Is it mathematically valid to average a series of non-linear relationships? With regard to a reel of paper, I have a series of reel diameters and their associated linear feet.
Can I divide each diameter by its associated linear feet, and then average the resulting ratio's? And finally divide 60" (the target diameter) by the resulting average of ratios to get an accurate average linear feet for a 60" diameter reel of paper?
For Example:
Diameter, Ln Ft, Dia/LnFT;
60, 12235, 0.004903964;
60, 12231, 0.004905568;
60, 12222, 0.00490918;
60.5, 12071, 0.005012012;
58.7, 11641, 0.005042522;
60.5, 12271, 0.004930324;
Average Dia/LnFT = 0.004950595
60/average = 12119.75533
If the paper is of length $L$ and spool is of radius $a$ and whole roll radius $b$ then the thickness of the paper is $d=\frac{\pi(b^2-a^2)}{L}$. Average the thicknesses and get the length from $L=\frac{\pi(3600-a^2)}{average(d)}$.