Conic Sections: Intro to Circles

WEBVTT
Kind: captions
Language: en

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Now

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that you know what the conic
sections are and why they're

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called conic sections, let's
see if we can understand the

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actual equations of the conic
sections a little bit better,

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and use that knowledge to be
able to at least recognize them

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when we see the equation, and
then if we see the equation,

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know how to plot them.

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So the first one we'll start
off with is the circle.

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And you've probably seen this
for many years already,

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but I'll review, if you
don't know it already.

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The general form of an
equation, or the general

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equation for a circle, is x
squared plus y squared is equal

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to r squared, where r is
the radius of the circle.

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And this would be a circle
centered at the point 0,0.

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So I'll just draw this kind of
a general graph right here.

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So that's the x-axis, that's
the y-axis, the circle will

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look like this-- let me do it
in a different color-- the

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circle would look something
like-- make sure it's a

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circle-- well, it's supposed to
be centered at 0,0, but that's

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close enough-- so its center
will be right here, and the

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radius if you go from the
center to any point along that

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circle will have a distance of
r, so if you go from there to

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there it's r, from there to
there it's r, from there to

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there it's r, and to some
degree, this formula, all it is

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is an extension of the distance
formula, which is really just

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the extension of the
Pythagorean Theorem.

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So for example, the distance
formula, if I want to know the

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distance between some point x,y
and the point 0,0, what you do

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is, you take the difference of
the x's-- so x minus 0-- you

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square that, and then you add
that to the distance between

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the y's squared-- so that's one
y point minus 0y-- y is equal

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to 0-- square that, and that is
equal to the distance squared.

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So if you simplify this, x
minus 0 squared, that's just

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x squared plus-- and this
is just y squared is equal

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to distance squared.

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So essentially, this equation
is the plot of all points that

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are exactly d away, a distance
of d away, from the point 0,0,

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and that's just a circle.

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And I'll let you think about
it, I think I actually showed

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this to you in a distance
formula video, but the distance

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formula just comes out of
the Pythagorean Theorem.

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And if that's not completely
obvious for you, just think

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about a little bit, and
it'll hopefully become a

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little bit more obvious.

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But anyway, this was probably--
you probably already knew this,

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and actually just to hit the
point home, if we had the

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equation like this x squared
plus y squared is equal to 9,

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the graph of this circle
would look like this.

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So that's the x-axis, that's
the y-axis, then draw the

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circle itself, circle looks
like-- close enough, and then

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the distance or the radius from
the center of the circle,

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that's going to be 3.

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There's a 9 here, why
isn't the radius 9?

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Oh, that's because this
is the radius squared.

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Remember the original formula I
just showed you. x squared plus

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y squared is equal
to r squared.

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So this right here is r
squared, so if r squared

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is equal to 9, than
r is equal to 3.

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It can't be minus 3.

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I mean, it could, you can't
have a negative radius, or if

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you did you're just going in
the other direction, but

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it's the same thing.

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So the radius is equal to 3.

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So that's a circle, and that's
pretty straightforward, but in

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a lot of algebra classes, they
complicate the issue a little

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bit by shifting the circle.

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So let's just shift
this circle.

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Instead of having-- so
let me just rewrite it.

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So the unshifted circle was x
squared plus y squared is equal

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to-- let me write it this way--
is equal to 3 squared, that's

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the same thing as 9, and let's
say that the new circle, the

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shifted circle, is x minus 1
squared plus y plus 2 squared

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is equal to 3 squared.

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Now all of a sudden this looks
really complicated and daunting

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and all the rest, but all you
have to recognize is, we just

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substituted an x minus 1 for
the-- whoops, messed

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up my pointer.

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We just substituted an x
minus 1 for the x, and

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we just substituted a
y plus 2 for the y.

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So it has the same basic
pattern as of this circle, and

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the fact that we added or
subtracted numbers from the x's

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and the y's tells us that we
shifted the circle, and so the

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next obvious question is,
where did you shift it to?

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And your impulse might be, oh,
well maybe I shifted it to,

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instead of the center being
0,0, your intuition might be

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to say that, well, now the
center is at negative 1,2.

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And you'd be almost right,
except you'd be exactly the

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opposite of the correct answer.

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The new center is now x is
equal to positive 1 and

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y is equal to minus 2.

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And that might be unintuitive
for you at first-- and you

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might want to watch some of the
videos, I think I've done them

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already, or I've always
intended to, on shifting

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functions-- but the way to
think about it is the center

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here is x is equal to 0.

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So when x and y is equal to 0,
x squared plus y squared is 0,

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you're exactly 0 away from the
center, or we're at the center.

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So now if we want to be-- if we
want x to be 0 away from our

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new center, this term
has to be equal to 0.

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And if-- just like when x was
equal to 0, this term equaled

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0, so now for us to be at the
center of our new circle, so to

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speak, this term has to be 0.

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So the new center has to
be at x is equal to 1.

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Likewise, this has to be
0, and so the center

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is at y is equal to 2.

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Another way to think about it
is this-- let's say when y,

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you know what happens here
when y is equal to 2.

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Whatever part of the circle
we're in when y is equal to 2.

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There's some part of the circle
we're at, in fact I could draw

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it, when y is equal to 2.

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Let's say this radius is 3 when
y is equal to 2, we're probably

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right around there
on the circle.

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We could be there or
we could be there.

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Now we're shifting the circle,
so now instead of being at 0,0,

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we're going to be at 1 minus 2.

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So now we're going to be at the
new center is x is equal to 1,

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y is equal to minus 2, the new
center is there, and if I were

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to draw the new circle, it
would look something like this.

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I'm going to try my best
to draw it still as a

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circle and show you
that it's been shifted.

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No, that's not good.

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Let me draw it like-- I
pressed the wrong button.

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That's not good.

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Let me draw it right there.

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That's close enough.

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I don't have to keep doing it.

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So what we did is we shifted
this circle down 2,

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and to the right 1.

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So if we take its center
point, we went down 2

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and to the right 1.

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And so if you think about up
here, when y was equal to 2, we

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could have been at this point
or this point, the kind of

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equivalent points of the new
circle are going to be here.

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Are going to be roughly here,
where you're shifting it

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down and to the right.

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And in order to have that
same behavior in the circle

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there, this whole thing
should be equal to 2.

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So that same point on the
circle, if this whole thing is

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going to be equal to 2--
because it's going to be the

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same kind of behavior in the
equation, and I hope I'm not

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confusing you there-- then the
new y has to be 0, and

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you see it there.

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At both of these points
now, y is equal to 0.

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So I know that's a little
unintuitive, but I want you to

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sit and think about that a lot.

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I mean, you could just memorize
it, that it's the opposite,

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when you have x minus 1 and y
plus 2 that it's actually you

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shifted to x is equal to-- the
center is now 1, minus 2, or

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you could memorize, if you
like, that what makes this 0

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and what makes this 0, and
that's your new center.

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But I really want you to
think about it, this

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is really a shift.

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And of course, if you were
to graph it, you were to

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get this thing up there.

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Anyway, let me see how
much time I have.

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Actually I didn't even
keep track of the time.

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I'll leave you there, I'll
continue this in the next video

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where I'll talk a little
bit about ellipses.


Introduction to conic sections

WEBVTT
Kind: captions
Language: en

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Let's see if we can learn
a thing or two about

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conic sections.

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So first of all, what are
they and why are they

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called conic sections?

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Actually, you probably
recognize a few of them

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already, and I'll
write them out.

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They're the circle, the
ellipse, the parabola,

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and the hyperbola.

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That's a p.

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Hyperbola.

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And you know what
these are already.

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When I first learned conic
sections, I was like, oh,

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I know what a circle is.

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I know what a parabola is.

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And I even know a little bit
about ellipses and hyperbolas.

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Why on earth are they
called conic sections?

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So to put things simply because
they're the intersection

00:00:46.450 --> 00:00:47.760
of a plane and a cone.

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And I draw you
that in a second.

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But just before I do that it
probably makes sense to just

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draw them by themselves.

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And I'll switch colors.

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Circle, we all know
what that is.

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Actually let me see if
I can pick a thicker

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line for my circles.

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so a circle looks
something like that.

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It's all the points that are
equidistant from some center,

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and that distance that they
all are that's the radius.

00:01:12.610 --> 00:01:17.130
So if this is r, and this is
the center, the circle is all

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the points that are exactly
r away from this center.

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We learned that early in our
education what a circle

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is; it makes the world
go round, literally.

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Ellipse in layman's terms is
kind of a squished circle.

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It could look
something like this.

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Let me do an ellipse
in another color.

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So an ellipse could
be like that.

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Could be like that.

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It's harder to draw using the
tool I'm drawing, but it could

00:01:41.680 --> 00:01:43.500
also be tilted and
rotated around.

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But this is a general sense.

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And actually, circles are a
special case of an ellipse.

