Common formula for two sets of linear data

Question

I've got two sets of data, both of which create a roughly linear line. I developed each set with a value of K, and I collected the data.

% of K  Set 1   Set 2
0%      14      40
25%     10      30
50%     8       15
75%     4       10
95%     1.5     2
97.5%   1       1.1

Set 1: K = 1,800

Set 2: K = 4,000

What would the common formula for a linear graph of both of these be, so that the Set will create values that are linear for any K?

So that for K = 3100, the values should be halfway between the values for each % in Set 1 and Set 2.

Answer 1

You can regard $K$ as a parameter and do a linear fit to each set of data. When I put it in Excel I get $y=-0.1304x+13.858$ for the $K=1800$ data and $y=-0.3949x+38.892$ for the $K=4000$ data, where $x$ is in percent in both cases. Now you can view the constants as points on a line with $K$ the independent variable, so the global fit is $y=\left( -0.1304 +\frac {K-1800}{4000-1800}(-0.3949+0.1304)\right)x+\left( 13.858 +\frac {K-1800}{4000-1800}(38.892+13.858)\right)$ , where I left it in that form to show the two-point form of the line.

Answer 2

May be you could consider that the model is $$y=a+b x$$ and assume a linear dependency of $a$ and $b$ to $K$. This will give $$y=(a_0+a_1 K)x+(b_0+b_1K)=\alpha+\beta x + \gamma K+\delta x K$$ and so perform a multi linear regression in which the independent variables would be $x$,$K$ and $xK$. Then, a global fit could possibly be better since taking into account both data sets simultaneously.

For the specific case where you just have two data sets, this will exactly lead to Ross Millikan's answer. Where the problem would be potentially quite different would be the inclusion of a third (or more) data sets.