# How to calculate error using logic?

The title may seem a bit off topic. I will explain my doubts with an example. Let there be a situation where we are measuring gravity using the formula

Now, if the least count or error in the measurement of $l$ and $T$ is given, I can easily find the net error or relative error by taking a natural log on both the sides of the equation and differentiation.

But how to include the number of trials? For example, I learned somewhere that if the least count of a stop watch is 1 second, and the number of trials, say to measure the oscillation period of a pendulum is 20, then should the error in measuring the time period be cut down to 1/20 seconds?

• Possible duplicate of How to combine measurement error with statistic error Apr 16, 2017 at 11:05
• By my reading, none of the proposed duplicates answer the question. They come close, but don't quite do it. If the OP is not satisfied with the answers he should say so in a comment. For now, I vote to reopen. Apr 16, 2017 at 13:13
• I did check the link. As @garyp said, yes I did not find the answer there. Maybe the answer there is too technical for me to understand or it is not answering my question at all. Apr 16, 2017 at 14:07
• There are two things to deal with. First: Either a.) your reaction time estimate is wrong, or b.) you have not captured all sources of error. In either case, you have to deal with finding the best estimate of the time, and the best estimate of its error given a set of measured times. The best estimate of the time is the mean. The best estimate of the uncertainty in that value is the standard error of the mean (or just standard error). If the question reopens, I'll elaborate. In the meantime, maybe searching on "standard error" will help. Apr 16, 2017 at 21:16

a) measure the time taken for $n$ oscillations and then your systematic error will indeed be reduced, e.g. minimum stopwatch interval $/ n$; and
b) do the $n$-oscillation measurement $N$ times to estimate the statistical uncertainty.
As described by the answers at How to combine measurement error with statistic error (thanks to Emilio for the link), these error sources should be added in quadrature. The statistical error will converge to zero as $N \to \infty$, but the systematic limitation on $T$ remains fixed... unless you make $n$ bigger, assuming that $T$ remains constant through a long "run".
• It depends on the nature of the timer limitations. If there's the possibility of a constant, unknown offset -- e.g. your reaction time always makes things later, never earlier -- then there's an uncertainty related to the unknown nature of that systematic offset. You could attempt to estimate (i.e. calibrate) or reduce that effect. On the other hand, if the inaccuracy of the timing is random, then it will reduce as $\sigma_\mathrm{timer} / \sqrt{N}$, along with other purely statistical effects. Apr 17, 2017 at 10:24