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It's an ellipse where it's not
stretched in one dimension

00:01:51.010 --> 00:01:51.730
more than the other.

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It's kind of perfectly
symmetric in every way.

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Parabola.

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You've learned that if you've
taken algebra two and you

00:01:59.610 --> 00:02:02.580
probably have if you care
about conic sections.

00:02:02.580 --> 00:02:07.610
But a parabola-- let me draw a
line here to separate things.

00:02:07.610 --> 00:02:12.300
A parabola looks something like
this, kind of a U shape and you

00:02:12.300 --> 00:02:13.650
know, the classic parabola.

00:02:13.650 --> 00:02:16.080
I won't go into the
equations right now.

00:02:16.080 --> 00:02:18.550
Well, I will because you're
probably familiar with it.

00:02:18.550 --> 00:02:20.370
y is equal to x squared.

00:02:20.370 --> 00:02:23.570
And then, you could shift it
around and then you can even

00:02:23.570 --> 00:02:25.170
have a parabola that
goes like this.

00:02:25.170 --> 00:02:27.520
That would be x is
equal to y squared.

00:02:27.520 --> 00:02:31.590
You could rotate these things
around, but I think you know

00:02:31.590 --> 00:02:33.090
the general shape
of a parabola.

00:02:33.090 --> 00:02:36.100
We'll talk more about how do
you graph it or how do you know

00:02:36.100 --> 00:02:38.780
what the interesting points
on a parabola actually are.

00:02:38.780 --> 00:02:40.730
And then the last one,
you might have seen this

00:02:40.730 --> 00:02:42.500
before, is a hyperbola.

00:02:42.500 --> 00:02:45.520
It almost looks like two
parabolas, but not quite,

00:02:45.520 --> 00:02:50.500
because the curves look a
little less U-ish and

00:02:50.500 --> 00:02:51.520
a little more open.

00:02:51.520 --> 00:02:54.010
But I'll explain what
I mean by that.

00:02:54.010 --> 00:02:56.170
So a hyperbola usually
looks something like this.

00:02:56.170 --> 00:03:02.900
So if these are the axes,
then if I were to draw-- let

00:03:02.900 --> 00:03:03.610
me draw some asymptotes.

00:03:07.512 --> 00:03:13.740
I want to go right through
the-- that's pretty good.

00:03:13.740 --> 00:03:14.230
These are asymptotes.

00:03:14.230 --> 00:03:17.130
Those aren't the
actual hyperbola.

00:03:17.130 --> 00:03:18.970
But a hyperbola would look
something like this.

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They get to be right here
and they get really

00:03:25.280 --> 00:03:26.090
close to the asymptote.

00:03:26.090 --> 00:03:30.020
They get closer and closer to
those blue lines like that and

00:03:30.020 --> 00:03:32.430
it happened on this side too.

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The graphs show up here and
then they pop over and

00:03:34.860 --> 00:03:36.000
they show up there.

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This magenta could be one
hyperbola; I haven't done

00:03:38.580 --> 00:03:39.840
true justice to it.

00:03:39.840 --> 00:03:42.290
Or another hyperbola could be
on, you could kind of call

00:03:42.290 --> 00:03:44.060
it a vertical hyperbola.

00:03:44.060 --> 00:03:45.720
That's not the exact word, but
it would look something like

00:03:45.720 --> 00:03:50.160
that where it's below
the asymptote here.

00:03:50.160 --> 00:03:54.160
It's above the asymptote there.

00:03:54.160 --> 00:03:57.060
So this blue one would be
one hyperbola and then the

00:03:57.060 --> 00:03:58.670
magenta one would be a
different hyperbola.

00:03:58.670 --> 00:04:00.050
So those are the
different graphs.

00:04:00.050 --> 00:04:03.525
So the one thing that I'm sure
you're asking is why are

00:04:03.525 --> 00:04:04.790
they called conic sections?

00:04:04.790 --> 00:04:08.430
Why are they not called
bolas or variations of

00:04:08.430 --> 00:04:09.810
circles or whatever?

00:04:09.810 --> 00:04:11.740
And in fact, wasn't
even the relationship.

00:04:11.740 --> 00:04:14.180
It's pretty clear that
circles and ellipses

00:04:14.180 --> 00:04:15.330
are somehow related.

00:04:15.330 --> 00:04:17.430
That an ellipse is just
a squished circle.

00:04:17.430 --> 00:04:20.250
And maybe it even seems that
parabolas and hyperbolas

00:04:20.250 --> 00:04:21.680
are somewhat related.

00:04:21.680 --> 00:04:23.340
This is a P once again.

00:04:23.340 --> 00:04:25.670
They both have bola in their
name and they both kind

00:04:25.670 --> 00:04:28.110
of look like open U's.

00:04:28.110 --> 00:04:30.930
Although a hyperbola has two of
these going and kind of opening

00:04:30.930 --> 00:04:32.380
in different directions,
but they look related.

00:04:32.380 --> 00:04:34.020
But what is the connection
behind all these?

00:04:34.020 --> 00:04:38.390
And that's frankly where
the word conic comes from.

00:04:38.390 --> 00:04:42.800
So let me see if I can draw
a three-dimensional cone.

00:04:42.800 --> 00:04:44.200
So this is a cone.

00:04:47.110 --> 00:04:47.900
That's the top.

00:04:52.990 --> 00:04:55.640
I could've used an
ellipse for the top.

00:04:55.640 --> 00:04:56.910
Looks like that.

00:04:56.910 --> 00:04:58.130
Actually, it has no top.

00:04:58.130 --> 00:05:01.690
It would actually keep going
on forever in that direction.

00:05:01.690 --> 00:05:04.080
I'm just kind of slicing it
so you see that it's a cone.

00:05:04.080 --> 00:05:06.920
This could be the
bottom part of it.

00:05:06.920 --> 00:05:10.760
So let's take different
intersections of a plane with

00:05:10.760 --> 00:05:13.930
this cone and see if we can at
least generate the different

00:05:13.930 --> 00:05:15.830
shapes that we talked
about just now.

00:05:15.830 --> 00:05:19.650
So if we have a plane that goes
directly-- I guess if you call

00:05:19.650 --> 00:05:23.030
this the axis of this
three-dimensional cone,

00:05:23.030 --> 00:05:24.460
so this is the axis.

00:05:24.460 --> 00:05:27.150
So if we have a plane that's
exactly perpendicular to that

00:05:27.150 --> 00:05:29.660
axis-- let's see if I can
draw it in three dimensions.

00:05:29.660 --> 00:05:32.265
The plane would look
something like this.

00:05:32.265 --> 00:05:34.940
So it would have a line.

00:05:34.940 --> 00:05:38.410
This is the front line that's
closer to you and then they

00:05:38.410 --> 00:05:43.370
would have another
line back here.

00:05:43.370 --> 00:05:45.370
That's close enough.

00:05:45.370 --> 00:05:47.443
And of course, you know these
are infinite planes, so it

00:05:47.443 --> 00:05:50.060
goes off in every direction.

00:05:50.060 --> 00:05:52.930
If this plane is directly
perpendicular to the axis

00:05:52.930 --> 00:05:55.400
of these and this is where
the plane goes behind it.

00:05:55.400 --> 00:05:58.010
The intersection of this
plane and this cone is

00:05:58.010 --> 00:06:01.230
going to look like this.

00:06:01.230 --> 00:06:03.790
We're looking at it from an
angle, but if you were looking

00:06:03.790 --> 00:06:06.170
straight down, if you were
listening here and you look at

00:06:06.170 --> 00:06:08.660
this plane-- if you were
looking at it right above.

00:06:08.660 --> 00:06:12.300
If I were to just flip this
over like this, so we're

00:06:12.300 --> 00:06:15.290
looking straight down on this
plane, that intersection

00:06:15.290 --> 00:06:17.700
would be a circle.

00:06:17.700 --> 00:06:23.070
Now, if we take the plane and
we tilt it down a little bit,

00:06:23.070 --> 00:06:27.810
so if instead of that we
have a situation like this.

00:06:27.810 --> 00:06:31.250
Let me see if I can
do it justice.

00:06:31.250 --> 00:06:35.690
We have a situation
where it's-- whoops.

00:06:35.690 --> 00:06:37.190
Let me undo that.

00:06:37.190 --> 00:06:37.570
Edit.

00:06:37.570 --> 00:06:38.870
Undo.

00:06:38.870 --> 00:06:44.940
Where it's like this and has
another side like this,

00:06:44.940 --> 00:06:45.900
and I connect them.

00:06:48.680 --> 00:06:49.950
So that's the plane.

00:06:49.950 --> 00:06:54.890
Now the intersection of this
plane, which is now not

00:06:54.890 --> 00:06:57.710
orthogonal or it's not
perpendicular to the axis of

00:06:57.710 --> 00:07:00.240
this three-dimensional cone.

