Estimating the slope from data

[Saladsamurai]

I had a thought and I am not sure how to answer it. Let's say that I have some data points (x1,y1), (x2,y2), (x3,y3) ... and I want to estimate the slope at x2. Would it be better to estimate it using (y2 - y1)/(x2 - x1) or (y3 - y1)/(x3 - x1) ?

That is, should the secant line that I draw have y(x2) as an endpoint or contain y(x2)? I feel like the latter would better describe the behavior since it is possible that drastic things could be happening AT y(x2) and so by drawing the secant through it, we get a better picture of the overall behavior around y(x2).

What do you think?

 

[HallsofIvy]

Just as you would expect to get the best "line" by drawing between two points that are farthest apart, you can argue that you will get the best approximation to the slope of a set of points by using the points that are farthest apart.

But that is assuming that there really is a "line" to have a slope of and that the given points would lie on that line if not for some error in their positioning. You should understand that, if the points to not happen to lie on a straight line, there is NO "correct" slope so there really is no such thing as a "good" approximation to it! Any of the three (y3- y1)/(x3- x1), (y3- y2)/(x3- x2), or (y2- y1)/(x2- x1), have equal claim to being a "slope". (Another thing that is sometimes done by draftsmen is to average those three "slopes".)