Hello!
It is obvious that there is no linear formula exactly connecting these x's and y's, if we consider the slopes between neighbor points:
`(155 - 134)/(31 - 30) = 21,` `(165 - 155)/(33 - 31) = 5,` `(167 - 165)/(35 - 33) = 1.`
For a single line, all these slopes must be the same.
But this isn't the whole story. We may seek such a line `y=ax+b` that would be the closest to...
Hello!
It is obvious that there is no linear formula exactly connecting these x's and y's, if we consider the slopes between neighbor points:
`(155 - 134)/(31 - 30) = 21,` `(165 - 155)/(33 - 31) = 5,` `(167 - 165)/(35 - 33) = 1.`
For a single line, all these slopes must be the same.
But this isn't the whole story. We may seek such a line `y=ax+b` that would be the closest to all these points. The simplest criteria of such a proximity is the least squares one, which means we try to minimize
`sum_(k=1)^n (y_n-(ax_n+b))^2.`
This problem has the exact unique answer (see for example the link attached). We have to compute the numbers
`p=sum_(k=1)^n x_k^2,` `q=sum_(k=1)^n x_k,` `r=sum_(k=1)^n x_k y_k` and `s=sum_(k=1)^n y_k.`
In our case `p=4175,` `q=129,` `r=20115` and `s=621.` Then we solve the linear system for the unknowns a and b,
`pa+qb=r,` `qa+nb=s` (here n=4).
I hope you know how to solve such systems, the solution for this is `a=351/59,` `b=-2160/59.`
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