Least Squares has the virtue that if you understand the idea and know a little math (say, first year calculus), you can usually "do it yourself" with pencil and paper (to get the formulas) and a "calculator" (which could be LabVIEW code) to do the arithmetic.
You have three data points, which I assume are measured at three different (known) values of an independent variable, which I will call X, and give you three measurements, Y. Imagine you plot the three sets of points x(i), y(i) on a piece of paper. You want to draw a (straight) line that best "fits" those points, i.e. is "closest" to them. But what does "closest" mean? [Note I'm assuming that they do not exactly lie on the line ...]. There are three obvious possibilities -- the line such that the vertical distances between the points (x, y) and the line are minimized, the horizontal distances are minimized, or the "perpendicular" (shortest) distances are minimized (draw a picture and you'll see what I mean).
[Before going further, a parenthetical remark, in square bracket. With three points, you can exactly fit a quadratic that will go through all three points. In general, with N points, you can exactly fit a polynomial of degree N-1, but this is rarely useful. So I'm going to go with a polynomial of degree 1, i.e. a Straight Line. Note you could also fit a polynomial of degree 0, namely a constant -- your choice of arithmetic mean, geometric mean, median, or something else.]
You say you are calibrating your instrument at what I assume are three different X settings. Since you set it, we'll assume you know its value exactly, i.e. there is no uncertainty in X, only in Y. So in our quest to find the line that "best fits" the points, what we want to minimize are the Y deviations, i.e. the vertical distances of our line at the X points from the Y values.
There are (mathematical/statistical) reasons to say what you want to minimize is the square of the distance between the line and the Y values (hence the name "Least Squares"). Let's see how you would do this. I'm going to generalize and say you have N points (here N=3).
Your model is y(i) = m x(i) + b (the equation for a straight line, high school algebra). We want to know m and b that minimize the distances from this line at the point x(i) from the measured value y(i). The value of the line at the point x(i) is m x(i) + b, and the value of the measured point here is simply y(i), so we need to find m and b that minimize (sum over N points) (y(i) - (m x(i) + b)**2. [Sorry, can't do math here, but I'm just summing the squared differences between my measured Y points and the Y I would get if I plugged in the X value in my equation for a straight line). I want to find values of m and b that minimize this sum.
Did someone say "minimize"? As in "take the derivative and set it = 0"? This is actually a useful exercise, and has a very simple solution. I'm going to give you the formula for the answer, but urge you to try to derive it yourself.
Let's define mean(Y) as the mean of the Y values, and mean(x) as the mean of the X values. Solve these equations:
- m = sum((x(i) - mean(X)) (y(i) - mean(Y))) / sum(sqr(x(i) - mean(x))), where sum means "sum over all the values of i" and sqr means "square".
- b = mean(Y) - m mean(X). Note similarity to y = m x + b -- we just took means.
So you can code these formulas up in LabVIEW and get the values of m and b that give you the "least-squares best straight line" through your data points. Note that it will work just as well (better, probably) for 100 data points -- the only assumption is that the model (namely a straight-line relationship between X and Y) is valid.
Bob Schor