00:07:00.240 --> 00:07:02.650
If you take the intersection of
that plane and that cone-- and

00:07:02.650 --> 00:07:04.280
in future videos, and you
don't do this in your

00:07:04.280 --> 00:07:04.870
algebra two class.

00:07:04.870 --> 00:07:06.470
But eventually we'll kind of
do the three-dimensional

00:07:06.470 --> 00:07:08.940
intersection and prove that
this is definitely the case.

00:07:08.940 --> 00:07:10.760
You definitely do get the
equations, which I'll show you

00:07:10.760 --> 00:07:12.880
in the not too far future.

00:07:12.880 --> 00:07:14.510
This intersection would
look something like this.

00:07:14.510 --> 00:07:16.400
I think you can
visualize it right now.

00:07:16.400 --> 00:07:17.490
It would look
something like this.

00:07:20.950 --> 00:07:24.120
And if you were to look
straight down on this plane, if

00:07:24.120 --> 00:07:27.200
you were to look right above
the plane, this would look

00:07:27.200 --> 00:07:29.440
something-- this figure I
just drew in purple-- would

00:07:29.440 --> 00:07:30.410
look something like this.

00:07:33.060 --> 00:07:34.290
Well, I didn't draw
it that well.

00:07:34.290 --> 00:07:35.290
It'd be an ellipse.

00:07:35.290 --> 00:07:36.800
You know what an
ellipse looks like.

00:07:36.800 --> 00:07:41.250
And if I tilted it the other
way, the ellipse would

00:07:41.250 --> 00:07:42.430
squeeze the other way.

00:07:42.430 --> 00:07:44.890
But that just gives you a
general sense of why both of

00:07:44.890 --> 00:07:46.330
these are conic sections.

00:07:46.330 --> 00:07:47.350
Now something very interesting.

00:07:47.350 --> 00:07:52.020
If we keep tilting this plane,
so if we tilt the plane so

00:07:52.020 --> 00:07:55.420
it's-- so let's say we're
pivoting around that point.

00:07:55.420 --> 00:07:59.870
So now my plane-- let me
see if I can do this.

00:07:59.870 --> 00:08:02.710
It's a good exercise in
three-dimensional drawing.

00:08:02.710 --> 00:08:05.730
Let's say it looks
something like this.

00:08:05.730 --> 00:08:09.310
I want to go through
that point.

00:08:09.310 --> 00:08:11.070
So this is my
three-dimensional plane.

00:08:13.690 --> 00:08:16.330
I'm drawing it in such a way
that it only intersects this

00:08:16.330 --> 00:08:20.920
bottom cone and the surface
of the plane is parallel to

00:08:20.920 --> 00:08:22.760
the side of this top cone.

00:08:22.760 --> 00:08:25.630
In this case the intersection
of the plane and the cone

00:08:25.630 --> 00:08:28.110
is going to intersect
right at that point.

00:08:28.110 --> 00:08:31.900
You can almost view that I'm
pivoting around this point, at

00:08:31.900 --> 00:08:35.850
the intersection of this point
and the plane and the cone.

00:08:35.850 --> 00:08:37.750
Well this now, the
intersection, would look

00:08:37.750 --> 00:08:39.470
something like this.

00:08:39.470 --> 00:08:41.230
It would look like that.

00:08:41.230 --> 00:08:42.490
And it would keep going down.

00:08:42.490 --> 00:08:44.980
So if I were to draw it,
it would look like this.

00:08:44.980 --> 00:08:46.840
If I was right above the
plane, if I were to

00:08:46.840 --> 00:08:49.390
just draw the plane.

00:08:49.390 --> 00:08:51.050
And there you get
your parabola.

00:08:51.050 --> 00:08:51.860
So that's interesting.

00:08:51.860 --> 00:08:54.830
If you keep kind of tilting--
if you start with a

00:08:54.830 --> 00:08:57.530
circle, tilt a little bit,
you get an ellipse.

00:08:57.530 --> 00:09:00.695
You get kind of a more
and more skewed ellipse.

00:09:00.695 --> 00:09:04.350
And at some point, the ellipse
keeps getting more and

00:09:04.350 --> 00:09:05.160
more skewed like that.

00:09:05.160 --> 00:09:10.660
It kind of pops right when you
become exactly parallel to

00:09:10.660 --> 00:09:11.990
the side of this top cone.

00:09:11.990 --> 00:09:14.260
And I'm doing it all very
inexact right now, but I

00:09:14.260 --> 00:09:15.060
think I want to give
you the intuition.

00:09:15.060 --> 00:09:17.400
It pops and it turns
into a parabola.

00:09:17.400 --> 00:09:19.190
So you can kind of view
a parabola-- there is

00:09:19.190 --> 00:09:20.470
this relationship.

00:09:20.470 --> 00:09:23.550
Parabola is what happens when
one side of an ellipse pops

00:09:23.550 --> 00:09:25.580
open and you get this parabola.

00:09:25.580 --> 00:09:30.240
And then, if you keep tilting
this plane, and I'll do it

00:09:30.240 --> 00:09:32.520
another color-- so
it intersects both

00:09:32.520 --> 00:09:36.140
sides of the cone.

00:09:36.140 --> 00:09:37.490
Let me see if I can draw that.

00:09:37.490 --> 00:09:42.620
So if this is my new
plane-- whoops.

00:09:42.620 --> 00:09:43.820
That's good enough.

00:09:43.820 --> 00:09:47.550
So if my plane looks like
this-- I know it's very hard to

00:09:47.550 --> 00:09:50.550
read now-- and you wanted the
intersection of this plane,

00:09:50.550 --> 00:09:52.700
this green plane and the cone--
I should probably redraw it

00:09:52.700 --> 00:09:55.950
all, but hopefully you're not
getting overwhelmingly

00:09:55.950 --> 00:09:59.190
confused-- the intersection
would look like this.

00:09:59.190 --> 00:10:01.490
It would intersect the bottom
cone there and it would

00:10:01.490 --> 00:10:04.760
intersect the top
cone over there.

00:10:04.760 --> 00:10:08.320
And then you would have
something like this.

00:10:08.320 --> 00:10:11.230
This would be intersection of
the plane and the bottom cone.

00:10:11.230 --> 00:10:13.200
And then up here would be
the intersection of the

00:10:13.200 --> 00:10:14.090
plane and the top one.

00:10:14.090 --> 00:10:18.810
Remember, this plane goes off
in every direction infinitely.

00:10:18.810 --> 00:10:21.920
So that's just a general sense
of what the conic sections are

00:10:21.920 --> 00:10:24.850
and why frankly they're
called conic sections.

00:10:24.850 --> 00:10:27.580
And let me know if this got
confusing because maybe I'll do

00:10:27.580 --> 00:10:29.350
another video while I redraw
it a little bit cleaner.

00:10:29.350 --> 00:10:32.650
Maybe I can find some kind of
neat 3D application that can do

00:10:32.650 --> 00:10:35.890
it better than I can do it.

00:10:35.890 --> 00:10:37.750
This is kind of just the reason
why they all are conic

00:10:37.750 --> 00:10:39.730
sections, and why they really
are related to each other.

00:10:39.730 --> 00:10:41.720
And will do that a little
more in depth mathematically

00:10:41.720 --> 00:10:42.860
in a few videos.

00:10:42.860 --> 00:10:45.080
But in the next video, now that
you know what they are and why

00:10:45.080 --> 00:10:47.340
they're all called conic
sections, I'll actually talk

00:10:47.340 --> 00:10:49.680
about the formulas about these
and how do you recognize

00:10:49.680 --> 00:10:50.370
the formulas.

00:10:50.370 --> 00:10:54.060
And given a formula, how do
you actually plot the graphs

00:10:54.060 --> 00:10:56.020
of these conic sections?

00:10:56.020 --> 00:10:57.820
See you in the next video.


Introduction to the normal distribution

WEBVTT
Kind: captions
Language: en

00:00:01.710 --> 00:00:05.120
The normal distribution is
arguably, the most important

00:00:05.120 --> 00:00:06.220
concept in statistics.

00:00:06.220 --> 00:00:08.660
Everything we do or almost
everything we do in inferential

00:00:08.660 --> 00:00:11.220
statistics which is
essentially, making inferences

00:00:11.220 --> 00:00:15.340
based on data points, is to
some degree, based on the

00:00:15.340 --> 00:00:16.460
normal distribution.

00:00:16.460 --> 00:00:23.470
So, what I want to do in this
video, in this spreadsheet, is

00:00:23.470 --> 00:00:26.100
to essentially give you as deep
an understanding of the normal

00:00:26.100 --> 00:00:27.850
distribution as possible.

00:00:27.850 --> 00:00:31.070
For the rest of your life if
someone says, we're assuming a

00:00:31.070 --> 00:00:32.810
normal distribution you can
say, oh I know what that is,

00:00:32.810 --> 00:00:34.190
this is a formula and I
understand how to

00:00:34.190 --> 00:00:35.310
use it et cetera.

