Hello! I have some points in the plane, with errors on both x and y coordinates. The goal of the experiment is to check if the points are consistent with a straight line or not i.e. if they can be described by a function of the form ##y = f(x)=a+bx## or if there is some nonlinearity involved (e.g. ##y = f(x)=a+bx+cx^2##). Assume first we have only 3 points measured. In this case, the approach is to calculate the area of the triangle formed and the associated error, so we get something of the form ##A\pm dA##. If ##dA>A##, then we are consistent with non-linearity and we can set a constraint (to some given confidence level) on the magnitude of a possible non-linearity (e.g. ##c<c_0##). If we have 4 points, we can do something similar and we can for example calculate the area of the triangle formed by the first 3 points (in order of the x coordinate), ##A_1\pm dA_1## and the area of the last 3 points ##A_2\pm dA_2## and then sum them add and do error propagation to get ##A\pm dA## then proceed as above (in the case of this experiment we expect to not see a non-linearity so we just aim for upper bounds). My question is, what is the advantage of having more points? Intuitively, I expect that the more points you have, the more information you gain and hence the better you can constrain the non-linearity. But it seems like the error gets bigger and bigger, simply because we have more points and error propagation (you can assume that the errors on x and y are the same, or at least very similar for different measurements). So, assuming the points are actually on the line, for 3 points we get ##0\pm dA_3## and for, say 10 points we get ##0\pm dA_{10}## with ##dA_{10}>dA_3##, so the upper bounds we can set on the non-linearity are better (smaller) in the case of 3 points. But intuitively that doesn't make sense. Can someone help me understand what I am doing wrong. Why is it better to have more points? Thank you!
