-
In this video we're going to
cover what is arguably, the
-
single most important concept
in all the statistics.
-
Well if you go into almost any
scientific field you might even
-
argue that it's the single
most important concept.
-
I've actually told people that
it's kind of sad they don't
-
cover this in the
core curriculum.
-
Everyone should know about is
because it touches on every
-
single aspect of our lives and
that's the normal distribution
-
or the Gaussian distribution
or the bell curve.
-
And just to kind of give you a
preview of what it is, my
-
preview will actually make it
seem pretty strange but as we
-
go through this video hopefully
you'll get a little bit more
-
intuition of what
it's all about.
-
The Gaussian distribution or
the normal distribution,
-
they're two words
for the same thing.
-
It was actually Gauss
who came up with it.
-
I think he was studying
astronomical phenomenon
-
when he did.
-
But it's a probability density
function just like we studies
-
the Poisson distribution.
-
It's just like that.
-
And just to give you the
preview it looks like this.
-
The probability of getting
any x, and it's a
-
class of probability
distribution functions.
-
Just like the binomial
distribution is and the Poisson
-
distribution, it's based on
a bunch of parameters.
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This is how you would
traditionally see it written in
-
a lot of textbooks and if we
have time, I'd like to
-
rearrange the algebra just to
get a little bit more intuition
-
of how it all works out.
-
Or maybe we could get
some insights on where
-
it all came from.
-
I'm not going to prove it in
this video, that's a little
-
bit beyond our scope.
-
Although, I do want to do it
and there's actually some
-
really neat mathematics
that might show up.
-
If you're a math lead there's
something called Sterling's
-
formula what you might want
to do a Wikipedia search on,
-
which is really fascinating.
-
It approximates factorials
with essentially a
-
continuous function.
-
But I won't go into
that right now.
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The normal distribution is 1
over -- this is how it's
-
normally written -- the
standard deviation times the
-
square root of 2 pi times
e to the minus 1/2.
-
Well, I like to write it this
way, it easier to remember,
-
times whatever value you're
trying to get minus the mean of
-
our distribution divided by the
standard deviation of our
-
distribution squared.
-
And so if you think about it,
actually this is a good thing
-
to just notice right now.
-
This is how far I'm from the
mean and we're dividing that
-
by the standard deviation of
whatever our distribution is.
-
This is a preview of actually a
normal distribution that I've
-
plotted, the purple line here
is a normal distribution.
-
Initially the whole exercise --
I know I jump around a little
-
bit -- is to show you that the
normal distribution is a good
-
approximation for the binomial
distribution and vice versa.
-
If you take enough trials in
your binomal distribution and
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we'll touch on that a second.
-
The intuition of this term
right here I think is
-
interesting because we're
saying, how far are we away
-
from the mean, we're dividing
by the standard deviation.
-
So this whole term right here
is how many standard deviations
-
we are away from the mean.
-
This is actually called
the standard z score.
-
One thing I found in statistics
is there's a lot of words a lot
-
of definitions and they all
sound very fancy, the
-
standard z score.
-
But the underlying concept
is pretty straightforward.
-
Let's say I had a probability
distribution and I get some x
-
value that's out here and it's
3 and a half the standard
-
deviations is away from the
mean, then it's standard
-
z score is 3 and a half.
-
Anyway let's focus on the
purpose of this video.
-
So that's what the normal
distribution, I guess
-
the probability density
function for the normal
-
distribution looks like.
-
But how did it get there?
-
By the end of this video you
should at least feel
-
comfortable that this is a good
approximation for the binomial
-
distribution if you're
taking enough trials.
-
And that's what's fascinating
about the normal distribution
-
is that if you have the sum --
and I'll do a whole other video
-
on the central limit theorem --
but if you have the sum of many
-
independent trials approaching
infinity, that the distribution
-
of those, even though the
distribution of each of those
-
trials might have been
non-normal but the distribution
-
of the sum of all those trials
approaches the normal
-
distribution.
-
I'll talk more
about that later.
-
But that's why it's such a good
distribution to kind of assume
-
for a lot of underlying
phenomenon.
-
If you're kind of modeling
weather patterns or drug
-
interactions and you we'll talk
about where it might work well
-
and where it might
not work so well.