00:00:35.310 --> 00:00:38.550
So this spreadsheet, just so
you know, is downloadable at

00:00:38.550 --> 00:00:43.880
www.khanacademy.org/downlads/
and if you just type that part

00:00:43.880 --> 00:00:45.550
in you'll see everything
that's downloadable.

00:00:45.550 --> 00:00:50.020
Then download/normalintro.xls
and you'll get this

00:00:50.020 --> 00:00:51.470
spreadsheet right here.

00:00:51.470 --> 00:00:53.360
I think I did this in
the right standard.

00:00:53.360 --> 00:00:55.920
Anyway, if you go onto
Wikipedia and if you were to

00:00:55.920 --> 00:00:59.420
type in normal distribution or
were to do a search for a

00:00:59.420 --> 00:01:02.910
normal distribution -- let me
actually get my pen tool going

00:01:02.910 --> 00:01:04.740
-- this is what you would see.

00:01:04.740 --> 00:01:06.950
I literally copied and pasted
this right here from Wikipedia.

00:01:06.950 --> 00:01:10.040
I know it looks daunting, you
have all these Greek letters

00:01:10.040 --> 00:01:13.620
there, but this is just -- this
sigma right here -- that is

00:01:13.620 --> 00:01:15.950
just the standard deviation
of the distribution.

00:01:15.950 --> 00:01:18.510
We'll play with that a little
bit in this chart and

00:01:18.510 --> 00:01:20.780
see what that means.

00:01:20.780 --> 00:01:22.610
You know what the standard
deviation means in general but

00:01:22.610 --> 00:01:25.800
this is the standard deviation
of this distribution, which is

00:01:25.800 --> 00:01:27.620
a probability density function.

00:01:27.620 --> 00:01:30.110
And I encourage you to rewatch
the video on probability

00:01:30.110 --> 00:01:32.280
density functions because it's
a little bit of a transition

00:01:32.280 --> 00:01:34.880
going from the binomial
distribution, which

00:01:34.880 --> 00:01:36.320
is discreet.

00:01:36.320 --> 00:01:39.180
The binomial distribution will
say, what is the probability of

00:01:39.180 --> 00:01:41.740
getting a 5, and you just kind
of look at that histogram or

00:01:41.740 --> 00:01:44.220
that bar chart and say oh,
that's the probability.

00:01:44.220 --> 00:01:48.220
But in a continuous probability
distribution or a continuous

00:01:48.220 --> 00:01:51.010
probability density function,
you can't just say what is the

00:01:51.010 --> 00:01:52.930
probability of me getting a 5.

00:01:52.930 --> 00:01:55.590
You have to say what is the
probability of me getting

00:01:55.590 --> 00:01:59.920
between, let's say
a 4.5 and a 5.5.

00:01:59.920 --> 00:02:02.180
You have to give it some range.

00:02:02.180 --> 00:02:04.700
And then, your probability
isn't given by just

00:02:04.700 --> 00:02:06.350
reading this graph.

00:02:06.350 --> 00:02:09.265
The probability is given by
the area under that curve.

00:02:11.780 --> 00:02:13.450
It be given by this area.

00:02:13.450 --> 00:02:17.260
For those of you who know
calculus, if p of x is our

00:02:17.260 --> 00:02:19.140
probability density function --
doesn't have to be a normal

00:02:19.140 --> 00:02:24.290
distribution although it often
is a normal distribution -- the

00:02:24.290 --> 00:02:28.740
way you actually figure out the
probability, let's say between

00:02:28.740 --> 00:02:29.890
4 and a half and 5 and half.

00:02:29.890 --> 00:02:33.570
What is the probability, the
odds of me getting between 4

00:02:33.570 --> 00:02:36.220
and a half and 5 and a half
inches of rain tomorrow?

00:02:36.220 --> 00:02:42.460
It'll actually be the integral
from 4 and a half to 5 and a

00:02:42.460 --> 00:02:47.060
half of this probability
density function or of this

00:02:47.060 --> 00:02:50.800
probably density
function, the x.

00:02:50.800 --> 00:02:51.950
So that's just the
area of the curve.

00:02:51.950 --> 00:02:54.370
For those of you who don't know
calculus yet, I encourage

00:02:54.370 --> 00:02:55.590
you to watch that playlist.

00:02:55.590 --> 00:02:58.030
But all this is saying is
the area of the curve

00:02:58.030 --> 00:03:00.660
from here to here.

00:03:00.660 --> 00:03:03.150
It turns out for the normal
distribution, this isn't an

00:03:03.150 --> 00:03:06.000
easy thing to evaluate
analytical and so you

00:03:06.000 --> 00:03:08.040
do it numerically.

00:03:08.040 --> 00:03:10.440
You don't have to feel bad
about doing it numerically

00:03:10.440 --> 00:03:13.570
because oh, how do I take
the integral of this?

00:03:13.570 --> 00:03:14.990
There's actually functions
for it and you can

00:03:14.990 --> 00:03:15.960
even approximate it.

00:03:15.960 --> 00:03:18.300
One way you can approximate it
is you could use it the way you

00:03:18.300 --> 00:03:20.000
approximate integrals
in general.

00:03:20.000 --> 00:03:21.630
You could say well, what
is the area of this?

00:03:21.630 --> 00:03:24.750
Well it's roughly the
area of this trapezoid.

00:03:24.750 --> 00:03:27.290
So you could figure out the
area of that trapezoid, taking

00:03:27.290 --> 00:03:29.880
the average of that point and
that point and multiplying

00:03:29.880 --> 00:03:30.750
it by the base.

00:03:30.750 --> 00:03:33.880
Or you can just take the level
of -- let me change colors

00:03:33.880 --> 00:03:36.830
because I think I'm overdoing
it with the green -- or you

00:03:36.830 --> 00:03:41.300
could just take the height of
this line right here and

00:03:41.300 --> 00:03:42.910
multiply it by the base.

00:03:42.910 --> 00:03:46.060
You'll get the area of this
rectangle, which might be a

00:03:46.060 --> 00:03:49.410
pretty good approximation for
the area under the curve.

00:03:49.410 --> 00:03:49.600
Right?

00:03:49.600 --> 00:03:52.060
Because you'll have a little
bit extra over here but you're

00:03:52.060 --> 00:03:53.680
going to miss a little bit
over there, so it might be a

00:03:53.680 --> 00:03:54.580
pretty good approximation.

00:03:54.580 --> 00:03:56.760
That's actually what I do in
the other video, just to

00:03:56.760 --> 00:03:59.870
approximate the area under the
curve and give you a good sense

00:03:59.870 --> 00:04:03.950
that the normal distribution is
what the binomial distribution

00:04:03.950 --> 00:04:08.450
becomes essentially, if
you have many trials.

00:04:08.450 --> 00:04:10.720
What's interesting about the
normal distribution -- I don't

00:04:10.720 --> 00:04:12.310
know if I mentioned this
already -- this right

00:04:12.310 --> 00:04:14.510
here, this is the graph.

00:04:17.340 --> 00:04:19.400
This is just another word,
people might talk about the

00:04:19.400 --> 00:04:20.410
central limit theorem.

00:04:20.410 --> 00:04:23.350
But this is really kind of one
of the most important or

00:04:23.350 --> 00:04:25.660
interesting things about our
universe, central

00:04:25.660 --> 00:04:27.810
limit theorem.

00:04:27.810 --> 00:04:30.560
I won't prove it here but it
essentially tells us, and you

00:04:30.560 --> 00:04:32.990
could kind of understand it by
looking at the other video

00:04:32.990 --> 00:04:34.690
where we talk about
flipping coins.

00:04:34.690 --> 00:04:38.230
If we were to do many flips of
coins -- those are independent

00:04:38.230 --> 00:04:40.570
trials of each other -- and if
you take the sum of all of your

00:04:40.570 --> 00:04:42.380
flips, if you were to give
yourself one point -- if you

00:04:42.380 --> 00:04:45.360
got ahead every time -- and if
you're take the sum of them as

00:04:45.360 --> 00:04:48.150
you approach an infinite number
flips, you approach the

00:04:48.150 --> 00:04:50.180
normal distribution.

00:04:50.180 --> 00:04:53.330
What's interesting about that
is each of those trials -- in

00:04:53.330 --> 00:04:55.480
the case of flipping the coin,
each trial is a flip of the

00:04:55.480 --> 00:04:58.240
coin -- each of those trials
don't have to have a

00:04:58.240 --> 00:04:59.580
normal distribution.

00:04:59.580 --> 00:05:03.950
So we could be talking about
molecular interactions and

00:05:03.950 --> 00:05:09.850
every time compound x interacts
with compound y what might it

00:05:09.850 --> 00:05:12.600
result doesn't have to have
to be normally distributed.

