I am given 2 columns of different numbers. How do I find the "mathmatical relationship" between the two. For...

Rearrange the data as follows:

x values   y values      xdiff      ydiff           slope=ydif/xdiff

x1=612     2052 =y1

x2=693     2779=y2     81          727               8.9753

x3=721     2170=y3     28         -609            -12.7747

It is possible to fit (i) a line of best fit of the form y=mx+c , but here as the y difference/xdifference is too much diffrent the slope is not constant and so the linear relationship is not suitable. (ii) a prarabola of 2nd degree of the form y=ax^2+bx+c between x and y.So, I chose the second form to fit here that establish the relation between x and y.

Since there are 3 pairs of entries we can find a solution of the form y=ax^2+bx+c. Then we get 3 equations if we substitute x=xi and y=yi for i=1,2 and 3 and i  is  a suffix to x or y.

axi^2+bxi+ c = yi  for i =1,2 and 3

Solve the 3 simultaneous equations and determine the values of a,b and c. Substituting the values for xi's and yi's we get:

a(612)^2+b(612)+c=2052         (1)

a(693)^2+b(693)+c= 2779        (2)

a(727)^2+b(727)+c=2170         (3)

(2)-(1) eliminates c and (3)-(2) also eliminates c. We get 2 equation with 2 unkwons in a,b to be determined.:

a{693^2-612^2)+b(693-612)=2779-2052

a(721^2-693^2)+b(721-693)=2170-2779

Equations  reduces to after dividing the former by (693-612) and the latter by (721-693):

1305a+b=8.97508642           (4)

1414a+b=-21.75                    (5)

(5)-(4) eliminates b , giving an equation with only a to be determined:

109a=30.72508642

a=-0.281881526

Substiture a=-0.281881526  in equation (5) to get b and

b=-21.75-1414(-0.281881526)=376.83047789

Having known a and b ,substitute the values of them in any of the original equations flagged at (1 ) or (2) or (3), to obtain c. So you will get the requred relation once you know all a,b and c  determined:

From equatio (1)

c= 2052-(a*612^2+b(612)

=2052-{-0.281881526*612^2+376.83047789*612}

=-122991.2181

Therefore the relation required is:

y=(-0.281881526)x^2+(376.83047789)x-122991.2181.

Hope this helps.