-
Like sometimes people might
assume things like a normal
-
distribution in finance and
we've see the financial crisis
-
that's led to a lot of
things blowing up but.
-
Anyway, let's go back to this.
-
This is a spreadsheet
right here.
-
I just made a black background
and you can downloaded it at
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khanacademy.org/downloads
Actually, if you just do that
-
you'll see all of
the downloads.
-
I haven't put it there yet, I'm
going to do it right after I
-
record the videos
downloads/normal
-
distribution.xls.
-
If you just go up to
khanacademy.org/download/
-
you'll see all the things
there and you'll see
-
this spreadsheet.
-
I encourage you to play with it
and maybe do other spreadsheets
-
were you experiment with it.
-
So this spreadsheet what we do
is we're doing a game or let's
-
say I'm sitting I'm on a street
and I flip a coin, I flip
-
a completely fair coin.
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If I get heads, this is heads,
I take a step backwards or
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let's say a step to the left.
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And if I get a tails I
take a step to the right.
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So in general I always have a
-- this is a completely fair
-
coin -- I have a 50 percent
chance of taking a step to the
-
left and I have a 50 percent
chance of taking a
-
step to the right.
-
So your intuition there is if
I told you I took a you a
-
thousand flips of the coin
you're going to keep
-
going left and right.
-
If by chance you get a bunch
of heads, you might end up
-
really kind of moving
over to the left.
-
If you get a bunch of tails you
might move over to the right.
-
And we learned already the odds
of getting a bunch of tales or
-
many more tails than heads is a
lot lower than things kind of
-
being equal or close to equal.
-
Right here what I've done --
let me scroll down a little
-
bit, I don't want to lose the
whole thing -- is I have this
-
little assumption here and I
encourage you to fill that out
-
and change it as you like.
-
This is the number
of steps I take.
-
This is the mean number of left
steps and all I did is I got
-
the probability and we
figured out the mean of the
-
binomial distribution.
-
The mean of the binomial
distribution is essentially the
-
probability of taking a left
step times the total
-
number of trials.
-
So that's equal to 5, that's
where that number comes from.
-
And then the variance -- and
I'm not sure if I went over
-
this and I need prove this to
you if I have and I'll make a
-
whole other video on the
variance of the binomial
-
distribution -- is essentially
equal to the number of trials,
-
10 times the probability of
taking the left step or kind of
-
a successful trial -- I'm
defining left as a successful
-
trial, that could be right as
well -- times the probability
-
of 1 minus the successful trial
or non successful trial.
-
In this case they're equally
probable and that's where
-
I got the 2.5 from.
-
And that's all on
the spreadsheet.
-
If you actually click on
the cell and look at the
-
actual formula I did that.
-
Although sometimes when you
see it in Excel it's a
-
little bit confusing.
-
And this is just the square
root of that number.
-
The standard deviation
is just the square
-
root of the variance.
-
That's just the
square root of 2.5.
-
And so if you look here
this says, OK what is
-
the probability that
I take 0 steps?
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So I take a total of 10 steps
-- just to understand this
-
spreadsheet -- what is the
probability that I
-
take 0 left steps?
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And just to make clear, if I
take 0 left steps that means I
-
must have taken 10 right steps.
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And I calculate this
probability -- I should have
-
drawn maybe a line here -- I
calculate this using the
-
binomial distribution.
-
And how do I do that?
-
Let me actually switch
colors just to make
-
things interesting.
-
Do they have a purple here?
-
I'll do a blue.
-
So blue for binomial.
-
So what I have here is
how many total steps?
-
There's a total of 10 steps.
-
So 10 factorial, that's kind of
the number of trials I have.
-
Of that I'm choosing
0 to go left.
-
So 0 factorial divided by
10 minus 0 factorial.
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This is 10 choose 0.
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I'm choosing 0 left steps of
the total 10 steps I'm taking
-
times the probability of 0 left
steps so, it's the probability
-
of a left step, I'm only taking
0 of them times the probability
-
of a right step, and I'm
taking 10 of those.
-
So that's where this number
came from, this .001.
-
That's what the binomial
distribution tells us.
-
And then this one similarly, is
10 factorial over 1 factorial
-
over 10 minus 1 factorial.
-
That's how I get that one.