00:05:12.600 --> 00:05:15.010
But what happens is, if you
take a sum of a ton of those

00:05:15.010 --> 00:05:18.340
interactions, then all of a
sudden the end result will

00:05:18.340 --> 00:05:20.100
be normally distributed.

00:05:20.100 --> 00:05:22.550
This is why this is such an
important distribution.

00:05:22.550 --> 00:05:27.540
It shows up in nature all of
the time and if you do take

00:05:27.540 --> 00:05:31.250
data points from something that
is very complex and it is the

00:05:31.250 --> 00:05:36.740
sum of arguably, many almost
infinite individual independent

00:05:36.740 --> 00:05:39.160
trials, it's a pretty good
assumption to assume the

00:05:39.160 --> 00:05:40.040
normal distribution.

00:05:40.040 --> 00:05:42.640
We'll do other videos where we
talk about when it is a good

00:05:42.640 --> 00:05:44.860
assumption, when it isn't
a good assumption.

00:05:44.860 --> 00:05:47.120
But anyway, just to digest
this a little bit and let

00:05:47.120 --> 00:05:48.880
me actually rewrite it.

00:05:48.880 --> 00:05:50.590
This is what you'll see on
Wikipedia but this could be

00:05:50.590 --> 00:05:58.080
rewritten as 1 over sigma times
the square root of 2 pi times

00:05:58.080 --> 00:06:00.250
x is just e to that power.

00:06:00.250 --> 00:06:05.386
So it's just e to the, this
whole thing over here, minus

00:06:05.386 --> 00:06:12.670
x minus the mean squared
over 2 sigma squared.

00:06:12.670 --> 00:06:14.200
This is the standard deviation.

00:06:14.200 --> 00:06:16.900
Standard deviation squared
is just the variance.

00:06:16.900 --> 00:06:18.330
Just so you know how to use
this, you're like, oh wow,

00:06:18.330 --> 00:06:20.770
there's so many Greek
letters here, what do I do?

00:06:20.770 --> 00:06:22.740
This tells you the
height of the normal

00:06:22.740 --> 00:06:24.100
distribution function.

00:06:24.100 --> 00:06:37.950
Let's say that this is the
distribution of people's

00:06:37.950 --> 00:06:39.200
heights above 5'9.

00:06:39.200 --> 00:06:41.060
Let's say that this
was 5'9 and not 0.

00:06:43.920 --> 00:06:51.980
What this tells you is, if you
wanted to figure out what is

00:06:51.980 --> 00:06:55.600
the probability of finding
someone who was roughly 5

00:06:55.600 --> 00:06:58.410
inches taller than the average
right here, what you would

00:06:58.410 --> 00:07:03.170
do is you would put in this
number here, this 5 into x.

00:07:03.170 --> 00:07:05.390
And then you know the standard
deviation, because you've

00:07:05.390 --> 00:07:07.020
taken a bunch of samples.

00:07:07.020 --> 00:07:09.200
You know the variance, which is
the standard deviation squared.

00:07:09.200 --> 00:07:11.860
You know the mean, and you just
put your x in there and it'll

00:07:11.860 --> 00:07:14.200
tell you the height
of the function.

00:07:14.200 --> 00:07:15.940
And then you have to
give it a range.

00:07:15.940 --> 00:07:19.290
You can't just say how many
people are exactly 5 inches

00:07:19.290 --> 00:07:20.850
taller than average.

00:07:20.850 --> 00:07:23.330
You would actually say how
many people are between 5.1

00:07:23.330 --> 00:07:25.567
inches and 4.9 nine inches
taller than the average.

00:07:25.567 --> 00:07:27.080
You have to give it a little
bit of range because no is

00:07:27.080 --> 00:07:30.880
exactly, it's almost infinitely
impossible to the atom

00:07:30.880 --> 00:07:32.410
to be exactly 5'9.

00:07:32.410 --> 00:07:35.230
Even the definition of
an inch isn't defined

00:07:35.230 --> 00:07:36.220
that particularly.

00:07:36.220 --> 00:07:39.100
So that's how you'd
use this function.

00:07:39.100 --> 00:07:42.760
This is so heavily used in,
one, it shows up in nature

00:07:42.760 --> 00:07:46.350
but in all of inferential
statistics, I think it behooves

00:07:46.350 --> 00:07:49.320
you to become as familiar with
this formula as possible.

00:07:49.320 --> 00:07:51.710
I guess to make that happen.

00:07:51.710 --> 00:07:54.250
Let me play around a little bit
with this formula just to kind

00:07:54.250 --> 00:07:57.890
of give you an intuition of how
everything works out et cetera.

00:07:57.890 --> 00:08:02.090
If I were to take this -- I'd
like to maybe help you memorize

00:08:02.090 --> 00:08:05.150
this -- this could be rewritten
as, if we take the sigma into

00:08:05.150 --> 00:08:07.440
the square root sign, if we
take the standard deviation in

00:08:07.440 --> 00:08:11.610
there, it becomes 1 over
the square root of 2

00:08:11.610 --> 00:08:13.800
pie sigma squared.

00:08:13.800 --> 00:08:15.570
I've never seen it written this
way but it gives me a little

00:08:15.570 --> 00:08:18.980
intuition that sigma squared,
it's always written as sigma

00:08:18.980 --> 00:08:20.790
squared, but it's really just
the variance and the variance

00:08:20.790 --> 00:08:22.990
is what you calculate before
you calculate the standard

00:08:22.990 --> 00:08:25.220
deviation, so that's
interesting.

00:08:25.220 --> 00:08:29.060
And then this top right here,
this could be written as e to

00:08:29.060 --> 00:08:35.910
the minus 1 half times and both
of these things here are

00:08:35.910 --> 00:08:40.850
squared so we could just say x
minus the mean over

00:08:40.850 --> 00:08:43.630
sigma squared.

00:08:43.630 --> 00:08:46.540
And this kind of clarifies
what's going on here

00:08:46.540 --> 00:08:47.570
a little bit better.

00:08:47.570 --> 00:08:50.010
Because what's this?

00:08:50.010 --> 00:08:52.970
x minus sigma is the distance
between whatever point

00:08:52.970 --> 00:08:54.680
we want to find.

00:08:54.680 --> 00:08:57.580
Let's say we're here.

00:08:57.580 --> 00:09:02.520
x minus mu is the mean so
that's here so that's this

00:09:02.520 --> 00:09:05.510
distance, and this is the
standard deviation which

00:09:05.510 --> 00:09:07.140
is this distance.

00:09:07.140 --> 00:09:11.680
So this in here tells me how
many standard deviations

00:09:11.680 --> 00:09:12.880
I am away from the mean.

00:09:12.880 --> 00:09:14.560
That's actually called the
standard z score, I talked

00:09:14.560 --> 00:09:15.700
about it in the other video.

00:09:15.700 --> 00:09:18.740
And then we square that
and then we take this

00:09:18.740 --> 00:09:20.340
to the minus 1/2.

00:09:20.340 --> 00:09:21.420
Well let me rewrite that.

00:09:21.420 --> 00:09:25.930
If I were to write e to the
minus 1/2 times a, that's the

00:09:25.930 --> 00:09:30.530
same thing as e to the a to
the minus 1/2 power right?

00:09:30.530 --> 00:09:32.480
If you take something to an
exponent and take that to

00:09:32.480 --> 00:09:35.200
an exponent you can just
multiply these exponents.

00:09:35.200 --> 00:09:39.460
So likewise, this could be
rewritten as, this is equal to

00:09:39.460 --> 00:09:46.500
1 over the square root of 2
pi sigma squared, which

00:09:46.500 --> 00:09:47.610
is just the variance.

00:09:47.610 --> 00:09:49.120
And I'm just playing around
with the formula because I

00:09:49.120 --> 00:09:51.165
really want you to see all
the ways, maybe you'll

00:09:51.165 --> 00:09:52.270
get a little intuition.

00:09:52.270 --> 00:09:54.300
I encourage you to email me
if you see some insight

00:09:54.300 --> 00:09:57.160
on why this exists.

00:09:57.160 --> 00:09:59.160
Once again, I think it is cool
then all of the sudden we have

00:09:59.160 --> 00:10:01.950
this other formula that
has pi and e in it.

00:10:04.610 --> 00:10:07.970
So many phenomenon are
described by this and once

00:10:07.970 --> 00:10:11.320
again pi and e show up together
just like e to the i pi

00:10:11.320 --> 00:10:12.120
is equal to negative 1.

00:10:12.120 --> 00:10:15.400
It tells you something
about our universe.

00:10:15.400 --> 00:10:24.870
Anyway, I could rewrite this as
e to the x minus mu over sigma

00:10:24.870 --> 00:10:28.690
squared and all of that
to the minus 1/2.

00:10:28.690 --> 00:10:30.970
Something in the minus 1/2
power, that's just 1 over

00:10:30.970 --> 00:10:33.360
the square root which is
already going on here.

