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let X = {Xi}/n where {} = sum of all Xi's and n = # of Xi, i=1,n
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Also, nX={Xi} and {Xi^2}/n = Xi^2.
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{XiYi}/n = Xi*{Yi}/n or {Xi}/n*Yi = Xi*Y or X*Yi
Note { a + b + c + d + e + f } = {a} + {b} + {c} + {d} {e} {f} _
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let x=Xi-X, SIGx^2={x^2}/n - we have to find X mean (X) first, then return to
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read each Xi again for Xi-X. Now expand the term and see what happens.
_ ! _ SIGx^2 ! _ _
{(Xi-X)(Xi-X)}={Xi^2 - 2X*Xi + X^2}/n
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{Xi^2}/n - {2X*Xi}/n + {X^2}/n
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{Xi^2}/n - 2X*{Xi}/n + nX^2/n
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Xi^2 - 2X*X + X^2
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Xi^2 - 2X^2 + X^2
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Xi^2 - X^2 = SIGx^2
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SIGx^2=(Xi^2-X^2) also
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SIGy^2=(Yi^2-Y^2)
Now we just sum Xi & Xi^2 till done reading, then get the mean of both and take the
square root of the difference between the mean of the squares and the square of the
mean for SIGx & SIGy.
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let x=Xi-X, y=Yi-Y & xy = (Xi-X)(Yi-Y)
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expanding xy = (Xi-X)(Yi-Y) = XiYi - XiY - XYi + X*Y
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{xy} = { XiYi - XiY - XYi + X*Y }
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= {XiYi} - {XiY} - {XYi} + {X*Y}
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{xy}/n = {XiYi}/n - {XiY}/n - {XYi}/n + {X*Y}/n
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= {XiYi}/n - {Xi}/n Y - X{Yi}/n + n(X*Y)/n
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= {XiYi}/n - X*Y - X*Y + X*Y
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= XiYi - X*Y
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Note: Just the same as SIGx except Yi -> Xi & Y -> X
To use the form y=ax^b, take logs ln(y) = ln(ax^b) = ln(a) + ln(x^b) = ln(a) + b ln(x).
Let a1 be the constant ln(a) so we have ln(y)= b ln(x) + a1 which is now in linear form
and would appear linear if plotted on a log-log scale. So now we have ln(Xi) and ln(Yi)
instead of just Xi & Yi. Now we can use
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b= {xy}/{x^2} and a1 = Yi - bXi again. Note, a=e^a1.
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Here we really let x=ln(Xi)-ln(X), y=ln(Yi) - ln(Y) & xy = (ln(Xi)-ln(X))*(ln(Yi)-ln(Y)).
Using these, we can also compute the correlation coefficient r as
r = {xy}/sqrt({x^2}*{y^2}) and another form for b as
b= r * SIGy/SIGx