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How to Calculate and Interpret a Correlation (Pearson's r) - YouTube
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okay now let's take a look at
calculating a correlation coefficient
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suppose we have the following values
with scores for five people on the
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variables x and y and it's a small
example because we're going to calculate
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this by hand so I want to keep it
relatively simple here we have for
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example the first person has a score of
1 on X and a score of 600 I the second
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person to on X 4 on Y and so on so we
have 5 different people here and the 5
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people produced a score on both x and y
so let's go ahead and calculate the
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correlation or Pearson's R on this
example the first thing we want to do
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though is state our null and alternative
hypotheses so the null hypothesis States
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this is called Rho here it's the
population correlation so Rho XY equals
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0 or in other words the correlation
between x and y in the population equals
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0
and that means that there is no
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relationship between x and y in the
population and once again this means row
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and the alternative hypothesis is really
just the opposite of that notice
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everything's the same except equals four
null not equals four alternative so the
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alternative hypothesis is that the
population correlation between x and y
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is not equal to zero as you can see here
and this is really saying that there is
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a relationship between x and y in the
population it could be positive
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relationship or a negative relationship
or in other words we're conducting a
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two-tailed test here in terms of the
calculations first we'll begin with
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calculating the mean of X and the mean
of Y so to find the mean of X and wife
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starting with X we're just going to add
all of these values together and we're
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going to divide by the number of values
that there are so a 1 plus 2 Plus 3 plus
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4 plus 5 divided by 5 total gives us a
mean of 3 for X and then we'll do the
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same for y so we add all these values
and divide by 5 and that gives us a mean
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of 4 okay next we need to calculate what
are known as deviation scores for each
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variable deviation scores subtract the
mean from each variable they're called
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deviation scores because they indicate
how far each value deviates or departs
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from the mean so let's take a look at
that now you know there's a lot of
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information here on the screen but let
me walk you through it it's really not
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that bad
recall that the mean of X was 3 in the
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mean of Y was 4 so here are my X values
and all I'm doing here notice the 3 for
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the mean is taking the x value and
subtracting the mean from it as I
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mentioned a few moments ago so here
we're just going to go 1 minus 3 so you
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see that here 1 minus 3 is equal to
negative 2 then we take 2 minus 3 or
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mean again equal to negative 1 3 minus 3
0 4 minus the mean of 3 gives us 1 and 5
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minus the mean of 3 gives us 2 an
interesting thing here if your
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calculations are correct the
these deviation scores should all add up
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to zero
so notice here we have negative 2
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negative 1 that's negative 3 0 we can
ignore that then we have positive 3 so
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negative 3 plus positive 3 is 0 notice
how these all add up to 0 and that's by
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definition because the mean of 3 here is
a balance point in the distribution and
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it always balances out the deviation
scores below the mean with the deviation
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scores above the mean so this should
equal 0 when you add these up if it does
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not that means that an error was made in
the calculations okay so now let's find
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the deviation scores for y so once again
we just take the value for Y and
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subtract the mean which is 4 in this
case from each value so 6 minus 4 is 2 4
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minus 4 is 0 5 minus 4 is 1 3 minus 4 is
negative 1 and 2 minus 4 is negative 2
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now let's take a look at this we have
positive 3 negative 3 notice how those
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add up to 0 again so that looks good ok
so that's it for the deviation scores so
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next what we need to do is we need to
square each of these values we need to
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square the deviation scores and the
reason for that is it gets rid of the
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negative numbers remember how when I
showed you that the deviation score is
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always sum or add up to 0 we can't
really do much with it if our answer is
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0 so when we square them that gets rid
of the negative values and then later
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we'll take care of that square by taking
the square root at the end I'll show you
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that later but for now step 3 is
squaring the deviation scores so these
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were our deviation scores from earlier
that we calculated so all we're doing
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now for X is squaring each one and that
gives us these values and for y we're
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squaring each of those deviation scores
and we get these values here okay so
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that's done we squared all those now all
we want to do is add up those squared
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values now technically speaking we do
call this the sum
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of the squared deviation scores and you
may see this in your text or in other
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places usually short-handed as SS or
some of the squares it's called as
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shorthand okay so some of the squares or
SS is equal to adding up all of the
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squared deviation scores so that's what
we'll do here so we have our square
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deviation scores here for X and for y so
all we do now is just add those up and
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SS or some of the squares for X when we
add these values together gives us 10
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and SS for Y also gives us 10 now it's
not always the case when you calculate a
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correlation that SS X and SS Y will be
equal to each other that just happened
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to occur in this example so don't expect
that you have to see these as equal they
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absolutely do not have to be equal
whatsoever but they are on occasion okay
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so finally what we need to do is find
what are called the cross products now
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cross products just multiply that word
product right multiply the two deviation
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scores together so let's take a look at
this next now when you first see this it
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may look a little intimidating but let
me walk you through it it's really not
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that bad because we've done most of this
already
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recall we found the deviation score for
X right these values here and we found
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the deviation scores for y we've already
done that now all we do to find the sum
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of the products is we multiply the
deviation score for X by the deviation
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score for y for a given person so here
we have negative two times positive 2
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you see that here this product is
negative for multiplying the deviation
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scores for the second person negative
one times zero you see that here that's
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zero third person 0 times 1 is 0 the
fourth person one times negative one is
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negative one and then finally for the
last person two times negative two is
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negative four
now those are the cross products and
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what we have to do is find the sum of
them so we have to add them up okay and
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that's what we do right here we have
negative 4 0 0 negative 1 negative 4 so
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that's what you see here all add it
together and that gives us the sum of
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the products of negative 9 a very quick
review we found the mean then we found
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the deviation scores these here and then
we squared them added them together and
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then our last step here we found the
cross products and then we sum those up
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so we've found the sum of the products
or SP so we have everything we need to
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calculate a correlation
so here's our formula for the
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correlation coefficient of Pearson's are
some of the products divided by square
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root of SS x times square root of SS y
and we found all of these values so
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we're ready to go recall that SS x and y
we're both 10 in this example and the SP
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was negative 9 so we'll just go ahead
and plug these values in we have
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negative 9 over square root of 10 times
square root of 10 and that gives us when
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we work it out an R of negative 0.9 okay
so once again our value of Pearson's R
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is negative 0.9 we can stop right here
and just report that our R was negative
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0.9 and we can be done however if we
want to know whether this value is
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statistically significant that is
whether it's significantly different
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from zero then we need to conduct a
hypothesis test we'll do that next
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