Sample variance

WEBVTT
Kind: captions
Language: en

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Let's say that you're
curious about people's TV

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watching habits.

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And in particular, how much TV
do people in the country watch?

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So what you are concerned
with, if we imagine

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the entire country-- and
we've already talked about--

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especially if we're talking
about a country like the United

00:00:15.960 --> 00:00:17.460
States, but pretty
much any country,

00:00:17.460 --> 00:00:18.940
is a very large population.

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In the United
States, we're talking

00:00:20.450 --> 00:00:24.030
about on the order of
300 million people.

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So ideally, if you could
somehow magically do it,

00:00:26.770 --> 00:00:31.400
you would survey or somehow
observe all 300 million people

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and take the mean of
how many hours of TV

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they watch on a given day.

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And then that will give you the
parameter, the population mean.

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But we've already
talked about, in a case

00:00:44.850 --> 00:00:46.570
like this, that's
a very impractical.

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Even if you tried to do
it, by the time you did it,

00:00:48.757 --> 00:00:51.340
your data might be stale because
some people might have passed

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away, other people
might have been born.

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Who knows what
might have happened.

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And so this is a truth
that is out there.

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There is a theoretical
population mean

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for the amount of the
average or the mean hours

00:01:06.050 --> 00:01:08.650
of TV watched per
day by Americans.

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There is a truth here at
any given point in time.

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It's just pretty much
impossible to come up

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with the exact answer, to
come up with this exact truth.

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But you don't give up.

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You say, well, maybe I don't
have to survey all 300 million

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or observe all 300 million.

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Instead, I'm just going
to observe a sample,

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right over here.

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And let's say, to make
the computation simple,

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you do a sample of six.

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And we'll talk about
later why six might not

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be as large of a sample
as you would like.

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But you survey how much
TV these folks watch.

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And you find one person who
watched 1 and 1/2 hours.

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Another person watched
2 and 1/2 hours.

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Another person watched 4 hours.

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And then you get one
person who watched 2 hours.

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And you get two people
who watched 1 hour each.

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So given this data
from your sample,

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what do you get as
your sample mean?

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Well, the sample mean, which
we would denote by lowercase

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x with a bar over
it, is just the sum

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of all of these divided by the
number of data points we have.

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So let's see we have 1.5
plus 2.5 plus 4 plus 2

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plus 1 plus 1.

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And all of that
divided by 6, which

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gives-- let's see, the numerator
1.5 plus 2.5 is 4, plus 4 is 8,

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plus 2 is 10, plus 2 more is 12.

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So it's going to
be 12 over 6, which

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is equal to 2 hours
of television.

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So at least for your
sample, you say,

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my sample mean is two
hours of television.

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It's an estimate.

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It's a statistic that
is trying to estimate

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this parameter, this thing
that's very hard to know.

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But it's our best shot.

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Maybe we get a better answer
if we get more data points.

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But this is we have so far.

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Now the next question
you ask yourself is,

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well, I don't want to just
estimate my population mean.

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I also want to estimate
another parameter.

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I also am interested in
estimating my population

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variance.

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So once again, since
we can't survey

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every one in the
population, this

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is pretty much
impossible to know.

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But we're going to attempt to
estimate of this parameter.

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We attempted to
estimate the mean.

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Now we will also
attempt to estimate

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this parameter, this
variance parameter.

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So how would you do it?

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Well, reasonable logic would
say, well, we maybe we'll

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do the same thing
with a sample as we

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would have done
with the population.

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When you're doing the
population variance,

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you would take each data
point in the population,

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find the distance between that
and the normal population mean,

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take the square of
that difference,

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and then add up all the
squares of those differences,

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and then divide by the number
of data points you have.

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So let's try that over here.

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So let's try to find-- take
each of these data points,

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and find the difference--
let me do that

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in a different color--
each of these data points,

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and find the difference
between that data point

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and our sample mean--
not the population mean,

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we don't know what
the population mean--

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the sample mean.

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So that's that first data
point plus the second data

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point-- so it's 4 minus 2
squared plus 1 minus 2 squared.

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And this is what
you would have done

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if you were taking a
population variance.

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If this was your
entire population,

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this is how you would you
find a population mean here,

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if this was your
entire population.

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And you find the
squared distances

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from each of those data
points and then divide

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by the number of data points.

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So let's just think
about this a little bit.

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1 minus 2 squared.

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Then you have 2.5
minus 2-- 2 being

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the sample mean-- squared.

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Let me see, this green color.

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Plus 2 minus 2 squared.

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Plus 1 minus 2 squared.

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And then maybe you would divide
by the number of data points

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that you have, where you have
the number of data points.

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So in this case,
we're dividing by 6.

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And what would we get
in this circumstance?

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Well, if we just do the
computation, 1.5 minus 2

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is negative 0.5.

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We square that.

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This becomes a positive 0.25.

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4 minus 2 squared is going
to be 2 squared, which is 4.

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1 minus 2 squared--
well, that's negative 1

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squared, which is just 1.

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2.5 minus 2 is 0.5
squared, is 0.25.

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2 minus 2 squared--
well, that's just 0.

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And then 1 minus 2 squared is
1, it's negative 1 squared.

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So we just get 1.

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And if we add all
of this up-- let

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me add the whole numbers first.

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4 plus 1 is 5, plus 1 is 6,
and then we have two 0.25s.

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So this is going to
be equal to 6.5--

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let me write this
in a neutral color.

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So this is going to be 6.5
over this 6 right over here.

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Well, there's a couple of
ways we could write this,

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but I'll just get
the calculator out

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and we can just calculate it.

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So 6.5 divided by 6
gets us-- if we round,

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it's approximately 1.08.

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So it's approximately
1.08 is this calculation.

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Now what we have
to think about is

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whether this is the best
calculation, whether this

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is the best estimate for the
population variance, given

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the data that we have.

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You can always argue that
we could have more data.

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But given the data we have,
is this the best calculation

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that we can make to estimate
the population variance?

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And I'll have you think
about that for a second.

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Well, it turns out
that this is close,

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this is close to the best
calculation, the best estimate

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that we can make,
given the data we have.

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And sometimes this will be
called the sample variance.

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But it's a particular
type of sample variance

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where we just divide by the
number of data points we have.

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And so people will write
just an n over here.

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So this is one way to define a
sample variance in an attempt

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to estimate our
population variance.

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But it turns out--
and in the next video

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I'll give you an
intuitive explanation

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of why it turns out this way.

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And then I would also like to
write a computer simulation

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that, at least
experimentally, makes

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you feel a little bit better.

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But it turns out, you're going
to get a better estimate--

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and it's a little bit weird
and voodooish at first

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when you first think
about it-- you're

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going to get a better estimate
for your population variance

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if you don't divide by
6, if you don't divide

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by the number of
data points you have

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but you divide by one less
than the number of data

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points you have.

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So how would we do that?

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And we can denote that
as sample variance.

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So when most people talk
about the sample variance,

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they're talking about
the sample variance

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where you do this calculation,
but instead of dividing by 6

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you were to divide by 5.

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You would divide by 5.

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So they would say you
divide by n minus 1.

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So what would we get
in those circumstances?

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Well, the top part is going
to be the exact same thing.

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We're going to get 6.5.

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But then our
denominator, our n is 6.

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We have 6 data points.

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But we're going to
divide by 1 less than 6.

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We're going to divide by 5.

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And 6.5 divided by
5 is equal to 1.3.

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So when we calculate our sample
variance with this technique,

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which is the more
mainstream technique--

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and it seems voodoo.

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Why are we dividing
by n minus 1,

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wherein for a population
variance we divide by n?

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But remember we're trying
to estimate the population

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variance.

00:09:02.720 --> 00:09:04.910
And it turns out that
this is a better estimate.

00:09:04.910 --> 00:09:08.310
Because this calculation
is underestimating

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what the population variance
is, this is a better estimate.

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We don't know for
sure what it is.

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These both could be way off.

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It could be just by chance
what we happen to sample.

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But over many samples--
and there's many ways

00:09:20.725 --> 00:09:22.100
to think about
it-- this is going

00:09:22.100 --> 00:09:23.700
to be a better calculation.

00:09:23.700 --> 00:09:26.350
It's going to give
you a better estimate.

00:09:26.350 --> 00:09:27.920
And so how would
we write this down?

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How would we write this down
with mathematical notation?

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Well, remember,
we're taking the sum.

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And we're taking each
of the data points.

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So we'll start with
the first data point

00:09:40.100 --> 00:09:42.480
all the way to the
nth data point.

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This lowercase n says that, hey,
we're looking at the sample.

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If I have an uppercase
N, that usually

00:09:47.200 --> 00:09:48.658
denotes that we're
trying to sum up

00:09:48.658 --> 00:09:50.190
everything in the population.

00:09:50.190 --> 00:09:53.350
Here we're looking at a
sample of size, lower case n.

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And we're taking each data
point, so each x sub i,

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and from it we're
subtracting the sample mean.

00:10:05.040 --> 00:10:06.810
And then we're squaring it.

00:10:06.810 --> 00:10:09.206
We're taking the sum of
the squared distances.

00:10:09.206 --> 00:10:11.580
And then we're dividing, not
by the number of data points

00:10:11.580 --> 00:10:16.187
we have, but by 1 less than the
number of data points we have.

00:10:16.187 --> 00:10:18.520
So this calculation, where
we just summed up all of this

00:10:18.520 --> 00:10:20.920
and then we divided
by 5, not by 6,

00:10:20.920 --> 00:10:27.500
this is the standard
definition of sample variance.

00:10:27.500 --> 00:10:28.820
So I'll leave you there.

00:10:28.820 --> 00:10:30.380
In the next video,
I will attempt

00:10:30.380 --> 00:10:34.650
to give you an intuition of
why we're dividing by n minus 1

00:10:34.650 --> 00:10:37.930
instead of dividing by n.


Variance of a population

WEBVTT
Kind: captions
Language: en

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Let's say I'm trying to judge
how many years of experience

00:00:04.760 --> 00:00:06.100
we have at the Khan Academy.

00:00:06.100 --> 00:00:09.050
Or on average, how many
years of experience we have.

00:00:09.050 --> 00:00:11.100
And in particular, the
particular type of average

00:00:11.100 --> 00:00:13.490
we'll focus on, is
the arithmetic mean.

00:00:13.490 --> 00:00:15.771
So I go and I survey
the folks there.

00:00:15.771 --> 00:00:18.020
And let's say this was when
Khan Academy was a smaller

00:00:18.020 --> 00:00:19.436
organization, when
there were only

00:00:19.436 --> 00:00:21.130
five people in the organization.

00:00:21.130 --> 00:00:24.240
And I find-- and I'm surveying
the entire population--

00:00:24.240 --> 00:00:27.237
so years of experience, the
entire population of Khan

00:00:27.237 --> 00:00:29.320
Academy, because that's
what I care about,

00:00:29.320 --> 00:00:32.930
years of experience at our
organization, at Khan Academy.

00:00:32.930 --> 00:00:34.940
And this was when
we had five people.

00:00:34.940 --> 00:00:36.870
And I were to go--
we're now 36 people,

00:00:36.870 --> 00:00:39.620
I don't want to date this video
too much-- but let's say I go,

00:00:39.620 --> 00:00:42.036
and I say, OK, there's one
person straight out of college,

00:00:42.036 --> 00:00:43.420
they have one year
of experience,

00:00:43.420 --> 00:00:44.920
or recently out of
college, somebody

00:00:44.920 --> 00:00:46.940
with three years of
experience, someone

00:00:46.940 --> 00:00:48.810
with five years of
experience, someone

00:00:48.810 --> 00:00:52.630
with seven years of experience,
and someone very experienced,

00:00:52.630 --> 00:00:56.100
or reasonably experienced,
with 14 years of experience.

00:00:56.100 --> 00:00:59.150
So based on this data point,
and this is our population,

00:00:59.150 --> 00:01:00.150
for years of experience.

00:01:00.150 --> 00:01:01.900
I'm assuming that we
only have five people

00:01:01.900 --> 00:01:04.340
in the organization,
at this point.

00:01:04.340 --> 00:01:07.930
What would be the
population mean

00:01:07.930 --> 00:01:09.710
for the years of experience?

00:01:09.710 --> 00:01:14.400
What is the mean years of
experience for my population?

