What is the meaning of "to within a factor of something"? so for example to within a factor of 3

It means the result of your calculation doesn't differ from the actual value by more than a factor of 3 (or "something").

For example, suppose you're estimating the volume of a lake. You estimate it to be 100,000 acre-feet. If its actual volume is 265,000 acre-feet, your estimate is within a factor of 3 of the correct volume. (You are too low by a factor of 2.65.)

If your estimate were 700,000 acre-feet, you would still be within a factor of 3 of the correct value. (You would be too high by a factor of 2.64.)

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This sort of tolerance allowance is usually specified for things that are difficult to estimate. A factor of 3 is generous, but requires you have a pretty good guess as to many of the parameters involved. For example, it may be unlikely you could estimate the number of grains of sand on a beach within such a factor. On the other hand, you could probably easily estimate the number of candies in a jar within that tolerance.