00:10:33.360 --> 00:10:40.650
So we could just rewrite this
over here as 1 over the square

00:10:40.650 --> 00:10:52.180
root of 2 pi times the variance
times e to essentially,

00:10:52.180 --> 00:10:54.080
our z score squared.

00:10:54.080 --> 00:10:58.600
If we say z is this thing in
here, z is how many standard

00:10:58.600 --> 00:11:02.370
deviations we are from the
mean, z score squared.

00:11:02.370 --> 00:11:05.500
And all of the sudden
this becomes very clean.

00:11:05.500 --> 00:11:09.240
We just say 2 pi times our
variance times e to the number

00:11:09.240 --> 00:11:11.700
standard deviations we are away
from the mean and

00:11:11.700 --> 00:11:12.760
you square that.

00:11:12.760 --> 00:11:16.590
You take the square root of
that thing and invert it and

00:11:16.590 --> 00:11:17.880
that's the normal distribution.

00:11:17.880 --> 00:11:18.930
Anyway, I wanted to do that.

00:11:18.930 --> 00:11:20.590
I thought it was neat
and it's interesting to

00:11:20.590 --> 00:11:21.290
play around with it.

00:11:21.290 --> 00:11:23.850
Then that way if you see it in
any of these other forms in the

00:11:23.850 --> 00:11:26.070
rest of your life your won't
say what's that, I thought the

00:11:26.070 --> 00:11:29.240
normal distribution was this or
it was this and now you know.

00:11:29.240 --> 00:11:31.480
With that said, let's play
around a little bit with

00:11:31.480 --> 00:11:34.060
this normal distribution.

00:11:34.060 --> 00:11:36.370
So this spreadsheet,
I've plotted the

00:11:36.370 --> 00:11:36.880
normal distribution.

00:11:36.880 --> 00:11:38.780
You can change the assumptions
that are in this kind

00:11:38.780 --> 00:11:40.670
of green, blue color.

00:11:40.670 --> 00:11:44.190
So right now it's plotting it
with a mean of zero and a

00:11:44.190 --> 00:11:45.360
standard deviation of 4.

00:11:45.360 --> 00:11:47.930
I just write the variance here
just for your information,

00:11:47.930 --> 00:11:50.090
the variance is just the
standard deviation squared.

00:11:50.090 --> 00:11:52.810
And so what happens when
you change the mean?

00:11:52.810 --> 00:11:57.170
So if the mean goes from 0
to let's say it goes to 5.

00:11:57.170 --> 00:12:00.740
Notice, this graph just
shifted to the right by 5.

00:12:00.740 --> 00:12:00.920
Right?

00:12:00.920 --> 00:12:03.400
It was centered here, now
it's centered over here.

00:12:03.400 --> 00:12:07.270
If we make it minus
5, what happens?

00:12:07.270 --> 00:12:10.070
The whole bell curve just
shifts 5 to the left

00:12:10.070 --> 00:12:11.270
from the center.

00:12:11.270 --> 00:12:13.320
Now what happens when you
change the standard deviation.

00:12:13.320 --> 00:12:20.490
The variance is the average
squared distance from the mean,

00:12:20.490 --> 00:12:22.010
the standard deviation is
the square root of that.

00:12:22.010 --> 00:12:25.150
So it's kind of, not exactly,
but kind of the average

00:12:25.150 --> 00:12:26.180
distance from the mean.

00:12:26.180 --> 00:12:29.790
So the smaller the standard
deviation the closer a

00:12:29.790 --> 00:12:31.440
lot of the points are
going to be the mean.

00:12:31.440 --> 00:12:33.763
We should get kind of a
narrower graph and let's

00:12:33.763 --> 00:12:34.740
see if that happens.

00:12:34.740 --> 00:12:38.630
When the standard deviation
is 2, we see that.

00:12:38.630 --> 00:12:41.550
The graph you're more likely
to be really close to the

00:12:41.550 --> 00:12:42.820
mean than further away.

00:12:42.820 --> 00:12:46.450
If you make the standard
deviation, if you make it 10,

00:12:46.450 --> 00:12:48.470
all of the sudden you got a
really flat graph and this

00:12:48.470 --> 00:12:50.140
thing keeps going on forever.

00:12:50.140 --> 00:12:52.880
And that's a key difference:
the binomial distribution

00:12:52.880 --> 00:12:54.000
is always finite.

00:12:54.000 --> 00:12:56.450
You can only have a finite
number of values while the

00:12:56.450 --> 00:12:59.640
normal distribution is
defined over the entire

00:12:59.640 --> 00:13:01.860
real number line.

00:13:01.860 --> 00:13:06.490
So the probability, if you
have a mean of minus 5 and a

00:13:06.490 --> 00:13:09.330
standard deviation of 10, the
probability of getting a

00:13:09.330 --> 00:13:15.200
thousand here is very low but
there's some probability.

00:13:15.200 --> 00:13:18.800
There's some probably that all
of the atoms in my body just

00:13:18.800 --> 00:13:20.720
arrange perfectly that I fall
through the seat

00:13:20.720 --> 00:13:21.220
I'm sitting on.

00:13:21.220 --> 00:13:23.850
It's very unlikely and it
probably won't happen in

00:13:23.850 --> 00:13:26.750
the life of the universe
but it can happen.

00:13:26.750 --> 00:13:29.260
And that can be described by
normal distribution because it

00:13:29.260 --> 00:13:31.880
says, anything can happen
although it could

00:13:31.880 --> 00:13:32.480
very unprobable.

00:13:35.090 --> 00:13:36.940
The thing I talked about at the
beginning of the video is when

00:13:36.940 --> 00:13:41.370
you figure out a normal
distribution you can't just

00:13:41.370 --> 00:13:44.260
look at this point on the graph
-- let me get the drop pen tool

00:13:44.260 --> 00:13:47.580
back -- you have to figure out
the area under the curve

00:13:47.580 --> 00:13:50.570
between two points.

00:13:50.570 --> 00:13:52.755
So if I wanted to say -- let's
say this is our distribution

00:13:52.755 --> 00:13:55.950
-- and I said what is the
probability that I get 0.

00:13:55.950 --> 00:13:57.550
I don't know what phenomena
this is describing

00:13:57.550 --> 00:13:59.210
but at 0 happened.

00:13:59.210 --> 00:14:04.880
If I say exactly 0, the
probability is 0 -- I shouldn't

00:14:04.880 --> 00:14:07.290
use 0 too much -- because
the area under the curve,

00:14:07.290 --> 00:14:09.960
just under 0, there's no
area, it's just a line.

00:14:09.960 --> 00:14:11.280
You have to say
between a range.

00:14:11.280 --> 00:14:17.290
So you have to say the
probability between -- and

00:14:17.290 --> 00:14:28.260
actually I can type it in here
-- between minus .005 and plus

00:14:28.260 --> 00:14:33.540
.05 is -- well it rounded -- it
says they're close to 0.

00:14:33.540 --> 00:14:38.340
Let me do it, between
minus 1 and between 1.

00:14:38.340 --> 00:14:40.450
It calculated at 7 percent
and I'll show you how I

00:14:40.450 --> 00:14:43.360
calculated this in a second.

00:14:43.360 --> 00:14:44.910
So let me get the
screen draw tool.

00:14:44.910 --> 00:14:46.440
So what did I just do?

00:14:46.440 --> 00:14:49.690
Between minus 1 and 1 -- and
I'll show you the behind the

00:14:49.690 --> 00:14:52.910
scenes, what excel is doing --
we're going from minus 1, which

00:14:52.910 --> 00:14:56.770
is roughly right here, to 1.

00:14:56.770 --> 00:15:01.090
And we're calculating the
area under the curve.

00:15:01.090 --> 00:15:05.220
We're calculating this area or
for those of you who know

00:15:05.220 --> 00:15:11.710
calculus, we're calculating the
integral from minus 1 to 1 of

00:15:11.710 --> 00:15:14.360
this function where the
standard deviation is right

00:15:14.360 --> 00:15:19.040
here, is 10 and the
mean is minus 5.

00:15:19.040 --> 00:15:20.070
Actually, let me put that in.

00:15:20.070 --> 00:15:23.200
So we're calculating, for this
example, the way it's drawn

00:15:23.200 --> 00:15:28.130
right here, the normal
distribution function, our

00:15:28.130 --> 00:15:34.590
standard deviation is 10 times
square root of 2 pi times e to

00:15:34.590 --> 00:15:40.180
the minus 1/2 times
x minus our mean.

00:15:40.180 --> 00:15:42.100
Our mean is negative right now.

00:15:42.100 --> 00:15:46.970
Our mean is minus 5 so it's x
plus 5 over the standard

00:15:46.970 --> 00:15:50.440
deviation squared which is the
variance, so that's

00:15:50.440 --> 00:15:55.730
100 squared dx.