00:01:14.400 --> 00:01:16.250
Well, we can just
calculate that.

00:01:16.250 --> 00:01:18.190
Our mean experience,
and I'm going

00:01:18.190 --> 00:01:20.166
to denote it with
mu, because we're

00:01:20.166 --> 00:01:21.540
talking about the
population now.

00:01:21.540 --> 00:01:24.590
This is a parameter
for the population.

00:01:24.590 --> 00:01:29.310
It's going to be equal to the
sum, from our first data point,

00:01:29.310 --> 00:01:34.200
so data point one all the way
to data point, in this case,

00:01:34.200 --> 00:01:39.520
data point five-- we have
five data points-- of each

00:01:39.520 --> 00:01:41.867
of-- so we're going to take
all, from the first data

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point, the second data
point, the third data

00:01:43.700 --> 00:01:45.033
point, all the way to the fifth.

00:01:45.033 --> 00:01:47.435
So this is going to be
equal to x1, plus x--

00:01:47.435 --> 00:01:49.893
and I'm going to divide it all
by the number of data points

00:01:49.893 --> 00:01:58.010
I have-- plus x2, plus x3, plus
x4, plus x sub 5, subscript 5.

00:01:58.010 --> 00:01:59.750
All of that over 5.

00:01:59.750 --> 00:02:01.950
And as we said, this is a
very fancy way of saying,

00:02:01.950 --> 00:02:05.400
I'm going to sum up
all of these things

00:02:05.400 --> 00:02:08.009
and then divide by the
number of things we have.

00:02:08.009 --> 00:02:09.039
So let's do that.

00:02:09.039 --> 00:02:11.540
Get the calculator out.

00:02:11.540 --> 00:02:18.595
So I'm going to add them
all up, 1 plus 3 plus 5--

00:02:18.595 --> 00:02:23.590
I really don't need a calculator
for this-- plus 7 plus 14.

00:02:23.590 --> 00:02:24.840
So that's five data points.

00:02:24.840 --> 00:02:27.150
And I'm going to divide by 5.

00:02:27.150 --> 00:02:28.930
And I get 6.

00:02:28.930 --> 00:02:31.470
So the population
mean, for years

00:02:31.470 --> 00:02:34.540
of experience at my
organization, is 6.

00:02:34.540 --> 00:02:36.280
6 years of experience.

00:02:36.280 --> 00:02:38.640
Well, that's, I
guess, interesting.

00:02:38.640 --> 00:02:40.730
But now I want to
ask another question.

00:02:40.730 --> 00:02:44.470
I want to get some
measure of how much spread

00:02:44.470 --> 00:02:45.900
there is around that mean.

00:02:45.900 --> 00:02:48.671
Or how much do the data
points vary around that mean.

00:02:48.671 --> 00:02:50.920
And obviously, I can give
someone all the data points.

00:02:50.920 --> 00:02:51.990
But instead, I actually
want to come up

00:02:51.990 --> 00:02:54.030
with a parameter that
somehow represents

00:02:54.030 --> 00:02:57.490
how much all of these things,
on average, are varying

00:02:57.490 --> 00:03:00.880
from this number right here.

00:03:00.880 --> 00:03:05.150
Or maybe I will call
that thing the variance.

00:03:05.150 --> 00:03:10.410
And so, what I do-- so the
variance-- and I will do--

00:03:10.410 --> 00:03:12.280
and this is a
population variance

00:03:12.280 --> 00:03:15.510
that I'm talking about, just
to be clear, it's a parameter.

00:03:15.510 --> 00:03:17.990
The population
variance I'm going

00:03:17.990 --> 00:03:22.040
to denote with the Greek letter
sigma, lowercase sigma-- this

00:03:22.040 --> 00:03:26.530
is capital sigma--
lowercase sigma squared.

00:03:26.530 --> 00:03:28.600
And I'm going to
say, well, I'm going

00:03:28.600 --> 00:03:32.310
to take the distance from each
of these points to the mean.

00:03:32.310 --> 00:03:35.064
And just so I get a positive
value, I'm going to square it.

00:03:35.064 --> 00:03:37.480
And then, I'm going to divide
by the number of data points

00:03:37.480 --> 00:03:38.046
that I have.

00:03:38.046 --> 00:03:39.420
So essentially,
I'm going to find

00:03:39.420 --> 00:03:41.370
the average squared distance.

00:03:41.370 --> 00:03:43.120
Now that might sound
very complicated,

00:03:43.120 --> 00:03:44.600
but let's actually work it out.

00:03:44.600 --> 00:03:48.930
So I'll take my first
data point and I

00:03:48.930 --> 00:03:51.624
will subtract our mean from it.

00:03:51.624 --> 00:03:53.540
So this is going to give
me a negative number.

00:03:53.540 --> 00:03:55.630
But if I square it, it's
going to be positive.

00:03:55.630 --> 00:03:57.005
So it's, essentially,
going to be

00:03:57.005 --> 00:03:59.930
the squared distance
between 1 and my mean.

00:03:59.930 --> 00:04:01.450
And then, to that,
I'm going to add

00:04:01.450 --> 00:04:03.840
the squared distance
between 3 and my mean.

00:04:07.520 --> 00:04:11.310
And to that, I'm going to add
the squared distance between 5

00:04:11.310 --> 00:04:13.974
and my mean.

00:04:13.974 --> 00:04:15.390
And since I'm
squaring, it doesn't

00:04:15.390 --> 00:04:17.603
matter if I do 5
minus 6, or 6 minus 5.

00:04:17.603 --> 00:04:19.769
When I square it, I'm going
to get a positive result

00:04:19.769 --> 00:04:20.990
regardless.

00:04:20.990 --> 00:04:22.780
And then, to that
I'm going to add

00:04:22.780 --> 00:04:25.590
the squared distance
between 7 and my mean.

00:04:25.590 --> 00:04:28.020
So 7 minus 6 squared.

00:04:28.020 --> 00:04:29.540
All of this, this
is my population

00:04:29.540 --> 00:04:31.750
mean that I'm finding
the difference between.

00:04:31.750 --> 00:04:36.880
And then, finally, the squared
difference between 14 and my

00:04:36.880 --> 00:04:37.380
mean.

00:04:42.042 --> 00:04:43.750
And then, I'm going
to find, essentially,

00:04:43.750 --> 00:04:45.600
the mean of these
squared distances.

00:04:45.600 --> 00:04:48.800
So I have five squared
distances right over here.

00:04:48.800 --> 00:04:56.450
So let me divide by 5.

00:04:56.450 --> 00:04:59.860
So what will I get when
I make this calculation,

00:04:59.860 --> 00:05:02.070
right over here?

00:05:02.070 --> 00:05:03.710
Well, let's figure this out.

00:05:03.710 --> 00:05:09.650
This is going to be equal
to 1 minus 6 is negative 5,

00:05:09.650 --> 00:05:13.200
negative 5 squared is 25.

00:05:13.200 --> 00:05:17.630
3 minus 6 is negative 3, now
if I square that, I get 9.

00:05:17.630 --> 00:05:21.820
5 minus 6 is negative 1, if I
square it, I get positive 1.

00:05:21.820 --> 00:05:26.440
7 minus 6 is 1, if I square
it, I get positive 1.

00:05:26.440 --> 00:05:31.240
And 14 minus 6 is 8, if
I square it, I get 64.

00:05:31.240 --> 00:05:36.335
And then, I'm going to
divide all of that by 5.

00:05:36.335 --> 00:05:38.080
And I don't need to
use a calculator,

00:05:38.080 --> 00:05:40.000
but I tend to make a
lot of careless mistakes

00:05:40.000 --> 00:05:43.140
when I do things
while making a video.

00:05:43.140 --> 00:05:53.860
So I get 25 plus 9 plus 1
plus 1 plus 64 divided by 5.

00:05:53.860 --> 00:05:55.630
So I get 20.

00:05:55.630 --> 00:05:59.570
So the average squared distance,
or the mean squared distance,

00:05:59.570 --> 00:06:05.452
from our population
mean is equal to 20.

00:06:05.452 --> 00:06:07.410
You may say, wait, these
things aren't 20 away.

00:06:07.410 --> 00:06:09.730
Remember, it's the
squared distance

00:06:09.730 --> 00:06:12.370
away from my population mean.

00:06:12.370 --> 00:06:13.954
So I squared each
of these things.

00:06:13.954 --> 00:06:15.620
I liked it, because
it made it positive.

00:06:15.620 --> 00:06:19.110
And we'll see later it has
other nice properties about it.

00:06:19.110 --> 00:06:20.760
Now the last thing
is, how can we

00:06:20.760 --> 00:06:22.680
represent this mathematically?

00:06:22.680 --> 00:06:26.640
We already saw that we know how
to represent a population mean,

00:06:26.640 --> 00:06:28.730
and a sample mean,
mathematically like this,

00:06:28.730 --> 00:06:31.480
and hopefully, we don't find
it that daunting anymore.

00:06:31.480 --> 00:06:34.080
But how would we do
the exact same thing?

00:06:34.080 --> 00:06:38.340
How would we denote what
we did, right over here?

00:06:38.340 --> 00:06:39.910
Well, let's just
think it through.

00:06:39.910 --> 00:06:44.440
We're just saying that
the population variance,

00:06:44.440 --> 00:06:50.895
we're taking the sum
of each-- so we're

00:06:50.895 --> 00:06:53.270
going to take each item, we'll
start with the first item.

00:06:53.270 --> 00:06:57.270
And we're going to go to the
n-th item in our population.

00:06:57.270 --> 00:06:59.582
We're talking about
a population here.

00:06:59.582 --> 00:07:01.040
And we're going to
take-- we're not

00:07:01.040 --> 00:07:04.040
going to just take the item,
this would just be the item--

00:07:04.040 --> 00:07:05.290
but we're going take the item.

00:07:05.290 --> 00:07:09.172
And from that, we're going to
subtract the population mean.

00:07:09.172 --> 00:07:10.630
We're going to
subtract this thing.

00:07:10.630 --> 00:07:12.300
We're going to
subtract this thing.

00:07:12.300 --> 00:07:13.620
We're going to square it.

00:07:13.620 --> 00:07:14.890
We're going to square it.

00:07:14.890 --> 00:07:16.190
So the way I've
written it right now,

00:07:16.190 --> 00:07:17.565
this would just
be the numerator.

00:07:17.565 --> 00:07:20.530
I've just taken the sum
of each of these things,

00:07:20.530 --> 00:07:23.180
the sum of the difference
between each data

00:07:23.180 --> 00:07:25.470
point and the population
mean and squared it.

00:07:25.470 --> 00:07:27.350
If I really want to get
the way I figure out

00:07:27.350 --> 00:07:28.891
this variance right
over here, I have

00:07:28.891 --> 00:07:33.815
to divide the whole thing by the
number of data points we have.

00:07:33.815 --> 00:07:35.190
So this might seem
very daunting,

00:07:35.190 --> 00:07:36.106
and very intimidating.

00:07:36.106 --> 00:07:39.300
But all it says is, take each
of your data points-- well, one,

00:07:39.300 --> 00:07:41.500
it says, figure out
your population mean.

00:07:44.520 --> 00:07:46.260
Figure that out first.

00:07:46.260 --> 00:07:51.230
And then, from each data
point, in your population,

00:07:51.230 --> 00:07:54.590
subtract out that
population mean, square it,

00:07:54.590 --> 00:07:56.510
take the sum of all
of those things,

00:07:56.510 --> 00:07:59.820
and then just divide by the
number of data points you have.

00:07:59.820 --> 00:08:03.867
And you will get your
population variance.


Inferring population mean from sample mean

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:02.660
Let's say you're
trying to design

00:00:02.660 --> 00:00:04.229
some type of a product for men.

00:00:04.229 --> 00:00:06.180
One that is somehow
based on their height.

00:00:06.180 --> 00:00:08.680
And the product is
for the United States.

00:00:08.680 --> 00:00:11.200
So ideally, you
would like to know

00:00:11.200 --> 00:00:15.030
the mean height of men
in the United States.

00:00:15.030 --> 00:00:16.340
Let me write this down.

00:00:27.580 --> 00:00:28.920
So how would you do that?

00:00:28.920 --> 00:00:31.390
And when I talk
about the mean, I'm

00:00:31.390 --> 00:00:33.285
talking about the
arithmetic mean.