-
And once again, if you click
on the actual cell you'll
-
see that explained.
-
We've done this multiple times.
-
This is just a
bionomial calculation.
-
Then right here, after
this line right here, you
-
can almost ignore it.
-
I did that so that I can do a
bunch of different scenarios.
-
For example, if I were to go to
my spreadsheet, and instead of
-
doing 10 I wanted to do 20
steps then everything changes.
-
And that's why down here after
you get to a certain point the
-
whole thing just
kind of repeats.
-
I'll let you think
about why I do that.
-
Maybe I should have made
a cleaner spreadsheet.
-
But it doesn't affect the
scatter plot chart that I did.
-
And so this plot in blue, and
you can't see it because the
-
purple is almost right over.
-
Actually let me make it
smaller so that you can see.
-
Let's say I only took 6 steps.
-
Well it's still hard to see the
difference between the two.
-
Once again the whole point
of this is to see that the
-
normal distribution is
a good approximation.
-
But they're so close that
you can't even see the
-
difference on mine.
-
If you only took four steps,
OK, I think you can see here.
-
Let me get my screen draw on.
-
The blue curve is
right around there.
-
This is the binomial.
-
There's only a few points
here, you the points
-
only go up to here.
-
This is if I take 0 steps left,
1 step left, 2 steps left,
-
3 steps left, 4 steps left.
-
And then I plot it and then I
say what's the probability
-
using the binomial
distribution?
-
And this is my final
position right?
-
If I take 0 steps to the left
then I take 4 steps to the
-
right so my final position
is at 4, so that's the
-
scenario right here.
-
Let me switch my color back to
yellow, it's easier to see.
-
If I take 4 steps to the left,
I take 0 steps to the right and
-
so my final position is
going to be at minus 4.
-
It's going to be here.
-
If I take an equal amount of
both, that's this scenario,
-
then I'm neutral.
-
I'm just stuck in the
middle right here.
-
I take 2 steps to the right and
then I take 2 steps to the left
-
or vice versa, I take 2 steps
to the left and then I take
-
2 steps right and I
end up right there.
-
Hopefully that makes
a little sense.
-
My phone is ringing.
-
I'll ignore that because
the normal distribution
-
is so important.
-
Actually, my 9 week old son is
watching so this is the first
-
time I have a live audience.
-
He might pick up something
about the normal distribution.
-
So the blue line right here --
I'll trace it maybe in yellow
-
so you can see it -- is the
plot of the binomial
-
distribution.
-
I connected the lines but you
see the binomial distribution
-
look something more like this.
-
This is the probability
of getting to minus 4.
-
This is the probability
of going to minus 2.
-
This right here is
the probability of
-
ending up nowhere.
-
Then this is the probability of
ending up 2 to the right and
-
this is the probability of
ending up 4 to the right.
-
This is the binomial
distribution, I just plotted
-
these points right here.
-
This is 0.375.
-
This is 0.375.
-
That's the height of that.
-
Now what I wanted to show
you is that the normal
-
distribution approximates
the binomial distribution.
-
So this right here, I wanted
to say what does the normal
-
distribution tell me is the
probability of ending up
-
with exactly 0 left steps?
-
This is a little bit tricky.
-
The binomial distribution
is a discreet probability
-
distribution.
-
You can just look at this chart
or look here and you say, what
-
is the probability of having
exactly 1 left step and 3 right
-
steps which puts me right here?
-
Well you just look at this
chart and you say oh, that
-
puts me right there.
-
I just read that probability,
it's actually .25.
-
And I say oh, I have a 25
percent chance of ending
-
up 2 steps to the right.
-
There's a 25 percent chance.
-
The normal distribution
function is a continuous
-
probability distribution so
it's a continuous curve.
-
It looks like that, it's a
bell curve and it goes off
-
to infinity and starts
approaching 0 on both sides.
-
It looks something like that.
-
This is a continuous
probability distribution.
-
You can't just take a point
here and say, what's the
-
probability that I end
up 2 feet to the right?
-
Because if you just say that
there's the actual the
-
probability of being exactly --
and you should watch with my
-
video on probability density
functions -- but the
-
probability of being exactly 2
feet to the right, exactly, I
-
mean I'm talking to the
atoms, is close to 0.