00:15:55.730 --> 00:15:59.040
This is what this number is
right here, this 7 percent

00:15:59.040 --> 00:16:02.150
or actually .07 is the
area right around there.

00:16:02.150 --> 00:16:04.710
Now, unfortunately for us in
the world, this isn't an

00:16:04.710 --> 00:16:07.400
easy integral to evaluate
analytical, even for those of

00:16:07.400 --> 00:16:08.870
us who know our calculus.

00:16:08.870 --> 00:16:11.600
So this tends to be
done numerically.

00:16:11.600 --> 00:16:14.850
And kind of an easy way to do
this is -- well not as easy way

00:16:14.850 --> 00:16:18.620
-- a function has been defined
called the cumuluative

00:16:18.620 --> 00:16:21.860
distribution function that is a
useful tool for figuring

00:16:21.860 --> 00:16:22.560
out this area.

00:16:22.560 --> 00:16:25.410
So what the cumulative
distribution function is

00:16:25.410 --> 00:16:28.350
essentially -- let me call it
the cumulative distribution

00:16:28.350 --> 00:16:32.220
function -- it's
a function of x.

00:16:32.220 --> 00:16:35.340
It gives us the area under
the curve, under this curve.

00:16:35.340 --> 00:16:39.040
So let's say that this is x
right here, that's our x.

00:16:39.040 --> 00:16:42.240
It tells you the area
under the curve up to x.

00:16:42.240 --> 00:16:44.130
So another way to think about
it, it tells you what is the

00:16:44.130 --> 00:16:46.980
probability that you land
at some value less

00:16:46.980 --> 00:16:47.990
than your x value.

00:16:47.990 --> 00:16:53.580
So it's the area from minus
infinity to x of our

00:16:53.580 --> 00:16:55.050
probability density function.

00:17:00.350 --> 00:17:04.220
When you actually use the Excel
normal distribution function,

00:17:04.220 --> 00:17:08.300
let me say norm distribution.

00:17:08.300 --> 00:17:11.630
You have to give it your
x value, you give it the

00:17:11.630 --> 00:17:14.230
mean, you give it the
standard deviation.

00:17:14.230 --> 00:17:17.240
And then you say whether you
want the cumulative

00:17:17.240 --> 00:17:19.660
distribution, in which case you
say true or you want just this

00:17:19.660 --> 00:17:21.610
normal distribution,
which you say false.

00:17:21.610 --> 00:17:24.670
So if you wanted to graph
this right here, you

00:17:24.670 --> 00:17:27.590
would say false in caps.

00:17:27.590 --> 00:17:29.390
If you wanted to graph the
cumulative distribution

00:17:29.390 --> 00:17:32.800
function which I do down here
-- let me move this down a

00:17:32.800 --> 00:17:45.580
little bit, let me get out of
pen tool -- then you say true

00:17:45.580 --> 00:17:46.670
when you make that Excel call.

00:17:46.670 --> 00:17:48.630
So this is a cumulative di
distribution function for this.

00:17:51.330 --> 00:17:53.260
This is a normal distribution,
here's a cumulative

00:17:53.260 --> 00:17:53.890
distribution.

00:17:53.890 --> 00:17:55.600
Just so you get the intuition.

00:17:55.600 --> 00:17:57.280
If you want to know, what
is the probability that I

00:17:57.280 --> 00:17:59.720
get a value less than 20?

00:17:59.720 --> 00:18:01.660
So I can get any value
less than 20 given

00:18:01.660 --> 00:18:04.320
this distribution.

00:18:04.320 --> 00:18:06.990
The cumulative distribution
right here, -- let me make it

00:18:06.990 --> 00:18:11.260
so you can see -- if you go to
20 you just go right to that

00:18:11.260 --> 00:18:13.510
point there and you say wow,
the probability of getting

00:18:13.510 --> 00:18:16.080
20 or less is pretty high.

00:18:16.080 --> 00:18:17.220
It's approaching 100 percent.

00:18:17.220 --> 00:18:19.210
That makes sense because
most of the area under this

00:18:19.210 --> 00:18:21.120
curve is less than 20.

00:18:21.120 --> 00:18:23.130
Or if you said what's the
probability of getting

00:18:23.130 --> 00:18:25.740
less than minus five.

00:18:25.740 --> 00:18:27.630
Well minus 5 is the mean
so half of your results

00:18:27.630 --> 00:18:29.160
should be above that and
half should be below.

00:18:29.160 --> 00:18:31.390
And if you go to this point
right here you could see that

00:18:31.390 --> 00:18:33.130
this right here is 50 percent.

00:18:33.130 --> 00:18:36.420
So the probability of getting
less than minus 5 is

00:18:36.420 --> 00:18:38.490
exactly 50 percent.

00:18:38.490 --> 00:18:43.350
So what you do is, if I wanted
to know the probability of

00:18:43.350 --> 00:18:47.410
getting between negative 1 and
1 what I do is -- let me get

00:18:47.410 --> 00:18:53.600
back to my pen tool -- I figure
out what is the probability

00:18:53.600 --> 00:18:57.050
of getting minus 1 or lower.

00:18:57.050 --> 00:19:00.560
So I figure out
this whole area.

00:19:00.560 --> 00:19:02.510
And then I figure out the
probability of getting one or

00:19:02.510 --> 00:19:05.356
lower which is this whole area
-- well I'm going to give it

00:19:05.356 --> 00:19:10.810
a different color -- 1 or
lower is everything there.

00:19:10.810 --> 00:19:14.740
And I subtract the yellow area
from the magenta area and I'll

00:19:14.740 --> 00:19:18.870
just get what's ever
left over here.

00:19:18.870 --> 00:19:20.670
That's exactly what I did
in the spread sheet.

00:19:23.292 --> 00:19:25.322
Let me scroll down.

00:19:25.322 --> 00:19:28.030
This might be taxing my
computer by taking the

00:19:28.030 --> 00:19:30.280
screen capture with it.

00:19:30.280 --> 00:19:33.220
So what I did is I evaluated
the cumulative distribution

00:19:33.220 --> 00:19:39.260
function at one to
be right there.

00:19:39.260 --> 00:19:42.740
And I evaluate the cumulative
distribution function at minus

00:19:42.740 --> 00:19:45.410
1, which is right there.

00:19:45.410 --> 00:19:48.490
And the difference between
these two, I subtract this

00:19:48.490 --> 00:19:52.560
number from this number and
that tells me essentially

00:19:52.560 --> 00:19:55.250
the probability that I'm
between those two numbers.

00:19:55.250 --> 00:20:02.490
Or another way to think about
it, the area right here.

00:20:02.490 --> 00:20:04.860
I really encourage you to play
with this and explore the excel

00:20:04.860 --> 00:20:07.130
formulas and everything.

00:20:07.130 --> 00:20:11.640
This area right here
would be minus 1 and 1.

00:20:18.260 --> 00:20:20.590
Just so you know this graph,
the central line right

00:20:20.590 --> 00:20:22.250
here, this is the mean.

00:20:22.250 --> 00:20:24.560
And then these two lines I drew
right here, these are 1

00:20:24.560 --> 00:20:27.230
standard deviation below and 1
standard deviation

00:20:27.230 --> 00:20:30.280
above the mean.

00:20:30.280 --> 00:20:32.430
Some people think what's the
probability that land within

00:20:32.430 --> 00:20:33.630
one standard deviation
of the mean?

00:20:33.630 --> 00:20:34.540
Well that's easy to do.

00:20:34.540 --> 00:20:36.850
What I can do is I'll
just click on this.

00:20:36.850 --> 00:20:39.810
And I'll just call this, what's
the probability that I land

00:20:39.810 --> 00:20:44.110
between 1 standard deviation --
the mean is minus 5 -- one

00:20:44.110 --> 00:20:47.720
standard deviation below the
mean is minus 15 and one

00:20:47.720 --> 00:20:52.580
standard deviation above the
mean is 10 plus minus 5 is 5.

00:20:52.580 --> 00:20:55.190
So that's between 5 and 15.

00:20:55.190 --> 00:20:56.740
So 68.3 percent.

00:20:56.740 --> 00:21:00.130
That's actually always the case
that you have a 68.3 percent

00:21:00.130 --> 00:21:04.800
probability of landing within
one standard deviation of the

00:21:04.800 --> 00:21:06.920
mean, assuming you have
a normal distribution.

00:21:06.920 --> 00:21:11.640
So once again, that number
represents the area under

00:21:11.640 --> 00:21:16.840
the curve here, this
area under the curve.

00:21:16.840 --> 00:21:19.670
And the way you get it
is with the cumulative

00:21:19.670 --> 00:21:20.660
distribution function.

00:21:20.660 --> 00:21:22.730
I'll go down here.

00:21:22.730 --> 00:21:27.410
Every time I move this I have
to get rid of the pen tool.