00:00:33.285 --> 00:00:35.410
If I were to talk about
some other types of means--

00:00:35.410 --> 00:00:36.570
and there are other
types of means,

00:00:36.570 --> 00:00:38.150
like the geometric mean--
I would specify it.

00:00:38.150 --> 00:00:39.840
But when people just
say mean, they're

00:00:39.840 --> 00:00:42.330
usually talking about
the arithmetic mean.

00:00:42.330 --> 00:00:43.830
So how would you
go about finding

00:00:43.830 --> 00:00:46.470
the mean height of men
in the United States?

00:00:46.470 --> 00:00:52.510
Well, the obvious one
is, is you go and ask

00:00:52.510 --> 00:00:55.540
every or measure every
man in the United States.

00:00:55.540 --> 00:00:57.750
Take their height,
add them all together,

00:00:57.750 --> 00:00:59.300
and then divide by
the number of men

00:00:59.300 --> 00:01:01.050
there are in the United States.

00:01:01.050 --> 00:01:03.020
But the question
you'd ask yourself

00:01:03.020 --> 00:01:06.280
is whether that is practical.

00:01:06.280 --> 00:01:08.290
Because you have on
the order-- let's see,

00:01:08.290 --> 00:01:10.960
there's about 300 million
people in the United States.

00:01:10.960 --> 00:01:13.410
Roughly half of them
will be men, or at least

00:01:13.410 --> 00:01:18.200
they'll be male, and so
you will have 150 million,

00:01:18.200 --> 00:01:21.670
roughly 150 million men
in the United States.

00:01:21.670 --> 00:01:24.370
So if you wanted
the true mean height

00:01:24.370 --> 00:01:26.170
of all of the men
United States, you

00:01:26.170 --> 00:01:28.659
would have to somehow
survey-- or not even survey.

00:01:28.659 --> 00:01:30.450
You would have to be
able to go and measure

00:01:30.450 --> 00:01:32.850
all 150 million men.

00:01:32.850 --> 00:01:35.890
And even if you did try to do
that, by the time you're done,

00:01:35.890 --> 00:01:38.670
many of them might
have passed away,

00:01:38.670 --> 00:01:42.120
new men will have been born,
and so your data will go stale

00:01:42.120 --> 00:01:43.170
immediately.

00:01:43.170 --> 00:01:47.210
So it is seemingly impossible,
or almost impossible,

00:01:47.210 --> 00:01:49.770
to get the exact height of
every man in the United States

00:01:49.770 --> 00:01:52.000
in a snapshot of time.

00:01:52.000 --> 00:01:54.770
And so, instead, what you
do is say, well, look, OK,

00:01:54.770 --> 00:01:57.600
I can't get every man, but
maybe I can take a sample.

00:01:57.600 --> 00:02:00.020
I could take a sample of the
men in the United States.

00:02:00.020 --> 00:02:04.320
And I'm going to make an effort
that it's a random sample.

00:02:04.320 --> 00:02:06.670
I don't want to just go
sample 100 people who

00:02:06.670 --> 00:02:08.729
happen to play basketball,
or played basketball

00:02:08.729 --> 00:02:09.750
for their college.

00:02:09.750 --> 00:02:12.315
I don't want to go
sample 100 people who

00:02:12.315 --> 00:02:13.540
are volleyball players.

00:02:13.540 --> 00:02:15.920
I want to randomly sample.

00:02:15.920 --> 00:02:17.520
Maybe the first
person who comes out

00:02:17.520 --> 00:02:19.810
of the mall in a random
town, or in several towns,

00:02:19.810 --> 00:02:20.768
or something like that.

00:02:20.768 --> 00:02:22.860
Something that should
not be based in any way,

00:02:22.860 --> 00:02:25.840
or skewed in any way, by height.

00:02:25.840 --> 00:02:30.410
So you take a sample
and from that sample

00:02:30.410 --> 00:02:34.180
you can calculate a mean
of at least the sample.

00:02:34.180 --> 00:02:37.580
And you'll hope that that is
indicative of-- especially

00:02:37.580 --> 00:02:40.050
if this was a reasonably
random sample-- you'll

00:02:40.050 --> 00:02:45.600
hope that was indicative of the
mean of the entire population.

00:02:45.600 --> 00:02:49.330
And what you're going to
see in much of statistics

00:02:49.330 --> 00:02:54.675
it is all about using
information, using things

00:02:54.675 --> 00:02:56.480
that we can calculate
about a sample,

00:02:56.480 --> 00:02:59.010
to infer things
about a population.

00:02:59.010 --> 00:03:04.230
Because we can't directly
measure the entire population.

00:03:04.230 --> 00:03:07.200
So for example, let's say--
And if you're actually

00:03:07.200 --> 00:03:08.700
trying to do this,
I would recommend

00:03:08.700 --> 00:03:11.570
doing at least 100
data points, or 1,000,

00:03:11.570 --> 00:03:13.250
and later on we'll
talk about how

00:03:13.250 --> 00:03:15.850
you can think about whether
you've measured enough

00:03:15.850 --> 00:03:17.410
or how confident you can be.

00:03:17.410 --> 00:03:19.243
But let's just say
you're a little bit lazy,

00:03:19.243 --> 00:03:21.600
and you just sample five men.

00:03:21.600 --> 00:03:24.140
And so you get
their five heights.

00:03:24.140 --> 00:03:26.950
Let's say one is 6.2 feet.

00:03:26.950 --> 00:03:33.300
Let's say one is 5.5 feet-- 5.5
feet would be 5 foot, 6 inches.

00:03:33.300 --> 00:03:40.390
Let's say one ends
up being 5.75 feet.

00:03:40.390 --> 00:03:44.560
Another one is 6.3 feet.

00:03:44.560 --> 00:03:48.460
Another is 5.9 feet.

00:03:48.460 --> 00:03:50.230
Now, if these are
the ones that you

00:03:50.230 --> 00:03:53.150
happen to sample,
what would you get

00:03:53.150 --> 00:03:54.985
for the mean of this sample?

00:03:57.524 --> 00:03:58.940
Well let's get our
calculator out.

00:03:58.940 --> 00:04:12.690
And we get 6.2 plus 5.5
plus 5.75 plus 6.3 plus 5.9.

00:04:12.690 --> 00:04:15.760
The sum is 29.65.

00:04:15.760 --> 00:04:18.740
And then we want to divide
by the number of data points

00:04:18.740 --> 00:04:19.240
we have.

00:04:19.240 --> 00:04:20.430
So we have five data points.

00:04:20.430 --> 00:04:29.710
So let's divide 29.65 divided
by 5, and we get 5.93 feet.

00:04:29.710 --> 00:04:31.990
So here, our sample
mean-- and I'm

00:04:31.990 --> 00:04:35.400
going to denote it with an
x with a bar over it, is--

00:04:35.400 --> 00:04:42.320
and I already forgot
the number-- 5.93 feet.

00:04:42.320 --> 00:04:50.760
This is our sample mean, or,
if we want to make it clear,

00:04:50.760 --> 00:04:53.540
sample arithmetic mean.

00:04:53.540 --> 00:04:56.850
And when we're taking this
calculation based on a sample,

00:04:56.850 --> 00:04:58.830
and somehow we're
trying to estimate it

00:04:58.830 --> 00:05:03.560
for the entire population,
we call this right over here,

00:05:03.560 --> 00:05:05.060
we call it a statistic.

00:05:09.140 --> 00:05:11.230
Now, you might be saying,
well, what notation

00:05:11.230 --> 00:05:13.269
do we use if, somehow,
we are able to measure it

00:05:13.269 --> 00:05:14.060
for the population?

00:05:14.060 --> 00:05:16.310
Let's say we can't even
measure it for the population,

00:05:16.310 --> 00:05:19.920
but we at least want to denote
what the population mean is.

00:05:19.920 --> 00:05:22.730
Well if you want to do
that, the population mean

00:05:22.730 --> 00:05:25.700
is usually denoted by
the Greek letter mu.

00:05:31.140 --> 00:05:33.180
And so in a lot of
statistics, it's

00:05:33.180 --> 00:05:35.750
calculating a sample
mean in an attempt

00:05:35.750 --> 00:05:39.060
to estimate this thing
that you might not know,

00:05:39.060 --> 00:05:41.010
the population mean.

00:05:41.010 --> 00:05:43.569
And these calculations
on the entire population,

00:05:43.569 --> 00:05:45.110
sometimes you might
be able to do it.

00:05:45.110 --> 00:05:47.480
Oftentimes, you will
not be able to do it.

00:05:47.480 --> 00:05:48.860
These are called parameters.

00:05:52.830 --> 00:05:55.350
So what you're going to
find in much of statistics,

00:05:55.350 --> 00:05:59.080
it's all about calculating
statistics for a sample,

00:05:59.080 --> 00:06:01.560
finding these sample
statistics in order

00:06:01.560 --> 00:06:07.100
to estimate parameters
for an entire population.

00:06:07.100 --> 00:06:09.440
Now the last thing I want
to do is introduce you

00:06:09.440 --> 00:06:12.160
to some of the notation that
you might see in a statistics

00:06:12.160 --> 00:06:15.530
textbook that looks very
math-y and very difficult.

00:06:15.530 --> 00:06:17.736
But hopefully, after
the next few minutes,

00:06:17.736 --> 00:06:19.610
you'll appreciate that
it's really just doing

00:06:19.610 --> 00:06:21.920
exactly what we did here--
adding up the numbers

00:06:21.920 --> 00:06:24.059
and dividing by the
number of numbers you add.

00:06:24.059 --> 00:06:25.600
If you had to do
the population mean,

00:06:25.600 --> 00:06:26.470
it's the exact same thing.

00:06:26.470 --> 00:06:28.553
It's just many, many more
numbers in this context.

00:06:28.553 --> 00:06:30.310
You have to add up
150 million numbers

00:06:30.310 --> 00:06:32.540
and divide by 150 million.

00:06:32.540 --> 00:06:35.670
So how do mathematicians
talk about an operation

00:06:35.670 --> 00:06:37.890
like that-- adding
up a bunch of numbers

00:06:37.890 --> 00:06:40.550
and then dividing by
the number of numbers?

00:06:40.550 --> 00:06:44.030
Let's first think
about the sample mean,

00:06:44.030 --> 00:06:46.560
because that's where we
actually did the calculation.

00:06:46.560 --> 00:06:48.550
So a mathematician
might call each

00:06:48.550 --> 00:06:51.590
of these data points-- let's
say they'll call this first one

00:06:51.590 --> 00:06:56.270
right over here x sub 1.

00:06:56.270 --> 00:06:58.200
They'll call this one x sub 2.

00:06:58.200 --> 00:07:00.375
They'll call this one x sub 3.

00:07:00.375 --> 00:07:02.000
They'll call this
one-- when I say sub,

00:07:02.000 --> 00:07:05.780
I'm really saying subscript
1, subscript 2, subscript 3.

00:07:05.780 --> 00:07:08.090
They could call
this x subscript 4.

00:07:08.090 --> 00:07:10.590
They could call
this x subscript 5.

00:07:10.590 --> 00:07:13.640
And so if you had n of these
you would just keep going.

00:07:13.640 --> 00:07:16.910
x subscript 6, x
subscript seven, all

00:07:16.910 --> 00:07:19.360
the way to x subscript n.

00:07:19.360 --> 00:07:21.720
And so to take the
sum of all of these,

00:07:21.720 --> 00:07:26.440
they would denote it as let
me write it right over here.

00:07:26.440 --> 00:07:34.400
So they will say that the sample
mean is equal to the sum of all

00:07:34.400 --> 00:07:39.890
my x sub i's-- so the way
you can conceptualize it,

00:07:39.890 --> 00:07:42.040
these i's will change.

00:07:42.040 --> 00:07:44.350
In this case, the
i started at 1.

00:07:44.350 --> 00:07:47.180
The i's are going to
start at 1 until the size

00:07:47.180 --> 00:07:49.020
of our actual sample.

00:07:49.020 --> 00:07:50.860
So all the way until n.

00:07:50.860 --> 00:07:53.780
In this case n was equal to 5.

00:07:53.780 --> 00:07:57.280
So this is literally
saying this is

00:07:57.280 --> 00:08:02.170
equal to x sub 1 plus x
sub 2 plus x sub 3, all

00:08:02.170 --> 00:08:05.140
the way to the nth one.