-
You actually have to specify
a range around this.
-
What I assume in this
within a half a foot
-
in either direction.
-
Right?
-
If we're talking about feet.
-
To figure that out what I did
here is I took the value of
-
the probability density
function there.
-
And I'll show you how
I evaluated that.
-
And then I multiply that by 1.
-
So it gives me this area.
-
And I use that as an
approximation for this area.
-
If you really want to be
particular about it what you
-
would do is you would take the
integral of this curve between
-
this point and this point
as a better approximation.
-
We'll do that in the future.
-
But right now I just want to
give you the intuition that the
-
binomial distribution really
does converge to the
-
normal distribution.
-
So how did I get this
number right here?
-
Well I said, what is the
probability that I think 1 left
-
step -- I kind of used less
steps as success -- of one?
-
And that equaled 1 over
the standard deviation.
-
When I only took 4 steps the
standard deviation was 1.
-
So 1 over 1.
-
Actually let me change this.
-
Let me change it to
a higher number.
-
We'll go back to the
example where I'm at 10.
-
So if this is at 10.
-
Let me go back to
my drawing tool.
-
Let me do this calculation.
-
Actually, even better let
me do this calculation.
-
So what's the probability
that I have 2 left steps?
-
If I have 2 left steps I took a
total of 10 steps so I'm going
-
to have 8 right steps and
that puts me 6 to the right.
-
So that's this
point right here.
-
So what's that probability?
-
How do I figure this out
using the probability
-
density function?
-
How do I figure this height?
-
Well I say the probability of
taking 2 left steps -- that's
-
how I calculate it, if you
actually click on the cell
-
you'll see that -- is equal to
1 over the standard deviation,
-
1.581 -- and I just directly
reference the cell there --
-
times the square root of 2 pi.
-
I'm always in awe of the whole
notion of e to the i pi is
-
equal to negative 1
and all of that.
-
But there's another
amazing thing.
-
That all of a sudden as we take
many trials we have this
-
formula that has e and pi in it
and square roots but once again
-
these two numbers just
keep showing up.
-
It tells you something about
the order of the universe
-
with a capital o.
-
But let's see, times e to
the minus 1/2 times x.
-
Well x is what we're trying
to calculate, two successes.
-
To to have exactly 2 left,
so it's 2 minus the mean.
-
So the mean is five, 2 minus
five divided by the standard
-
deviation, divided by 1.581,
all of that squared.
-
That's where this
calculation came from.
-
So I told you in the last one
this right here just tells
-
me this value up here.
-
If I want to know this
exact probability,
-
it's the area of this.
-
And if I just take one
line the are is 0.
-
Remember, in this case you can
only be 2 feet away because
-
we're taking very exact steps.
-
But what the normal
distribution is it's the
-
continuous probability density
function so it can tell us
-
what's the probability of
being 2.183 feet away?
-
Which obviously can only happen
if we're taking infinitely
-
small steps every time.
-
But that's what it's use is.
-
It happens when you start
taking an infinite
-
number of steps.
-
But it can approximate
the discreet.
-
And the way I approximate it is
I say oh, what's probability of
-
being within a foot of that.
-
And so I multiply
this height, which I
-
calculate here, times 1.
-
So let's say this has a base of
1, to calculate this area which
-
I use as an approximation.
-
So you just multiply that times
1 and that's what you get here.
-
And I just want to show you.
-
Even with just 10 trials, the
curves, the normal distribution
-
here is in purple and the
binomial distribution
-
is in blue.
-
So they're almost right
on top of each other.
-
As you can take many more steps
they almost converge right on
-
top of each other and I
encourage you to play
-
with the spreadsheets.
-
Actually, let me show
you that they converge.
-
There's a convergence worksheet
on this spreadsheet as well if
-
you click on the bottom
tab on convergence.
-
This is the same thing but I
just want to show you what
-
happens at any given point.
-
Let me explain this
spreadsheet to you.
-
So this is what's
the probability of
-
moving left, right?
-
So this is just saying, I'm
just fixing a point what's
-
the probability -- and you
can change this -- of my
-
final position being 10.
-
And this essentially tells you
that if I take 10 moves, for my
-
final position to be 10 to the
right, I have to take 10 right
-
moves and 0 left moves.
