:11:44.707
due to the electrical force
not due to electricity.

00:11:44.707 --> 00:11:46.086
We haven't done that yet.

00:11:46.086 --> 00:11:48.957
The static electric force
between those two particles

00:11:48.988 --> 00:11:52.615
is equal to the constant,
nine times ten to the ninth.

00:11:52.615 --> 00:11:57.226
Nine times ten to the
ninth times the first chage

00:11:57.226 --> 00:12:01.238
times five times ten to the minus three

00:12:01.238 --> 00:12:02.496
times the second charge.

00:12:02.496 --> 00:12:04.289
Let me do that in a different color.

00:12:04.289 --> 00:12:09.127
Times minus ten times
ten to the minus two.

00:12:09.127 --> 00:12:11.120
I just rewrote that although
you probably can't see it.

00:12:11.120 --> 00:12:15.266
Divided by the distance
squared, so point five squared.

00:12:15.266 --> 00:12:16.696
We just plugged into this formula.

00:12:16.696 --> 00:12:18.675
So, that equals, let me see.

00:12:18.675 --> 00:12:20.667
So, nine times point five times ten.

00:12:20.667 --> 00:12:23.049
I'm just gonna do the ten separately.

00:12:23.049 --> 00:12:24.182
So, that's times minus ten.

00:12:24.182 --> 00:12:27.923
So, it's minus, this is point five times

00:12:27.923 --> 00:12:32.923
minus ten is minus five, times
nine is minus forty five,

00:12:32.937 --> 00:12:35.516
and then ten to the ninth minus three,

00:12:35.516 --> 00:12:38.836
so ten to the sixth and then minus two.

00:12:38.836 --> 00:12:43.404
So, ten to the fourth
times ten to the fourth

00:12:43.404 --> 00:12:48.404
divided by, and what's point
five squared is point two five.

00:12:48.409 --> 00:12:50.317
All right, and this is equal to what?

00:12:50.317 --> 00:12:53.058
It's equal to four times is not one sixty

00:12:53.058 --> 00:12:57.563
plus this is equal to minus
one hundred and eighty

00:12:57.563 --> 00:13:02.428
times ten to the fourth newtons.

00:13:05.472 --> 00:13:07.146
And actually, this might
seem like a large number

00:13:07.146 --> 00:13:08.525
but these charges that I put here

00:13:08.525 --> 00:13:10.404
are actually fairly large charges

00:13:10.404 --> 00:13:13.102
and we'll, hopefully
you'll get a sense for

00:13:13.102 --> 00:13:14.686
what's a big or small charge later

00:13:14.686 --> 00:13:16.505
but these are reasonably large charges

00:13:16.505 --> 00:13:19.664
and so that's why there's
a relatively large force

00:13:19.664 --> 00:13:22.024
exerting between these two particles.

00:13:22.024 --> 00:13:23.567
Now, we got a negative number,

00:13:23.567 --> 00:13:24.078
so what does that mean?

00:13:24.078 --> 00:13:27.960
Well, we know that unlike
particles attract, right?

00:13:27.960 --> 00:13:29.600
Almost by definition.

00:13:29.600 --> 00:13:31.478
In this case we had a
positive and negative.

00:13:31.478 --> 00:13:33.517
So, when we have a negative...

00:13:33.517 --> 00:13:35.773
When we end up with a negative force

00:13:35.773 --> 00:13:36.844
when we use Coulomb's law,

00:13:36.844 --> 00:13:40.448
that means that the force
will draw the two particles

00:13:40.463 --> 00:13:42.880
to each other along their,

00:13:42.880 --> 00:13:44.015
the shortest distance between them.

00:13:44.015 --> 00:13:45.796
I mean, it's not it's gonna
make them go in a curve.

00:13:45.796 --> 00:13:47.536
It kinda makes sense.

00:13:47.536 --> 00:13:49.234
If we had a positive
there, then that means

00:13:49.234 --> 00:13:51.893
that the force was
repelling the two particles.

00:13:51.893 --> 00:13:52.974
And if you ever get confused,

00:13:52.974 --> 00:13:53.527
just think about it.

00:13:53.527 --> 00:13:55.488
If they've both negative,
they're gonna repel.

00:13:55.488 --> 00:13:57.707
If they're both positive,
they're going to attract.

00:13:57.707 --> 00:14:01.337
I will see you in the next video.