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Chi-square distribution introduction | Probability and Statistics | Khan Academy - YouTube
Channel: Khan Academy
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In this video, we'll
just talk a little bit
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about what the chi-square
distribution is, sometimes
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called the chi-squared
distribution.
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And then in the next
few videos, we'll
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actually use it to
really test how well
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theoretical distributions
explain observed ones,
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or how good a fit
observed results
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are for theoretical
distributions.
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So let's just think
about it a little bit.
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So let's say I have
some random variables.
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And each of them
are independent,
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standard, normal, normally
distributed random variables.
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So let me just remind
you what that means.
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So let's say I have the random
variable X. If X is normally
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distributed, we
could write that X
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is a normal random
variable with a mean of 0
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and a variance of 1.
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Or you could say that
the expected value of X,
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is equal to 0, or
in that the variance
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of our random variable
X is equal to 1.
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Or just to visualize
it is that, when
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we take an instantiation
of this very variable,
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we're sampling from a
normal distribution,
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a standardized normal
distribution that
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looks like this.
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Mean of 0 and then a variance
of 1, which would also mean,
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of course, a standard
deviation of 1.
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So that could be the standard
deviation, or the variance,
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or the standard deviation,
that would be equal to 1.
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So a chi-square
distribution, if you just
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take one of these
random variables--
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and let me define it this way.
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Let me define a new
random variable.
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Let me define a
new random variable
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Q that is equal to--
you're essentially
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sampling from this the
standard normal distribution
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and then squaring
whatever number you got.
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So it is equal to this
random variable X squared.
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The distribution for this
random variable right
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here is going to be an example
of the chi-square distribution.
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Actually what we're going
to see in this video is
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that the chi-square, or the
chi-squared distribution
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is actually a set
of distributions
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depending on how
many sums you have.
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Right now, we only have
one random variable
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that we're squaring.
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So this is just one
of the examples.
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And we'll talk more
about them in a second.
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So this right
here, this we could
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write that Q is a chi-squared
distributed random variable.
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Or that we could use
this notation right here.
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Q is-- we could
write it like this.
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So this isn't an X anymore.
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This is the Greek
letter chi, although it
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looks a lot like a curvy X. So
it's a member of chi-squared.
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And since we're only
taking one sum over here--
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we're only taking the
sum of one independent,
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normally distributed, standard
or normally distributed
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variable, we say that this
only has 1 degree of freedom.
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And we write that over here.
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So this right here is
our degree of freedom.
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We have 1 degree of
freedom right over there.
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So let's call this Q1.
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Let's say I have
another random variable.
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Let's call this Q-- let me
do it in a different color.
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Let me do Q2 in blue.
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Let's say I have another
random variable, Q2,
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that is defined as-- let's say I
have one independent, standard,
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normally distributed variable.
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I'll call that X1.
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And I square it.
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And then I have another
independent, standard,
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normally distributed
variable, X2.
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And I square it.
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So you could imagine
both of these guys
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have distributions like this.
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And they're independent.
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So get to sample Q2,
you essentially sample
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X1 from this distribution,
square that value, sample X2
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from the same distribution,
essentially, square that value,
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and then add the two.
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And you're going to get Q2.
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This over here-- here we
would write-- so this is Q1.
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Q2 here, Q2 we would
write is a chi-squared,
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distributed random variable
with 2 degrees of freedom.
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Right here.
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2 degrees of freedom.
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And just to visualize
kind of the set
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of chi-squared distributions,
let's look at this over here.
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So this, I got this
off of Wikipedia.
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This shows us some of the
probability density functions
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for some of the
chi-square distributions.
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This first one over here,
for k of equal to 1,
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that's the degrees of freedom.
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So this is essentially our Q1.
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This is our probability
density function for Q1.
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And notice it really
spikes close to 0.
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And that makes sense.
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Because if you are
sampling just once
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from this standard
normal distribution,
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there's a very high
likelihood that you're
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going to get something
pretty close to 0.
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And then if you square
something close to 0-- remember,
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these are decimals, they're
going to be less than 1,
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pretty close to 0-- it's
going to become even smaller.
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So you have a high probability
of getting a very small value.
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You have high probabilities
of getting values
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less than some threshold,
this right here, less than,
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I guess, this is 1 right here.
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So the less than 1/2.