00:08:05.140 --> 00:08:08.200
Once again, in this
case, we only had five.

00:08:08.200 --> 00:08:09.320
Now, are we done?

00:08:09.320 --> 00:08:12.189
Is this what the sample mean is?

00:08:12.189 --> 00:08:13.230
Well, no, we aren't done.

00:08:13.230 --> 00:08:16.680
We don't just add up
all of the data points.

00:08:16.680 --> 00:08:19.640
We then have to divide by
the number of data points

00:08:19.640 --> 00:08:20.220
there are.

00:08:28.060 --> 00:08:30.200
So this might look like
very fancy notation,

00:08:30.200 --> 00:08:32.720
but it's really just saying,
add up your data points

00:08:32.720 --> 00:08:36.570
and divide by the number
of data points you have.

00:08:36.570 --> 00:08:40.885
And this capital Greek letter
sigma literally means sum.

00:08:40.885 --> 00:08:45.950
Sum all of the x i's, from x
sub 1 all the way to x sub n,

00:08:45.950 --> 00:08:49.290
and then divide by the number
of data points you have.

00:08:49.290 --> 00:08:51.700
Now let's think about how we
would denote the same thing

00:08:51.700 --> 00:08:53.810
but, instead of for the
sample mean, doing it

00:08:53.810 --> 00:08:55.840
for the population mean.

00:08:55.840 --> 00:08:59.500
So the population mean,
they will denote it with mu,

00:08:59.500 --> 00:09:01.010
we already talked about that.

00:09:01.010 --> 00:09:03.520
And here, once again you're
going to take the sum,

00:09:03.520 --> 00:09:05.510
but this time it's going
to be the sum of all

00:09:05.510 --> 00:09:07.670
of the elements in
your population.

00:09:07.670 --> 00:09:12.607
So your x sub i's-- and you'll
still start at i equals 1.

00:09:12.607 --> 00:09:14.440
But it usually gets
denoted that, hey you're

00:09:14.440 --> 00:09:16.960
taking the whole
population, so they'll often

00:09:16.960 --> 00:09:19.670
put a capital N right over
here to somehow denote

00:09:19.670 --> 00:09:24.130
that this is a bigger number
than maybe this smaller n.

00:09:24.130 --> 00:09:26.200
But once again, we are not done.

00:09:26.200 --> 00:09:28.400
We have to divide by the
number of data points

00:09:28.400 --> 00:09:30.470
that we are actually summing.

00:09:30.470 --> 00:09:32.390
And so this, once
again, is the same thing

00:09:32.390 --> 00:09:40.260
as x sub 1 plus x sub 2 plus
x sub 3-- all the way to x sub

00:09:40.260 --> 00:09:45.820
capital N, all of that
divided by capital N.

00:09:45.820 --> 00:09:47.330
And once again,
in this situation,

00:09:47.330 --> 00:09:49.530
we found this practical.

00:09:49.530 --> 00:09:51.530
We found this impractical.

00:09:51.530 --> 00:09:53.500
We can debate whether
we took enough data

00:09:53.500 --> 00:09:56.170
points on our sample
mean right over here.

00:09:56.170 --> 00:09:58.220
But we're hoping that
it's at least somehow

00:09:58.220 --> 00:10:01.731
indicative of our
population mean.


Introduction to the unit circle

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:07.870
What I have attempted to
draw here is a unit circle.

00:00:07.870 --> 00:00:10.460
And the fact I'm
calling it a unit circle

00:00:10.460 --> 00:00:13.190
means it has a radius of 1.

00:00:13.190 --> 00:00:15.290
So this length from
the center-- and I

00:00:15.290 --> 00:00:19.230
centered it at the origin--
this length, from the center

00:00:19.230 --> 00:00:23.560
to any point on the
circle, is of length 1.

00:00:23.560 --> 00:00:26.890
So what would this coordinate
be right over there, right

00:00:26.890 --> 00:00:30.150
where it intersects
along the x-axis?

00:00:30.150 --> 00:00:33.910
Well, x would be
1, y would be 0.

00:00:33.910 --> 00:00:35.770
What would this
coordinate be up here?

00:00:35.770 --> 00:00:38.420
Well, we've gone 1
above the origin,

00:00:38.420 --> 00:00:40.450
but we haven't moved to
the left or the right.

00:00:40.450 --> 00:00:41.860
So our x value is 0.

00:00:41.860 --> 00:00:43.420
Our y value is 1.

00:00:43.420 --> 00:00:44.930
What about back here?

00:00:44.930 --> 00:00:47.970
Well, here our x value is -1.

00:00:47.970 --> 00:00:49.200
We've moved 1 to the left.

00:00:49.200 --> 00:00:52.140
And we haven't moved up or
down, so our y value is 0.

00:00:52.140 --> 00:00:53.520
And what about down here?

00:00:53.520 --> 00:00:58.670
Well, we've gone a unit
down, or 1 below the origin.

00:00:58.670 --> 00:01:00.800
But we haven't moved
in the xy direction.

00:01:00.800 --> 00:01:04.810
So our x is 0, and
our y is negative 1.

00:01:04.810 --> 00:01:08.230
Now, with that out of the way,
I'm going to draw an angle.

00:01:08.230 --> 00:01:10.510
And the way I'm going
to draw this angle--

00:01:10.510 --> 00:01:13.160
I'm going to define a
convention for positive angles.

00:01:13.160 --> 00:01:16.200
I'm going to say a
positive angle-- well,

00:01:16.200 --> 00:01:18.110
the initial side
of the angle we're

00:01:18.110 --> 00:01:20.500
always going to do along
the positive x-axis.

00:01:20.500 --> 00:01:24.320
So you can kind of view
it as the starting side,

00:01:24.320 --> 00:01:26.805
the initial side of an angle.

00:01:26.805 --> 00:01:31.090
And then to draw a positive
angle, the terminal side,

00:01:31.090 --> 00:01:35.820
we're going to move in a
counterclockwise direction.

00:01:35.820 --> 00:01:38.470
So positive angle means
we're going counterclockwise.

00:01:41.739 --> 00:01:43.780
And this is just the
convention I'm going to use,

00:01:43.780 --> 00:01:46.160
and it's also the convention
that is typically used.

00:01:46.160 --> 00:01:47.860
And so you can imagine
a negative angle

00:01:47.860 --> 00:01:51.950
would move in a
clockwise direction.

00:01:51.950 --> 00:01:54.602
So let me draw a positive angle.

00:01:54.602 --> 00:01:59.510
So a positive angle might
look something like this.

00:01:59.510 --> 00:02:00.770
This is the initial side.

00:02:00.770 --> 00:02:05.510
And then from that, I go in
a counterclockwise direction

00:02:05.510 --> 00:02:07.530
until I measure out the angle.

00:02:07.530 --> 00:02:09.250
And then this is
the terminal side.

00:02:09.250 --> 00:02:11.720
So this is a
positive angle theta.

00:02:11.720 --> 00:02:14.830
And what I want to do is
think about this point

00:02:14.830 --> 00:02:16.920
of intersection
between the terminal

00:02:16.920 --> 00:02:19.620
side of this angle
and my unit circle.

00:02:19.620 --> 00:02:23.200
And let's just say it has
the coordinates a comma b.

00:02:23.200 --> 00:02:25.370
The x value where
it intersects is a.

00:02:25.370 --> 00:02:28.550
The y value where
it intersects is b.

00:02:28.550 --> 00:02:31.330
And the whole point
of what I'm doing here

00:02:31.330 --> 00:02:33.720
is I'm going to see how
this unit circle might

00:02:33.720 --> 00:02:37.720
be able to help us extend our
traditional definitions of trig

00:02:37.720 --> 00:02:38.220
functions.

00:02:38.220 --> 00:02:39.970
And so what I want
to do is I want

00:02:39.970 --> 00:02:43.900
to make this theta part
of a right triangle.

00:02:43.900 --> 00:02:45.760
So to make it part
of a right triangle,

00:02:45.760 --> 00:02:49.660
let me drop an altitude
right over here.

00:02:49.660 --> 00:02:54.550
And let me make it clear that
this is a 90-degree angle.

00:02:54.550 --> 00:02:58.030
So this theta is part
of this right triangle.

00:02:58.030 --> 00:03:00.050
So let's see what
we can figure out

00:03:00.050 --> 00:03:03.350
about the sides of
this right triangle.

00:03:03.350 --> 00:03:05.630
So the first question
I have to ask

00:03:05.630 --> 00:03:11.740
you is, what is the
length of the hypotenuse

00:03:11.740 --> 00:03:15.130
of this right triangle that
I have just constructed?

00:03:15.130 --> 00:03:19.500
Well, this hypotenuse is just
a radius of a unit circle.

00:03:19.500 --> 00:03:21.250
The unit circle
has a radius of 1.

00:03:21.250 --> 00:03:24.770
So the hypotenuse has length 1.

00:03:24.770 --> 00:03:29.480
Now, what is the length of
this blue side right over here?

00:03:29.480 --> 00:03:33.530
You could view this as the
opposite side to the angle.

00:03:33.530 --> 00:03:38.570
Well, this height is
the exact same thing

00:03:38.570 --> 00:03:42.880
as the y-coordinate of
this point of intersection.

00:03:42.880 --> 00:03:46.380
So this height right over here
is going to be equal to b.

00:03:46.380 --> 00:03:48.370
The y-coordinate
right over here is b.

00:03:48.370 --> 00:03:50.690
This height is equal to b.

00:03:50.690 --> 00:03:54.060
Now, exact same logic--
what is the length

00:03:54.060 --> 00:03:55.570
of this base going to be?

00:03:55.570 --> 00:03:59.540
The base just of
the right triangle?

00:03:59.540 --> 00:04:02.780
Well, this is going
to be the x-coordinate

00:04:02.780 --> 00:04:04.082
of this point of intersection.

00:04:04.082 --> 00:04:05.540
If you were to drop
this down, this

00:04:05.540 --> 00:04:07.500
is the point x is equal to a.

00:04:07.500 --> 00:04:14.780
Or this whole length between the
origin and that is of length a.

00:04:14.780 --> 00:04:17.460
Now that we have
set that up, what

00:04:17.460 --> 00:04:24.580
is the cosine-- let me
use the same green-- what

00:04:24.580 --> 00:04:29.510
is the cosine of my angle going
to be in terms of a's and b's

00:04:29.510 --> 00:04:32.460
and any other numbers
that might show up?

00:04:32.460 --> 00:04:34.630
Well, to think
about that, we just

00:04:34.630 --> 00:04:36.310
need our soh cah toa definition.

00:04:36.310 --> 00:04:37.870
That's the only one we have now.

00:04:37.870 --> 00:04:41.380
We are actually in the process
of extending it-- soh cah toa

00:04:41.380 --> 00:04:44.370
definition of trig functions.

00:04:44.370 --> 00:04:46.770
And the cah part is what
helps us with cosine.

00:04:46.770 --> 00:04:48.880
It tells us that the
cosine of an angle

00:04:48.880 --> 00:04:51.220
is equal to the length
of the adjacent side

00:04:51.220 --> 00:04:52.750
over the hypotenuse.

00:04:52.750 --> 00:04:54.310
So what's this going to be?

00:04:54.310 --> 00:04:56.410
The length of the
adjacent side--

00:04:56.410 --> 00:04:59.280
for this angle, the
adjacent side has length a.

00:04:59.280 --> 00:05:02.050
So it's going to be
equal to a over-- what's

00:05:02.050 --> 00:05:03.580
the length of the hypotenuse?

00:05:03.580 --> 00:05:05.090
Well, that's just 1.

00:05:05.090 --> 00:05:09.740
So the cosine of theta
is just equal to a.

00:05:09.740 --> 00:05:11.320
Let me write this down again.

00:05:11.320 --> 00:05:15.450
So the cosine of theta
is just equal to a.

00:05:15.450 --> 00:05:18.070
It's equal to the x-coordinate
of where this terminal

00:05:18.070 --> 00:05:21.340
side of the angle
intersected the unit circle.

00:05:21.340 --> 00:05:25.320
Now let's think about
the sine of theta.

00:05:25.320 --> 00:05:28.600
And I'm going to do it in-- let
me see-- I'll do it in orange.

00:05:28.600 --> 00:05:34.286
So what's the sine
of theta going to be?

