Originally posted by m42man This is a total non-sequitur (it's nothing to do with statistics, for a start). Please re-read what I've written on the subject.

It is no non-sequitur. This problem has everything to do with statistics - the goal is to minimize the SR error for a random selection of focal lengths between some max and minimum.

In the math I gave earlier I stopped before deriving the whole concept of least squared error (or maximum likelyhood). It boils down to minimizing the average of the squares of the errors. Recall that the square of each error is (angle(actual.focal.length-SR.focal.length))^2.

The "mean square error" is the (integral of the squared errors for the range) divided by the range. You might look it up. In this case or in any case where a random error is proportional to the deviation from a value, the optimum estimate of that value is the mean of the range. Also see "Central limit theorem".

You say

Originally posted by m42man (and using my prescribed formula you get the same SR at both ends of the zoom range). If you set the value to "max", then whenever you set the zoom lens to a focal length less than max/2, you'll end up with more blur than you would have had if you'd turned SR off.

Turning off SR is the same as setting the SR focal length to zero. If you set it to the middle of the range, the errors both above and below the SR chosen are proportional to (focal.length - SR).

Say you set the SR to zero like you said, the error you get for focal lengths less than max/2 are less than if you'd used max as the SR focal length. But that doesn't mean it is the best choice. The best choice is the one that minimizes the (focal.length-SR) error; if min is not zero then SR = (max+min)/2 gives a smaller error for max/2 than if SR=0

Case 1. SR=(max+min)/2, f.actual = max/2, error is max/2- (max + min)/2 = -min/2

Case 2. SR=0, f.actual = max/2, error is max/2

Since min < max the absolute error is less when SR=(max+min)/2

Go back and look at what your formula recommends. For a 28-250mm lens (a common extended zoom range), your recommendation is:

SR.focal.length = 2*28*250/(28+250) = 50mm !!

That doesn't make sense; if you pick up a 28-250 focal length lens and randomly pick a focal length the most probable choice of focal length is (250+28)/2 = 139mm, far from 50mm. Surely you are best off using a SR focal length close to the most probable focal length used. You would not recommend that 50mm be used as the SR parameter for a 139mm fixed focal length lens, so why tell someone to use a SR parameter of 50mm for a lens that will most probably be used at 139mm?

I will not let this go; people are at best being confused, at worst, mislead.