00:21:27.410 --> 00:21:33.370
You evaluate it at plus
5, which is right here.

00:21:33.370 --> 00:21:36.120
This is 1 standard deviation
above the mean, that's a

00:21:36.120 --> 00:21:37.370
number right around there.

00:21:37.370 --> 00:21:40.220
It looks like it's like 80
something percent, maybe

00:21:40.220 --> 00:21:41.920
90 percent roughly.

00:21:41.920 --> 00:21:44.050
And then you evaluate it at
1 standard deviation below

00:21:44.050 --> 00:21:46.120
the mean which is minus 15.

00:21:46.120 --> 00:21:51.230
And this one looks like roughly
15 percent or so, 15, 16, maybe

00:21:51.230 --> 00:21:54.220
17 percent, I'll
say 18 percent.

00:21:54.220 --> 00:21:57.390
But the big picture is when
you subtract this value

00:21:57.390 --> 00:22:00.740
from this value you get
the probability that you

00:22:00.740 --> 00:22:01.910
land between those two.

00:22:01.910 --> 00:22:05.010
And that's because this
value tells the probability

00:22:05.010 --> 00:22:06.600
that you're less.

00:22:06.600 --> 00:22:08.080
So when you go to the
cumulative distribution

00:22:08.080 --> 00:22:09.540
function you get
that right there.

00:22:12.630 --> 00:22:15.630
It keeps crawling
back and forth.

00:22:21.280 --> 00:22:27.660
So when you go to five -- and
you just go right over here --

00:22:27.660 --> 00:22:29.670
this is essentially tells you
this area under the curve, the

00:22:29.670 --> 00:22:31.220
probability that you're
less than or equal to

00:22:31.220 --> 00:22:32.500
5, everything up there.

00:22:32.500 --> 00:22:35.370
And then when you evaluate it
at minus 15 down here, it

00:22:35.370 --> 00:22:38.430
tells you the probability
that you're back here.

00:22:38.430 --> 00:22:41.690
So when you subtract this from
the larger thing you're just

00:22:41.690 --> 00:22:46.190
left with what's under
the curve right there.

00:22:46.190 --> 00:22:49.530
Just to understand this
spreadsheet a little bit better

00:22:49.530 --> 00:22:53.930
because I really want you to
play with it and see what

00:22:53.930 --> 00:22:56.750
happens if I make this
distribution, the mean was

00:22:56.750 --> 00:22:58.605
minus 5 now let me make it 5.

00:22:58.605 --> 00:23:00.950
It just shifted to the right.

00:23:00.950 --> 00:23:03.410
It just moved over
to the right by 5.

00:23:03.410 --> 00:23:03.890
Whoops.

00:23:03.890 --> 00:23:04.920
Let me use the pen tool.

00:23:04.920 --> 00:23:06.595
It just moved over
to the right by 5.

00:23:11.440 --> 00:23:13.610
If I were to try to make the
standard deviation smaller

00:23:13.610 --> 00:23:15.670
we'll see that the whole thing
just gets a little bit tighter.

00:23:15.670 --> 00:23:17.670
Let's make it 6.

00:23:17.670 --> 00:23:20.430
And all of a sudden this looks
a like a tighter curve, we make

00:23:20.430 --> 00:23:24.370
it 2, it becomes even tighter.

00:23:24.370 --> 00:23:27.290
I really want you to play with
this and play with the formula

00:23:27.290 --> 00:23:29.620
and get an intuitive feeling
for this, the cumulative

00:23:29.620 --> 00:23:31.665
distribution function and think
a lot about how it relates to

00:23:31.665 --> 00:23:35.900
the binomial distribution and I
cover that in the last video.

00:23:35.900 --> 00:23:37.890
To plot this I just took
each of these points.

00:23:37.890 --> 00:23:42.170
I went to plot the points
between minus 20 and 20 and

00:23:42.170 --> 00:23:43.990
I just incremented by 1.

00:23:43.990 --> 00:23:45.460
I just decided to
increment by 1.

00:23:45.460 --> 00:23:47.720
So this isn't a continues
curve, it's actually just

00:23:47.720 --> 00:23:49.270
plotting a point at each
point and connecting

00:23:49.270 --> 00:23:51.220
it with the line.

00:23:51.220 --> 00:23:52.890
Then I did the distance
between each of those

00:23:52.890 --> 00:23:55.110
points and the mean.

00:23:55.110 --> 00:24:01.020
So I just took 0 minus 5,
this is this distance.

00:24:01.020 --> 00:24:04.270
So this just tells you
the point minus 20 is

00:24:04.270 --> 00:24:07.150
25 less than the mean.

00:24:07.150 --> 00:24:08.400
That's all I did there.

00:24:08.400 --> 00:24:10.560
Then I divided that by the
standard deviation and this

00:24:10.560 --> 00:24:15.000
is the z score, the
standard z score.

00:24:15.000 --> 00:24:18.420
This tells me how many standard
deviations is minus 20

00:24:18.420 --> 00:24:19.720
away from the mean.

00:24:19.720 --> 00:24:24.250
It's 12 and a half standard
deviations below the mean.

00:24:24.250 --> 00:24:26.410
Then I use that and I just
plugged it into essentially

00:24:26.410 --> 00:24:29.270
this formula to figure out
the height of the function.

00:24:29.270 --> 00:24:33.590
Let's say at minus 20
the height is very low.

00:24:33.590 --> 00:24:36.130
Let's say at minus 2 the
heights a little bit better,

00:24:36.130 --> 00:24:38.100
the heights going to be
some place, it's going

00:24:38.100 --> 00:24:42.130
to be right there.

00:24:42.130 --> 00:24:44.180
And so that gives
me that value.

00:24:44.180 --> 00:24:48.470
But then to actually figure out
the probability of that, what I

00:24:48.470 --> 00:24:50.940
do is I calculate the
cumulative distribution

00:24:50.940 --> 00:24:52.960
function.

00:24:52.960 --> 00:24:56.470
Well this is the probability
that you're less than that.

00:24:56.470 --> 00:24:59.520
So the area under the curve
below that which is very small.

00:24:59.520 --> 00:25:02.270
It's not zero, I know it
looks like 0 here but that's

00:25:02.270 --> 00:25:03.720
only because I round it.

00:25:03.720 --> 00:25:06.320
It's going to be 0001,
it's going to be a

00:25:06.320 --> 00:25:07.150
really small number.

00:25:07.150 --> 00:25:10.200
There's some probability that
we even get minus a thousand.

00:25:10.200 --> 00:25:12.760
Another intuitive thing that
you really should have a sense

00:25:12.760 --> 00:25:18.030
for is the integral over this
or the entire area of the curve

00:25:18.030 --> 00:25:20.620
has to be 1 because that
takes into account all

00:25:20.620 --> 00:25:22.210
possible circumstances.

00:25:22.210 --> 00:25:24.810
And that should happen if we
put a suitably smaller number

00:25:24.810 --> 00:25:27.410
here and a suitably
large number here.

00:25:27.410 --> 00:25:28.530
There you go, we
get 100 percent.

00:25:28.530 --> 00:25:29.540
Although, this isn't
a 100 percent.

00:25:29.540 --> 00:25:32.180
We'd have to go from minus
infinity to plus infinity to

00:25:32.180 --> 00:25:33.370
really get a 100 percent.

00:25:33.370 --> 00:25:34.840
It's just rounding
to 100 percent.

00:25:34.840 --> 00:25:39.720
It's probably 99.99999 percent
or something like that.

00:25:39.720 --> 00:25:43.900
And so to actually calculate
this, what I do is I take the

00:25:43.900 --> 00:25:46.920
cumulative distribution
function of this point and I

00:25:46.920 --> 00:25:48.730
subtract from that the
cumulative distribution

00:25:48.730 --> 00:25:49.680
function of that point.

00:25:49.680 --> 00:25:52.270
And that's where I got
this 100 percent from.

00:25:52.270 --> 00:25:57.760
Anyway, hopefully that will
give you a good feel for

00:25:57.760 --> 00:25:59.570
the normal distribution.

00:25:59.570 --> 00:26:01.830
I really encourage you to play
with the spreadsheet and to

00:26:01.830 --> 00:26:03.320
even make a spreadsheet
like this yourself.

00:26:03.320 --> 00:26:06.230
In a future exercise we'll
actually use this type of

00:26:06.230 --> 00:26:09.370
spreadsheet as an input
into other models.

00:26:09.370 --> 00:26:12.420
If we're doing a financial
model and if we say our revenue

00:26:12.420 --> 00:26:15.220
has a normal distribution
around some expected value,

00:26:15.220 --> 00:26:18.000
what is the distribution
of our net income?

00:26:18.000 --> 00:26:20.050
Or we could think of a
hundred other different

00:26:20.050 --> 00:26:20.930
types of examples.

00:26:20.930 --> 00:26:23.570
Anyway, see you in
the next video.