00:05:34.286 --> 00:05:37.330
Well, we just have to look at
the soh part of our soh cah toa

00:05:37.330 --> 00:05:38.300
definition.

00:05:38.300 --> 00:05:41.470
It tells us that sine is
opposite over hypotenuse.

00:05:41.470 --> 00:05:44.950
Well, the opposite
side here has length b.

00:05:44.950 --> 00:05:48.460
And the hypotenuse has length 1.

00:05:48.460 --> 00:05:54.890
So our sine of
theta is equal to b.

00:05:54.890 --> 00:05:57.540
So an interesting
thing-- this coordinate,

00:05:57.540 --> 00:06:00.050
this point where our
terminal side of our angle

00:06:00.050 --> 00:06:01.940
intersected the
unit circle, that

00:06:01.940 --> 00:06:05.344
point a, b-- we could
also view this as a

00:06:05.344 --> 00:06:07.363
is the same thing
as cosine of theta.

00:06:10.250 --> 00:06:16.300
And b is the same
thing as sine of theta.

00:06:16.300 --> 00:06:17.810
Well, that's interesting.

00:06:17.810 --> 00:06:20.650
We just used our soh
cah toa definition.

00:06:20.650 --> 00:06:24.190
Now, can we in some way use
this to extend soh cah toa?

00:06:24.190 --> 00:06:26.020
Because soh cah
toa has a problem.

00:06:26.020 --> 00:06:30.090
It works out fine if our angle
is greater than 0 degrees,

00:06:30.090 --> 00:06:31.840
if we're dealing with
degrees, and if it's

00:06:31.840 --> 00:06:33.370
less than 90 degrees.

00:06:33.370 --> 00:06:36.620
We can always make it
part of a right triangle.

00:06:36.620 --> 00:06:39.190
But soh cah toa
starts to break down

00:06:39.190 --> 00:06:43.130
as our angle is either 0 or
maybe even becomes negative,

00:06:43.130 --> 00:06:46.330
or as our angle is
90 degrees or more.

00:06:46.330 --> 00:06:48.890
You can't have a right triangle
with two 90-degree angles

00:06:48.890 --> 00:06:49.390
in it.

00:06:49.390 --> 00:06:50.750
It starts to break down.

00:06:50.750 --> 00:06:52.540
Let me make this clear.

00:06:52.540 --> 00:06:55.540
So sure, this is
a right triangle,

00:06:55.540 --> 00:06:57.040
so the angle is pretty large.

00:06:57.040 --> 00:06:59.322
I can make the angle even
larger and still have

00:06:59.322 --> 00:07:00.030
a right triangle.

00:07:00.030 --> 00:07:03.741
Even larger-- but I can never
get quite to 90 degrees.

00:07:03.741 --> 00:07:05.240
At 90 degrees, it's
not clear that I

00:07:05.240 --> 00:07:06.531
have a right triangle any more.

00:07:06.531 --> 00:07:07.979
It all seems to break down.

00:07:07.979 --> 00:07:09.520
And especially the
case, what happens

00:07:09.520 --> 00:07:12.380
when I go beyond 90 degrees.

00:07:12.380 --> 00:07:16.230
So let's see if we can
use what we said up here.

00:07:16.230 --> 00:07:19.230
Let's set up a new definition
of our trig functions

00:07:19.230 --> 00:07:23.290
which is really an
extension of soh cah toa

00:07:23.290 --> 00:07:25.050
and is consistent
with soh cah toa.

00:07:25.050 --> 00:07:28.530
Instead of defining cosine as
if I have a right triangle,

00:07:28.530 --> 00:07:30.780
and saying, OK, it's the
adjacent over the hypotenuse.

00:07:30.780 --> 00:07:32.660
Sine is the opposite
over the hypotenuse.

00:07:32.660 --> 00:07:34.560
Tangent is opposite
over adjacent.

00:07:34.560 --> 00:07:37.520
Why don't I just
say, for any angle,

00:07:37.520 --> 00:07:41.280
I can draw it in the unit circle
using this convention that I

00:07:41.280 --> 00:07:42.270
just set up?

00:07:42.270 --> 00:07:48.160
And let's just say that
the cosine of our angle

00:07:48.160 --> 00:07:56.270
is equal to the x-coordinate
where we intersect,

00:07:56.270 --> 00:07:59.402
where the terminal
side of our angle

00:07:59.402 --> 00:08:00.950
intersects the unit circle.

00:08:17.180 --> 00:08:20.990
And why don't we
define sine of theta

00:08:20.990 --> 00:08:25.350
to be equal to the
y-coordinate where the terminal

00:08:25.350 --> 00:08:28.060
side of the angle
intersects the unit circle?

00:08:28.060 --> 00:08:30.720
So essentially, for
any angle, this point

00:08:30.720 --> 00:08:34.237
is going to define cosine
of theta and sine of theta.

00:08:34.237 --> 00:08:36.070
And so what would be a
reasonable definition

00:08:36.070 --> 00:08:37.240
for tangent of theta?

00:08:37.240 --> 00:08:40.350
Well, tangent of theta--
even with soh cah toa--

00:08:40.350 --> 00:08:42.620
could be defined
as sine of theta

00:08:42.620 --> 00:08:45.750
over cosine of theta,
which in this case

00:08:45.750 --> 00:08:48.990
is just going to be the
y-coordinate where we intersect

00:08:48.990 --> 00:08:53.650
the unit circle over
the x-coordinate.

00:08:53.650 --> 00:08:56.310
In the next few videos,
I'll show some examples

00:08:56.310 --> 00:08:59.210
where we use the unit
circle definition to start

00:08:59.210 --> 00:09:03.194
evaluating some trig ratios.


Radian and degree conversion practice

WEBVTT
Kind: captions
Language: en

00:00:00.656 --> 00:00:03.140
Lets see if we can give ourselves a little bit

00:00:03.140 --> 00:00:05.460
of practice converting between radians and degrees

00:00:05.460 --> 00:00:07.974
and degrees and radians.

00:00:07.974 --> 00:00:10.944
And just as a review lets just remind ourselves the relationship

00:00:10.944 --> 00:00:14.125
and I always do this before I have to convert between the two.

00:00:14.125 --> 00:00:19.620
If I do one revolution of a circle how many radians is that going to be?

00:00:19.620 --> 00:00:26.727
Well we know one revolution of a circle is (2)(pi) radians

00:00:26.727 --> 00:00:32.522
and how many degrees is that if I do one revolution around a circle?

00:00:32.522 --> 00:00:34.975
Well we know that is 360 degrees.....

00:00:34.975 --> 00:00:40.224
I can either write it with the little degree symbol like that or I can write it just like that.

00:00:40.224 --> 00:00:43.013
And this is really enough information for us to think about

00:00:43.013 --> 00:00:45.040
how to convert between radians and degrees

00:00:45.040 --> 00:00:47.623
if we want to simplify this a little bit we can divide both sides

00:00:47.623 --> 00:00:49.061
by 2.

00:00:49.061 --> 00:00:55.826
And you can have (pi) radians are equal to 180 degrees

00:00:55.826 --> 00:00:57.277
or another way to think about it

00:00:57.277 --> 00:01:02.094
going halfway around a circle in radians is (pi) radians.

00:01:02.094 --> 00:01:06.438
The arc that subtends that angle is (pi) radiusssses

00:01:06.438 --> 00:01:09.645
and that's also 180 degrees.

00:01:09.645 --> 00:01:14.804
And if you wanna really think about, well how many degrees are there per radian

00:01:14.804 --> 00:01:18.095
you can divide both sides of this by (pi).

00:01:18.095 --> 00:01:20.373
So if you divide both sides of this by (pi),

00:01:20.373 --> 00:01:24.041
you would get one radian

00:01:24.041 --> 00:01:26.179
it would have to go from plural to singular

00:01:26.179 --> 00:01:29.789
one radian is equal to 180/pi degrees.

00:01:29.789 --> 00:01:35.155
So all I did is I divided both sides by (pi)

00:01:35.155 --> 00:01:37.929
and if you wanted to figure out how many radians

00:01:37.929 --> 00:01:41.405
are there per degree you can divide both sides by 180.

00:01:41.405 --> 00:01:45.277
So you would get pi/180 radians

00:01:45.277 --> 00:01:51.676
is equal to 1 degree.

00:01:51.676 --> 00:01:55.029
So this is....now I think we are ready to start converting.

00:01:55.029 --> 00:02:02.242
So lets convert 30 degrees....to radians.

00:02:02.242 --> 00:02:04.460
So lets think about it.

00:02:04.460 --> 00:02:09.039
I'm going to write it out and actually this might remind you of unit analysis

00:02:09.039 --> 00:02:11.359
that you might do when you first did unit conversion

00:02:11.359 --> 00:02:12.793
but it also works out here.

00:02:12.793 --> 00:02:14.298
So if I were to write 30 degrees

00:02:14.298 --> 00:02:16.415
and this is how my brain likes to work with it

00:02:16.415 --> 00:02:19.206
I like to write out the word degrees.

00:02:19.206 --> 00:02:22.211
Well I wanna convert to radians so I really want to figure out

00:02:22.211 --> 00:02:25.191
how many radians are there per degree?

00:02:25.191 --> 00:02:35.539
So let me write this down. I wanna figure out how many radians do we have per degree?

00:02:35.539 --> 00:02:37.930
And I haven't filled out how many that is

00:02:37.930 --> 00:02:39.809
but we see the units will cancel out

00:02:39.809 --> 00:02:42.441
if we have degrees times radians/degree

00:02:42.441 --> 00:02:45.706
the degrees will cancel out and I'll be just left with radians.

00:02:45.706 --> 00:02:49.475
If I multiply the number of degrees I have times the number of radians per degree

00:02:49.475 --> 00:02:51.160
we're gonna get radians.

00:02:51.160 --> 00:02:53.007
And hopefully that makes intuitive sense as well.

00:02:53.007 --> 00:02:54.495
And here we just have to think about

00:02:54.495 --> 00:02:59.288
well if I have...think of it this way, if I have

00:02:59.288 --> 00:03:04.237
pi radians how many degrees is that?

00:03:04.237 --> 00:03:08.909
Well that's 180 degrees, comes straight out of this right over here

00:03:08.909 --> 00:03:13.755
pi radians for every 180 degrees or pi/180 radians/degree.

00:03:13.755 --> 00:03:20.464
This is going to get us to...we're going to get 30 times pi/180

00:03:20.464 --> 00:03:26.188
30 times pi/180

00:03:26.188 --> 00:03:27.646
which will simplify to

00:03:27.646 --> 00:03:29.830
30/180 is 1/6

00:03:29.830 --> 00:03:33.161
so this is equal to pi/6

00:03:33.161 --> 00:03:35.626
let me write the units out

00:03:35.626 --> 00:03:43.727
this is 30 radians which is equal to pi/6 radians.

00:03:43.727 --> 00:03:45.411
Now lets go the other way

00:03:45.411 --> 00:03:53.697
lets think about if we have pi/3 radians

00:03:53.697 --> 00:03:55.758
and I wanna convert that to degrees.

00:03:55.758 --> 00:03:58.831
So what am I going to get if I convert that to degrees?

00:03:58.831 --> 00:04:00.591
Well here we're gonna want to figure out

00:04:00.591 --> 00:04:05.174
how many degrees are there per radian?

00:04:05.174 --> 00:04:07.313
One way to think about it is

00:04:07.313 --> 00:04:10.642
think about the pi and the 180

00:04:10.642 --> 00:04:15.043
for every 180 degrees you have pi radians.

00:04:15.043 --> 00:04:17.195
180 degrees/pi radians

00:04:17.195 --> 00:04:18.944
these are essentially the equivalent thing

00:04:18.944 --> 00:04:21.027
essentially you're just multiplying this quantity by 1

00:04:21.027 --> 00:04:22.876
but you're changing the units

00:04:22.876 --> 00:04:26.465
the radians cancel out and then the pi's cancel out

00:04:26.465 --> 00:04:29.706
and you're left with 180/3 degrees

00:04:29.706 --> 00:04:34.577
180/3 is 60 and we can either write out the word degrees or you can write degrees

00:04:34.577 --> 00:04:36.379
just like that.

00:04:36.379 --> 00:04:39.856
Now lets think about 45 degrees.

00:04:39.856 --> 00:04:44.464
So what about 45 degrees?