-
That's a typo right there, it
should be moves not movest.
-
If I take 20 moves to end up 10
moves to the right then I have
-
to make 15 right moves
and 5 left moves.
-
Likewise if I take a total of
80 moves, if I think 80 flips
-
of my coin to make me go left
or right, in order end up 10 to
-
the right, I to take 45 right
moves and 35 left moves in any
-
order and it will end up
with 10 to the right.
-
So what I want to figure out
is, as I start taking a bunch
-
of total moves -- here I max it
out at 170 -- if I start
-
flipping this coin an infinite
number of times, I want to
-
figure out what's the
probability that my final
-
position is 10 to the right.
-
And I want to show you that as
you take more and more moves
-
the normal distribution becomes
a better and better
-
approximation for the
binomial distribution.
-
So right here, this calculates
the binomial probability, just
-
the way we did before and you
can look at the cell
-
to figure it out.
-
I used left moves as a success.
-
So this is 10 choose 0 and
we know what that is.
-
It's 10 factorial over 0
factorial over 10 minus 0
-
factorial times 0.5 to the
0 times 0.5 to the 10th.
-
That's where that
number comes from.
-
If I go to this one right
here is calculated.
-
Actually let me write
it out because I think
-
it's interesting.
-
I have a total of 60 total
moves, so it's 60 factorial
-
over, I have to have 25 left
moves so 25 factorial.
-
So that's I'm 60 minus 25
factorial times the probability
-
of a left move and have 25 of them, times the
-
probability of a right move
and I have 35 of those.
-
So that's just what the
binomial probability
-
distribution will tell us.
-
And then it figures out the
mean and the variance for each
-
of those circumstances and you
could look at the formula.
-
But the mean is just the
probability of having
-
a left move times the
total number of moves.
-
The variance is probability of
left times probability of right
-
times total number of moves.
-
And then the normal
probability, once again, I just
-
use the normal probability.
-
So I approximate
it the same way.
-
And Excel has a normal
distribution function but I
-
actually typed in the formula
because I wanted you kind of
-
see what was under the covers
for that function that
-
Excel actually has.
-
So I actually say what's the
probability of 25 left moves?
-
No, 45 left moves.
-
So I say the probability of
45 left moves is equal to 1
-
over the standard deviation.
-
So in this situation the
standard deviation is
-
the square root of 25.
-
So it's five times 2 pi times e
to the minus 1/2 times 45 minus
-
the mean, minus 50 over the
standard deviation, which we
-
figured out was 5, squared.
-
So that tells me the value of
what the normal distribution
-
tells me for this situation
with this standard deviation
-
and this mean and then I
multiply that by 1 -- you don't
-
see that in the formula, I
don't actually write times 1 --
-
to figure out the area
under the curve.
-
Because remember it's a
continuous distribution
-
function.
-
This right here just gives me
the value but to figure out the
-
probability of being within a
foot of it I have
-
to multiply by 1.
-
I'm approximating really.
-
I really should take the
integral from there to there
-
but this little rectangle is
a pretty good approximation.
-
In this chart I show you that
as the total number of moves
-
gets larger and larger the
difference between what the
-
normal probability distribution
tells us and the binomial
-
probability of distribution
tells us, gets smaller and
-
smaller in terms of the
probability of you ending up
-
to 10 moves to the right.
-
And you can change
this number here.
-
Let me change it
just to show you.
-
You could say what's the
probability of being
-
15 moves to the right?
-
I think that something is
happening with the floating
-
point error because when you
get to large factorials I
-
think something weird
happens out here.
-
You may just have to
get even further out.
-
For 10 you can see clearly that
it converges and I'll trying to
-
figure out why I was getting
those weird saw tooth patterns.
-
Maybe while I do screen capture
something weird is happening.
-
The whole point of this was to
show you that if you want to
-
figure out the probability of
being 10 moves to the right, as
-
you take more and more flips
of your coin the normal
-
distribution becomes a much
better approximation for the
-
actual binomial distribution.
-
And as you approach infinity
they actually converge
-
to each other.
-
Anyway, that's all
for this video.
-
I'll actually do several
more videos on the normal
-
distributions because it is
such an important concept.
-
See you soon.