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And you have a very
low probability
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of getting a large number.
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I mean, to get a 4, you
would have to sample a 2
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from this distribution.
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And we know that 2 is--
actually it's 2 variances
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or 2 standard deviations
from the mean.
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So it's less likely.
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And actually that's to get a 4.
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So to get even
larger numbers are
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going to be even less likely.
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So that's why you see
this shape over here.
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Now when you have 2
degrees of freedom,
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it moderates a little bit.
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This is the shape,
this blue line right
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here is the shape of Q2.
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And notice you're a little
bit less likely to get values
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close to 0 and a little bit
more likely to get numbers
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further out.
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But it still is kind of
shifted or heavily weighted
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towards small numbers.
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And then if we had
another random variable,
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another chi-squared
distributed random variable--
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so then we have, let's say, Q3.
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And let's define
it as the sum of 3
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of these independent
variables, each of them
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that have a standard
normal distribution.
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So X1, X2 squared
plus X3 squared.
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Then all of a sudden, our
Q3-- this is Q2 right here--
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has a chi-squared distribution
with 3 degrees of freedom.
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And so this guy
right over here--
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that will be this green line.
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Maybe I should have
done this in green.
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This will be this
green line over here.
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And then notice,
now it's starting
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to become a little
bit more likely
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that you'd get values
in this range over here
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because you're taking the sum.
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Each of these are going
to be pretty small values,
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but you're taking the sum.
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So it starts to shift it a
little over to the right.
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And so the more degrees
of freedom you have,
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the further this lump
starts to move to the right
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and, to some degree, the
more symmetric it gets.
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And what's interesting
about this,
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I guess it's different than
almost every other distribution
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we've looked at, although
we've looked at others that
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have this property as well,
is that you can't have
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a value below 0 because
we're always just squaring
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these values.
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Each of these guys can
have values below 0.
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They're normally distributed.
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They could have negative values.
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But since we're squaring and
taking the sum of squares,
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this is always going
to be positive.
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And the place that this
is going to be useful--
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and we're going to see
in the next few videos--
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is in measuring essentially
error from an expected value.
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And if you took take
this total error,
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you can figure out
the probability
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of getting that total error
if you assume some parameters.
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And we'll talk more about
it in the next video.
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Now with that said, I
just want to show you
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how to read a chi-squared
distribution table.
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So if I were to ask you, if
this is our distribution--
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let me pick this
blue one right here.
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So over here, we have
2 degrees of freedom
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because we're adding 2
of these guys right here.
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If I were to ask you, what is
the probability of Q2 being
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greater than-- or, let
me put it this way.
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What is the probability of
Q2 being greater than 2.41?
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And I'm picking that
value for a reason.
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So I want the probability of
Q2 being greater than 2.41.
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What I want to do is I'll
look at a chi-square table
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like this.
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Q2 is a version of chi-squared
with 2 degrees of freedom.
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So I look at this row right
here under 2 degrees of freedom.
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And I want the probability of
getting a value above 2.41.
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And I picked 2.41 because
it's actually at this table.
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And so most of these
chi-squared-- the reason
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why we have these weird
numbers like this instead
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of whole numbers or
easy-to-read fractions
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is it is actually
driven by the p value.
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It's driven by the
probability of getting
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something larger
than that value.
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So normally you would
look at the other way.
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You'd say, OK, if I want to
say, what chi-squared value
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for 2 degrees of
freedom, there's
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a 30% chance of getting
something larger than that?
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Then I would look up 2.41.
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But I'm doing it
the other way just
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for the sake of this video.
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So if I want the probability
of this random variable
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right here being greater
than 2.41, or its p value,
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we read it right here.
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It is 30%.
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And just to visualize
it on this chart,
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this chi-square distribution--
this was Q2, the blue one,
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over here-- 2.41 is
going to sit-- let's see.
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This is 3.
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This is 2.5.
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So 2.41 is going to be
someplace right around here.
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So essentially,
what that table is
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telling us is, this entire
area under this blue line right
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here, what is that?
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And that right there is
going to be 30% of-- well,
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it's going to be 0.3.
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Or you could view it as
30% of the entire area
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under this curve, because
obviously all the probabilities
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have to add up to 1.
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So that's our intro to the
chi-square distribution.
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In the next video,
we're actually
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going to use it to make some,
or to test some, inferences.
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