00:04:44.464 --> 00:04:49.061
And I'll write it like that just so you can figure it out with that notation as well.

00:04:49.061 --> 00:04:53.464
How many radians will this be equal to?

00:04:53.464 --> 00:04:59.186
Well once again we're gonna want to think about how many radians do we have per degree?

00:04:59.186 --> 00:05:01.994
So we're going to multiply this times

00:05:01.994 --> 00:05:06.765
well we know we have pi radians

00:05:06.765 --> 00:05:12.756
for every 180 degrees

00:05:12.756 --> 00:05:14.965
or we can even write it this way

00:05:14.965 --> 00:05:18.710
pi radians for every 180 degrees.

00:05:18.710 --> 00:05:21.044
And here this might be a little less intuitive

00:05:21.044 --> 00:05:23.289
the degrees cancel out and that's why I usually like to

00:05:23.289 --> 00:05:24.542
write out the word

00:05:24.542 --> 00:05:28.430
and you're left with 45 pi/180 radians.

00:05:28.430 --> 00:05:30.797
Actually let me write this with the words written out

00:05:30.797 --> 00:05:32.715
for me that's more intuitive when I'm thinking about it

00:05:32.715 --> 00:05:35.863
in terms of using the notation.

00:05:35.863 --> 00:05:51.577
So 45 degrees times, we have pi radians for every 180 degrees.

00:05:51.577 --> 00:05:54.072
So we are left with when you multiply

00:05:54.072 --> 00:05:59.207
45 times pi over 180

00:05:59.207 --> 00:06:01.695
the degrees have canceled out and you're just left with

00:06:01.695 --> 00:06:04.510
radians.

00:06:04.510 --> 00:06:06.278
Which is equal to what?

00:06:06.278 --> 00:06:09.798
45 is half of 90 which is half of 180 so this is 1/4

00:06:09.798 --> 00:06:15.489
this is equal to pi over 4 radians.

00:06:15.489 --> 00:06:19.675
Lets do 1 more over here.

00:06:19.675 --> 00:06:28.197
So lets say that we had negative pi/2 radians.

00:06:28.197 --> 00:06:32.676
What's that going to be in degrees?

00:06:32.676 --> 00:06:34.280
Well once again we have to figure out

00:06:34.280 --> 00:06:35.427
how many degrees

00:06:35.427 --> 00:06:37.243
are each of these radians.

00:06:37.243 --> 00:06:41.691
We know that there 180 degrees

00:06:41.691 --> 00:06:46.448
for every pi radians

00:06:46.448 --> 00:06:47.762
so we're gonna get

00:06:47.762 --> 00:06:49.182
the radians cancel out

00:06:49.182 --> 00:06:50.911
the pi's cancel out

00:06:50.911 --> 00:06:51.561
and so you have

00:06:51.561 --> 00:06:53.333
-180/2

00:06:53.333 --> 00:06:58.257
this is -90 degrees

00:06:58.257 --> 00:07:02.660
or we can write it as -90 degrees.

00:07:02.660 --> 00:07:04.315
Anyway hopefully you found that helpful

00:07:04.315 --> 00:07:06.513
and I'll do a couple more example problems here

00:07:06.513 --> 00:07:07.831
because the more example for this

00:07:07.831 --> 00:07:08.868
the better and hopefully it will become

00:07:08.868 --> 00:07:12.000
a little bit intuitive


Introduction to radians

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:04.420
You are by now probably
used to the idea

00:00:04.420 --> 00:00:08.240
of measuring angles in degrees.

00:00:08.240 --> 00:00:10.060
We use it in everyday language.

00:00:10.060 --> 00:00:13.504
We've done some examples
on this playlist where

00:00:13.504 --> 00:00:14.920
if you had an angle
like that, you

00:00:14.920 --> 00:00:19.010
might call that a
30-degree angle.

00:00:19.010 --> 00:00:21.970
If you have an
angle like this, you

00:00:21.970 --> 00:00:23.490
could call that a
90-degree angle.

00:00:23.490 --> 00:00:26.910
And we'd often use this
symbol, just like that.

00:00:26.910 --> 00:00:29.450
If you were to go
180 degrees, you'd

00:00:29.450 --> 00:00:34.680
essentially form
a straight line.

00:00:34.680 --> 00:00:36.520
Let me make these proper angles.

00:00:36.520 --> 00:00:40.260
If you go 360 degrees,
you have essentially

00:00:40.260 --> 00:00:41.600
done one full rotation.

00:00:41.600 --> 00:00:43.690
And if you watch figure
skating on the Olympics,

00:00:43.690 --> 00:00:45.190
and someone does a
rotation, they'll

00:00:45.190 --> 00:00:46.330
say, oh, they did a 360.

00:00:46.330 --> 00:00:49.370
Or especially in some
skateboarding competitions,

00:00:49.370 --> 00:00:50.604
and things like that.

00:00:50.604 --> 00:00:53.020
But the one thing to realize--
and it might not be obvious

00:00:53.020 --> 00:00:55.680
right from the get-go-- but
this whole notion of degrees,

00:00:55.680 --> 00:00:57.640
this is a
human-constructed system.

00:00:57.640 --> 00:01:00.066
This is not the only way
that you can measure angles.

00:01:00.066 --> 00:01:01.440
And if you think
about it, you'll

00:01:01.440 --> 00:01:05.762
say, well, why do we call a
full rotation 360 degrees?

00:01:05.762 --> 00:01:07.220
And there's some
possible theories.

00:01:07.220 --> 00:01:08.886
And I encourage you
to think about them.

00:01:08.886 --> 00:01:14.580
Why does 360 degrees show up in
our culture as a full rotation?

00:01:14.580 --> 00:01:16.610
Well, there's a couple
of theories there.

00:01:16.610 --> 00:01:18.766
One is ancient calendars.

00:01:18.766 --> 00:01:20.390
And even our calendar
is close to this,

00:01:20.390 --> 00:01:24.850
but ancient calendars were
based on 360 days in a year.

00:01:24.850 --> 00:01:27.160
Some ancient
astronomers observed

00:01:27.160 --> 00:01:33.010
that things seemed to move
1/360 of the sky per day.

00:01:33.010 --> 00:01:35.350
Another theory is the
ancient Babylonians

00:01:35.350 --> 00:01:37.450
liked equilateral
triangles a lot.

00:01:37.450 --> 00:01:40.290
And they had a base
60 number system.

00:01:40.290 --> 00:01:41.310
So they had 60 symbols.

00:01:41.310 --> 00:01:42.710
We only have 10.

00:01:42.710 --> 00:01:43.530
We have a base 10.

00:01:43.530 --> 00:01:44.570
They had 60.

00:01:44.570 --> 00:01:46.930
So in our system, we like
to divide things into 10.

00:01:46.930 --> 00:01:48.890
They probably liked to
divide things into 60.

00:01:48.890 --> 00:01:51.110
So if you had a circle,
and you divided it

00:01:51.110 --> 00:01:56.420
into 6 equilateral
triangles, and each

00:01:56.420 --> 00:01:59.450
of those equilateral triangles
you divided into 60 sections,

00:01:59.450 --> 00:02:01.560
because you have a
base 60 number system,

00:02:01.560 --> 00:02:05.774
then you might end
up with 360 degrees.

00:02:05.774 --> 00:02:07.440
What I want to think
about in this video

00:02:07.440 --> 00:02:10.069
is an alternate way
of measuring angles.

00:02:10.069 --> 00:02:12.110
And that alternate way--
even though it might not

00:02:12.110 --> 00:02:15.420
seem as intuitive to you from
the get-go-- in some ways

00:02:15.420 --> 00:02:18.620
is much more mathematically
pure than degrees.

00:02:18.620 --> 00:02:21.580
It's not based on these cultural
artifacts of base 60 number

00:02:21.580 --> 00:02:24.000
systems or
astronomical patterns.

00:02:24.000 --> 00:02:27.760
To some degree, an
alien on another planet

00:02:27.760 --> 00:02:30.530
would not use degrees,
especially if the degrees are

00:02:30.530 --> 00:02:33.190
motivated by these
astronomical phenomena.

00:02:33.190 --> 00:02:36.780
But they might use what we're
going to define as a radian.

00:02:36.780 --> 00:02:42.800
There's a certain degree
of purity here-- radians.

00:02:42.800 --> 00:02:46.980
So let's just cut to the chase
and define what a radian is.

00:02:46.980 --> 00:02:49.810
So let me draw a
circle here, my best

00:02:49.810 --> 00:02:52.380
attempt at drawing a circle.

00:02:52.380 --> 00:02:54.210
Not bad.

00:02:54.210 --> 00:02:58.150
And let me draw the
center of the circle.

00:02:58.150 --> 00:03:00.800
And now let me draw this radius.

00:03:00.800 --> 00:03:03.050
And let's say that this
radius-- and you might already

00:03:03.050 --> 00:03:06.030
notice the word radius is very
close to the word radians.

00:03:06.030 --> 00:03:07.810
And that's not a coincidence.

00:03:07.810 --> 00:03:12.650
So let's say that this circle
has a radius of length r.

00:03:12.650 --> 00:03:14.600
Now let's construct an angle.

00:03:14.600 --> 00:03:16.370
I'll call that angle theta.

00:03:16.370 --> 00:03:19.300
So let's construct
an angle theta.

00:03:19.300 --> 00:03:22.760
So let's call this angle
right over here theta.

00:03:22.760 --> 00:03:25.730
And let's just say, for
the sake of argument,

00:03:25.730 --> 00:03:30.160
that this angle is just
the exact right measure

00:03:30.160 --> 00:03:34.200
so that if you look at the
arc that subtends this angle--

00:03:34.200 --> 00:03:35.790
and that seems like
a very fancy word.

00:03:35.790 --> 00:03:37.600
But let me draw the angle.

00:03:37.600 --> 00:03:40.090
So if you were to draw
the angle-- so if you

00:03:40.090 --> 00:03:43.350
look at the arc that subtends
the angle, that's a fancy word.

00:03:43.350 --> 00:03:46.730
That's really just
talking about the arc

00:03:46.730 --> 00:03:50.370
along the circle that intersects
the two sides of the angles.

00:03:50.370 --> 00:03:55.610
So this arc right over here
subtends the angle theta.

00:03:55.610 --> 00:03:56.930
So let me write that down.

00:03:56.930 --> 00:04:03.310
Subtends this arc,
subtends angle theta.

00:04:03.310 --> 00:04:06.680
Let's say theta is the exact
right size so that this arc is

00:04:06.680 --> 00:04:09.810
also the same length as
the radius of the circle.

00:04:09.810 --> 00:04:13.450
So this arc is also of length r.

00:04:13.450 --> 00:04:18.200
So given that, if you were
defining a new type of angle

00:04:18.200 --> 00:04:20.980
measurement, and you
wanted to call it

00:04:20.980 --> 00:04:25.110
a radian, which is very close
to a radius, how many radians

00:04:25.110 --> 00:04:28.662
would you define
this angle to be?

00:04:28.662 --> 00:04:30.620
Well, the most obvious
one, if you kind of view

00:04:30.620 --> 00:04:32.730
a radian as another
way of saying

00:04:32.730 --> 00:04:35.750
radiuses, or I guess radii.

00:04:35.750 --> 00:04:38.470
Well, you say, look,
this is subtended

00:04:38.470 --> 00:04:41.510
by an arc of one radius.

00:04:41.510 --> 00:04:48.060
So why don't we call this right
over here one radian, which

00:04:48.060 --> 00:04:50.400
is exactly how a
radian is defined.

00:04:50.400 --> 00:04:53.910
When you have a circle, and you
have an angle of one radian,

00:04:53.910 --> 00:04:58.307
the arc that subtends it
is exactly one radius long.

00:04:58.307 --> 00:05:00.390
Which you can imagine might
be a little bit useful

00:05:00.390 --> 00:05:03.920
as we start to interpret more
and more types of circles.

00:05:03.920 --> 00:05:05.390
When you give a
degrees, you really

00:05:05.390 --> 00:05:06.880
have to do a little
bit of math and think

00:05:06.880 --> 00:05:08.505
about the circumference
and all of that

00:05:08.505 --> 00:05:11.400
to think about how many radiuses
are subtending that angle.

00:05:11.400 --> 00:05:16.420
Here, the angle in radians
tells you exactly the arc length

00:05:16.420 --> 00:05:18.210
that is subtending the angles.

00:05:18.210 --> 00:05:20.750
So let's do a couple of
thought experiments here.

00:05:20.750 --> 00:05:26.790
So given that, what would
be the angle in radians

00:05:26.790 --> 00:05:31.370
if we were to go-- so let
me draw another circle here.

00:05:38.015 --> 00:05:41.660
So that's the center, and
we'll start right over there.

00:05:41.660 --> 00:05:45.510
So what would happen if I
had an angle-- if I wanted

00:05:45.510 --> 00:05:49.000
to measure in radians, what
angle would this be in radians?

00:05:49.000 --> 00:05:51.764
And you could almost
think of it as radiuses.

00:05:51.764 --> 00:05:52.930
So what would that angle be?

00:05:52.930 --> 00:05:55.600
Going one full
revolution in degrees,

00:05:55.600 --> 00:05:57.610
that would be 360 degrees.

00:05:57.610 --> 00:06:01.320
Based on this definition,
what would this be in radians?

00:06:01.320 --> 00:06:05.300
Well, let's think about the
arc that subtends this angle.

00:06:05.300 --> 00:06:09.290
The arc that subtends this angle
is the entire circumference

00:06:09.290 --> 00:06:10.130
of this circle.

00:06:19.290 --> 00:06:21.260
Well, what's the
circumference of a circle

00:06:21.260 --> 00:06:24.380
in terms of radiuses?

00:06:24.380 --> 00:06:28.100
So if this has length r,
if the radius is length r,

00:06:28.100 --> 00:06:31.170
what's the circumference of
the circle in terms of r?

00:06:31.170 --> 00:06:31.980
Well, we know that.

00:06:31.980 --> 00:06:33.470
That's going to be 2 pi r.

00:06:36.190 --> 00:06:41.220
So going back to this
angle, the length

00:06:41.220 --> 00:06:45.870
of the arc that subtends this
angle is how many radiuses?

00:06:45.870 --> 00:06:47.760
Well, it's 2 pi radiuses.

00:06:47.760 --> 00:06:50.040
It's 2 pi times r.

00:06:50.040 --> 00:06:54.220
So this angle right
over here, I'll

00:06:54.220 --> 00:06:57.492
call this a different-- well,
let's call this angle x.

00:06:57.492 --> 00:07:03.200
x in this case is going
to be 2 pi radians.

00:07:03.200 --> 00:07:08.080
And it is subtended by an
arc length of 2 pi radiuses.

00:07:08.080 --> 00:07:09.830
If the radius was
one unit, then this

00:07:09.830 --> 00:07:14.460
would be 2 pi times
1, 2 pi radiuses.

00:07:14.460 --> 00:07:16.970
So given that,
let's start to think

00:07:16.970 --> 00:07:21.610
about how we can convert
between radians and degrees,

00:07:21.610 --> 00:07:23.520
and vice versa.

00:07:23.520 --> 00:07:26.330
If I were to have-- and we
can just follow up over here.

00:07:26.330 --> 00:07:28.650
If we do one full
revolution-- that is,

00:07:28.650 --> 00:07:37.986
2 pi radians-- how many degrees
is this going to be equal to?

00:07:37.986 --> 00:07:39.110
Well, we already know this.

00:07:39.110 --> 00:07:42.020
A full revolution in
degrees is 360 degrees.

00:07:45.341 --> 00:07:47.340
Well, I could either write
out the word degrees,

00:07:47.340 --> 00:07:49.580
or I can use this little
degree notation there.

00:07:49.580 --> 00:07:51.784
Actually, let me write
out the word degrees.

00:07:51.784 --> 00:07:53.950
It might make things a
little bit clearer that we're

00:07:53.950 --> 00:07:56.300
kind of using units
in both cases.

00:07:56.300 --> 00:07:58.300
Now, if we wanted to
simplify this a little bit,

00:07:58.300 --> 00:08:00.010
we could divide both sides by 2.

00:08:00.010 --> 00:08:02.680
In which case, on
the left-hand side,

00:08:02.680 --> 00:08:07.760
we would get pi radians would
be equal to how many degrees?

00:08:07.760 --> 00:08:09.810
Well, it would be
equal to 180 degrees.

00:08:13.770 --> 00:08:19.950
And I could write it that way,
or I could write it that way.

00:08:19.950 --> 00:08:22.530
And you see over here,
this is 180 degrees.

00:08:22.530 --> 00:08:26.790
And you also see if you were
to draw a circle around here,

00:08:26.790 --> 00:08:30.340
we've gone halfway
around the circle.

00:08:30.340 --> 00:08:33.020
So the arc length, or the
arc that subtends the angle,

00:08:33.020 --> 00:08:34.220
is half the circumference.

00:08:34.220 --> 00:08:37.850
Half the circumference
is pi radiuses.

00:08:37.850 --> 00:08:39.830
So we call this pi radians.

00:08:39.830 --> 00:08:42.750
Pi radians is 180 degrees.

00:08:42.750 --> 00:08:45.750
And from this, we can
come up with conversions.

00:08:45.750 --> 00:08:50.470
So one radian would
be how many degrees?

00:08:50.470 --> 00:08:52.250
Well, to do that,
we would just have

00:08:52.250 --> 00:08:54.412
to divide both sides by pi.

00:08:54.412 --> 00:08:55.870
And on the left-hand
side, you'd be

00:08:55.870 --> 00:08:58.120
left with 1-- I'll just
write it singular now.

00:08:58.120 --> 00:09:00.534
1 radian is equal to-- I'm
just dividing both sides.

00:09:00.534 --> 00:09:02.950
Let me make it clear what I'm
doing here, just to show you

00:09:02.950 --> 00:09:04.610
this isn't some voodoo.

00:09:04.610 --> 00:09:08.380
So I'm just dividing
both sides by pi here.

00:09:08.380 --> 00:09:10.574
On the left-hand side,
you're left with 1.

00:09:10.574 --> 00:09:11.990
And on the right-hand
side, you're

00:09:11.990 --> 00:09:14.580
left with 180/pi degrees.

00:09:14.580 --> 00:09:23.732
So 1 radian is equal to
180/pi degrees, which

00:09:23.732 --> 00:09:25.440
is starting to make
it an interesting way

00:09:25.440 --> 00:09:26.910
to convert them.

00:09:26.910 --> 00:09:28.920
Let's think about
it the other way.

00:09:28.920 --> 00:09:32.710
If I were to have 1 degree,
how many radians is that?

00:09:32.710 --> 00:09:34.690
Well, let's start
off with-- let me

00:09:34.690 --> 00:09:37.240
rewrite this thing over here.

00:09:37.240 --> 00:09:46.595
We said pi radians is
equal to 180 degrees.

00:09:49.605 --> 00:09:51.230
So now we want to
think about 1 degree.

00:09:51.230 --> 00:09:52.580
So let's solve for 1 degree.

00:09:52.580 --> 00:09:56.540
1 degree, we can divide
both sides by 180.

00:09:56.540 --> 00:10:01.790
We are left with pi/180
radians is equal to 1 degree.

00:10:01.790 --> 00:10:10.450
So pi/180 radians is
equal to 1 degree.

00:10:10.450 --> 00:10:12.154
This might seem
confusing and daunting.

00:10:12.154 --> 00:10:13.570
And it was for me
the first time I

00:10:13.570 --> 00:10:14.690
was exposed to this,
especially because we're not

00:10:14.690 --> 00:10:16.640
exposed to this in
our everyday life.

00:10:16.640 --> 00:10:18.890
But what we're going to see
over the next few examples

00:10:18.890 --> 00:10:21.560
is that as long as we keep
in mind this whole idea

00:10:21.560 --> 00:10:24.160
that 2 pi radians is
equal to 360 degrees,

00:10:24.160 --> 00:10:27.250
or that pi radians is equal to
180 degrees, which is the two

00:10:27.250 --> 00:10:29.160
things that I do
keep in my mind.

00:10:29.160 --> 00:10:31.299
We can always re-derive
these two things.

00:10:31.299 --> 00:10:32.840
You might say, hey,
how do I remember

00:10:32.840 --> 00:10:36.150
if it's pi/180 or 180/pi
to convert the two things?

00:10:36.150 --> 00:10:39.150
Well, just remember, which
is hopefully intuitive,

00:10:39.150 --> 00:10:42.550
that 2 pi radians is
equal to 360 degrees.

00:10:42.550 --> 00:10:45.180
And we'll work through a bunch
of examples in the next video,

00:10:45.180 --> 00:10:46.580
to just make sure
that we're used

00:10:46.580 --> 00:10:50.130
to converting one
way or the other.


Statistics intro: Mean, median, and mode

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:06.650
We will now begin our journey
into the world of statistics,

00:00:06.650 --> 00:00:09.750
which is really a way
to understand or get

00:00:09.750 --> 00:00:11.520
our head around data.

00:00:11.520 --> 00:00:14.670
So statistics is all about data.

00:00:14.670 --> 00:00:19.000
And as we begin our journey
into the world of statistics,

00:00:19.000 --> 00:00:20.610
we will be doing
a lot of what we

00:00:20.610 --> 00:00:23.210
can call descriptive statistics.

00:00:23.210 --> 00:00:25.470
So if we have a bunch
of data, and if we

00:00:25.470 --> 00:00:27.990
want to tell something
about all of that data

00:00:27.990 --> 00:00:29.890
without giving them
all of the data,

00:00:29.890 --> 00:00:33.870
can we somehow describe it
with a smaller set of numbers?

00:00:33.870 --> 00:00:35.720
So that's what we're
going to focus on.

00:00:35.720 --> 00:00:37.360
And then once we
build our toolkit

00:00:37.360 --> 00:00:39.260
on the descriptive
statistics, then we

00:00:39.260 --> 00:00:41.710
can start to make
inferences about that data,

00:00:41.710 --> 00:00:44.200
start to make conclusions,
start to make judgments.

00:00:44.200 --> 00:00:49.430
And we'll start to do a lot
of inferential statistics,

00:00:49.430 --> 00:00:51.160
make inferences.

00:00:51.160 --> 00:00:53.110
So with that out of
the way, let's think

00:00:53.110 --> 00:00:56.390
about how we can describe data.

00:00:56.390 --> 00:01:00.710
So let's say we have
a set of numbers.

00:01:00.710 --> 00:01:02.360
We can consider this to be data.

00:01:02.360 --> 00:01:04.580
Maybe we're measuring
the heights of our plants

00:01:04.580 --> 00:01:05.740
in our garden.

00:01:05.740 --> 00:01:07.400
And let's say we
have six plants.

00:01:07.400 --> 00:01:13.870
And the heights are 4 inches,
3 inches, 1 inch, 6 inches,

00:01:13.870 --> 00:01:17.990
and another one's 1 inch,
and another one is 7 inches.

00:01:17.990 --> 00:01:20.934
And let's say someone just
said-- in another room, not

00:01:20.934 --> 00:01:22.350
looking at your
plants, just said,

00:01:22.350 --> 00:01:24.657
well, you know, how
tall are your plants?

00:01:24.657 --> 00:01:26.240
And they only want
to hear one number.

00:01:26.240 --> 00:01:30.560
They want to somehow
have one number that

00:01:30.560 --> 00:01:33.410
represents all of these
different heights of plants.

00:01:33.410 --> 00:01:36.580
How would you do that?

00:01:36.580 --> 00:01:38.810
Well, you'd say, well,
how can I find something

00:01:38.810 --> 00:01:40.990
that-- maybe I want
a typical number.

00:01:40.990 --> 00:01:44.060
Maybe I want some number that
somehow represents the middle.

00:01:44.060 --> 00:01:46.250
Maybe I want the
most frequent number.

00:01:46.250 --> 00:01:48.830
Maybe I want the number
that somehow represents

00:01:48.830 --> 00:01:51.270
the center of all
of these numbers.

00:01:51.270 --> 00:01:53.220
And if you said any
of those things,

00:01:53.220 --> 00:01:55.189
you would actually have
done the same things

00:01:55.189 --> 00:01:57.730
that the people who first came
up with descriptive statistics

00:01:57.730 --> 00:01:58.230
said.

00:01:58.230 --> 00:02:00.150
They said, well,
how can we do it?

00:02:00.150 --> 00:02:04.960
And we'll start by thinking
of the idea of average.

00:02:04.960 --> 00:02:07.610
And in every day
terminology, average

00:02:07.610 --> 00:02:09.720
has a very particular
meaning, as we'll see.

00:02:09.720 --> 00:02:11.570
When many people
talk about average,

00:02:11.570 --> 00:02:13.070
they're talking
about the arithmetic

00:02:13.070 --> 00:02:14.960
mean, which we'll see shortly.

00:02:14.960 --> 00:02:18.100
But in statistics, average
means something more general.

00:02:18.100 --> 00:02:22.980
It really means
give me a typical,

00:02:22.980 --> 00:02:29.810
or give me a middle number,
or-- and these are or's.

00:02:29.810 --> 00:02:31.930
And really it's
an attempt to find

00:02:31.930 --> 00:02:33.490
a measure of central tendency.

00:02:38.550 --> 00:02:40.560
So once again, you have
a bunch of numbers.

00:02:40.560 --> 00:02:42.970
You're somehow trying
to represent these

00:02:42.970 --> 00:02:45.840
with one number we'll call
the average, that's somehow

00:02:45.840 --> 00:02:49.130
typical, or middle,
or the center somehow

00:02:49.130 --> 00:02:50.450
of these numbers.

00:02:50.450 --> 00:02:54.110
And as we'll see, there's
many types of averages.

00:02:54.110 --> 00:02:56.690
The first is the one that you're
probably most familiar with.

00:02:56.690 --> 00:02:58.398
It's the one-- and
people talk about hey,

00:02:58.398 --> 00:03:00.840
the average on this exam
or the average height.

00:03:00.840 --> 00:03:02.970
And that's the arithmetic mean.

00:03:02.970 --> 00:03:05.470
Just let me write it in.

00:03:05.470 --> 00:03:13.100
I'll write in yellow,
arithmetic mean.

00:03:13.100 --> 00:03:16.010
When arithmetic is a noun,
we call it arithmetic.

00:03:16.010 --> 00:03:19.960
When it's an adjective like
this, we call it arithmetic,

00:03:19.960 --> 00:03:21.620
arithmetic mean.

00:03:21.620 --> 00:03:25.300
And this is really just the
sum of all the numbers divided

00:03:25.300 --> 00:03:28.180
by-- this is a human-constructed
definition that we've

00:03:28.180 --> 00:03:31.630
found useful-- the sum of
all these numbers divided

00:03:31.630 --> 00:03:34.460
by the number of
numbers we have.

00:03:34.460 --> 00:03:36.830
So given that, what
is the arithmetic mean

00:03:36.830 --> 00:03:39.114
of this data set?

00:03:39.114 --> 00:03:40.280
Well, let's just compute it.

00:03:40.280 --> 00:03:46.160
It's going to be 4 plus
3 plus 1 plus 6 plus 1

00:03:46.160 --> 00:03:51.210
plus 7 over the number
of data points we have.

00:03:51.210 --> 00:03:53.210
So we have six data points.

00:03:53.210 --> 00:03:54.860
So we're going to divide by 6.

00:03:54.860 --> 00:04:01.840
And we get 4 plus 3 is 7,
plus 1 is 8, plus 6 is 14,

00:04:01.840 --> 00:04:04.934
plus 1 is 15, plus 7.

00:04:04.934 --> 00:04:07.927
15 plus 7 is 22.

00:04:07.927 --> 00:04:09.135
Let me do that one more time.

00:04:09.135 --> 00:04:15.180
You have 7, 8, 14, 15,
22, all of that over 6.

00:04:15.180 --> 00:04:17.070
And we could write
this as a mixed number.

00:04:17.070 --> 00:04:21.120
6 goes into 22 three times
with a remainder of 4.

00:04:21.120 --> 00:04:25.200
So it's 3 and 4/6, which is
the same thing as 3 and 2/3.

00:04:25.200 --> 00:04:28.670
We could write this as a
decimal with 3.6 repeating.

00:04:28.670 --> 00:04:32.360
So this is also 3.6 repeating.

00:04:32.360 --> 00:04:34.380
We could write it any
one of those ways.

00:04:34.380 --> 00:04:36.700
But this is kind of a
representative number.

00:04:36.700 --> 00:04:39.820
This is trying to get
at a central tendency.

00:04:39.820 --> 00:04:41.620
Once again, these are
human-constructed.

00:04:41.620 --> 00:04:43.590
No one ever-- it's
not like someone just

00:04:43.590 --> 00:04:46.140
found some religious
document that said,

00:04:46.140 --> 00:04:47.990
this is the way that
the arithmetic mean

00:04:47.990 --> 00:04:49.180
must be defined.

00:04:49.180 --> 00:04:52.700
It's not as pure
of a computation

00:04:52.700 --> 00:04:55.005
as, say, finding the
circumference of the circle,

00:04:55.005 --> 00:04:56.880
which there really is--
that was kind of-- we

00:04:56.880 --> 00:04:57.840
studied the universe.

00:04:57.840 --> 00:05:00.600
And that just fell out of
our study of the universe.

00:05:00.600 --> 00:05:02.250
It's a human-constructed
definition

00:05:02.250 --> 00:05:04.110
that we found useful.

00:05:04.110 --> 00:05:07.260
Now there are other ways
to measure the average

00:05:07.260 --> 00:05:10.130
or find a typical
or middle value.

00:05:10.130 --> 00:05:14.470
The other very typical
way is the median.

00:05:14.470 --> 00:05:15.667
And I will write median.

00:05:15.667 --> 00:05:16.750
I'm running out of colors.

00:05:16.750 --> 00:05:18.660
I will write median in pink.

00:05:18.660 --> 00:05:21.280
So there is the median.

00:05:21.280 --> 00:05:25.160
And the median is literally
looking for the middle number.

00:05:25.160 --> 00:05:27.350
So if you were to order
all the numbers in your set

00:05:27.350 --> 00:05:31.460
and find the middle one,
then that is your median.

00:05:31.460 --> 00:05:34.050
So given that, what's the
median of this set of numbers

00:05:34.050 --> 00:05:35.806
going to be?

00:05:35.806 --> 00:05:36.930
Let's try to figure it out.

00:05:36.930 --> 00:05:38.170
Let's try to order it.

00:05:38.170 --> 00:05:39.810
So we have 1.

00:05:39.810 --> 00:05:41.010
Then we have another 1.

00:05:41.010 --> 00:05:42.860
Then we have a 3.

00:05:42.860 --> 00:05:46.630
Then we have a 4, a 6, and a 7.

00:05:46.630 --> 00:05:48.700
So all I did is
I reordered this.

00:05:48.700 --> 00:05:50.890
And so what's the middle number?

00:05:50.890 --> 00:05:52.320
Well, you look here.

00:05:52.320 --> 00:05:54.960
Since we have an even number of
numbers, we have six numbers,

00:05:54.960 --> 00:05:57.260
there's not one middle number.

00:05:57.260 --> 00:05:59.650
You actually have two
middle numbers here.

00:05:59.650 --> 00:06:02.050
You have two middle
numbers right over here.

00:06:02.050 --> 00:06:03.160
You have the 3 and the 4.

00:06:03.160 --> 00:06:05.940
And in this case, when you
have two middle numbers,

00:06:05.940 --> 00:06:09.640
you actually go halfway
between these two numbers.

00:06:09.640 --> 00:06:12.080
You're essentially taking the
arithmetic mean of these two

00:06:12.080 --> 00:06:14.272
numbers to find the median.

00:06:14.272 --> 00:06:16.230
So the median is going
to be halfway in-between

00:06:16.230 --> 00:06:19.190
3 and 4, which is
going to be 3.5.

00:06:19.190 --> 00:06:24.424
So the median in
this case is 3.5.

00:06:24.424 --> 00:06:26.590
So if you have an even
number of numbers, the median

00:06:26.590 --> 00:06:28.714
or the middle two, the--
essentially the arithmetic

00:06:28.714 --> 00:06:31.329
mean of the middle two, or
halfway between the middle two.

00:06:31.329 --> 00:06:32.870
If you have an odd
number of numbers,

00:06:32.870 --> 00:06:34.270
it's a little bit
easier to compute.

00:06:34.270 --> 00:06:35.644
And just so that
we see that, let

00:06:35.644 --> 00:06:36.920
me give you another data set.

00:06:36.920 --> 00:06:39.030
Let's say our data
set-- and I'll

00:06:39.030 --> 00:06:41.740
order it for us--
let's say our data set

00:06:41.740 --> 00:06:55.689
was 0, 7, 50, I don't know,
10,000, and 1 million.

00:06:55.689 --> 00:06:56.980
Let's say that is our data set.

00:06:56.980 --> 00:06:58.450
Kind of a crazy data set.

00:06:58.450 --> 00:07:02.400
But in this situation,
what is our median?

00:07:02.400 --> 00:07:04.045
Well, here we have five numbers.

00:07:04.045 --> 00:07:05.420
We have an odd
number of numbers.

00:07:05.420 --> 00:07:07.200
So it's easier to
pick out a middle.

00:07:07.200 --> 00:07:12.040
The middle is the number that is
greater than two of the numbers

00:07:12.040 --> 00:07:13.540
and is less than
two of the numbers.

00:07:13.540 --> 00:07:14.760
It's exactly in the middle.

00:07:14.760 --> 00:07:18.840
So in this case,
our median is 50.

00:07:18.840 --> 00:07:20.742
Now, the third measure
of central tendency,

00:07:20.742 --> 00:07:22.200
and this is the
one that's probably

00:07:22.200 --> 00:07:26.426
used least often in
life, is the mode.

00:07:26.426 --> 00:07:27.800
And people often
forget about it.

00:07:27.800 --> 00:07:29.852
It sounds like
something very complex.

00:07:29.852 --> 00:07:31.310
But what we'll see
is it's actually

00:07:31.310 --> 00:07:33.080
a very straightforward idea.

00:07:33.080 --> 00:07:36.180
And in some ways, it
is the most basic idea.

00:07:36.180 --> 00:07:40.510
So the mode is actually the most
common number in a data set,

00:07:40.510 --> 00:07:41.885
if there is a most
common number.

00:07:41.885 --> 00:07:43.801
If all of the numbers
are represented equally,

00:07:43.801 --> 00:07:45.760
if there's no one single
most common number,

00:07:45.760 --> 00:07:47.320
then you have no mode.

00:07:47.320 --> 00:07:50.240
But given that
definition of the mode,

00:07:50.240 --> 00:07:54.190
what is the single most common
number in our original data

00:07:54.190 --> 00:07:58.300
set, in this data
set right over here?

00:07:58.300 --> 00:08:00.100
Well, we only have one 4.

00:08:00.100 --> 00:08:01.490
We only have one 3.

00:08:01.490 --> 00:08:03.370
But we have two 1's.

00:08:03.370 --> 00:08:04.880
We have one 6 and one 7.

00:08:04.880 --> 00:08:08.730
So the number that shows up
the most number of times here

00:08:08.730 --> 00:08:11.060
is our 1.

00:08:11.060 --> 00:08:14.070
So the mode, the most typical
number, the most common number

00:08:14.070 --> 00:08:17.610
here is a 1.

00:08:17.610 --> 00:08:19.590
So, you see, these
are all different ways

00:08:19.590 --> 00:08:23.320
of trying to get at a typical,
or middle, or central tendency.

00:08:23.320 --> 00:08:25.600
But they do it in very,
very different ways.

00:08:25.600 --> 00:08:27.350
And as we study more
and more statistics,

00:08:27.350 --> 00:08:29.760
we'll see that they're
good for different things.

00:08:29.760 --> 00:08:31.730
This is used very frequently.

00:08:31.730 --> 00:08:34.574
The median is really good if you
have some kind of crazy number

00:08:34.574 --> 00:08:35.990
out here that could
have otherwise

00:08:35.990 --> 00:08:38.100
skewed the arithmetic mean.

00:08:38.100 --> 00:08:41.449
The mode could also be useful
in situations like that,

00:08:41.449 --> 00:08:43.240
especially if you do
have one number that's

00:08:43.240 --> 00:08:45.960
showing up a lot
more frequently.

00:08:45.960 --> 00:08:47.570
Anyway, I'll leave you there.

00:08:47.570 --> 00:08:51.710
And we'll-- the next few videos,
we will explore statistics even

00:08:51.710 --> 00:08:53.260
deeper.