Which way to obtain ##f(x)## from measuring ##x_i##

[ChrisVer]

I am having a question concerning multiple measurements of the period of a pendulum, for let's say the determination of g.
Let's say I am using a stopwatch with an uncertainty in time \delta t and I measure 10 times the period of the pendulum: T_1, T_2,...,T_{10}.
Are the two following approaches equivalent?
WAY 1: determine the g_1, g_2, ...,g_{10} from the periods and the \delta g and then take the average+std.
WAY 2: obtain the average period T and its std from the different measurements and calculate g \pm \delta g from error propagation.

 

[mathman]

I don't know the relationship between g and T. If it's linear, the methods are equivalent. If not, the first way is better.

 

[ChrisVer]

I don't know the relationship between g and T. If it's linear, the methods are equivalent. If not, the first way is better.

Sorry the relationship for the specific case is g(T) = \frac{(2 \pi)^2 l}{T^2}
How did you deduce that for linear relationship the two ways are equivalent vs the 1st is better?

 

[mfb]

If g and T would be proportional, the conversion is just a constant scale factor, and it does not matter at which point you apply it.
If they are not, the transformation is nonlinear, if your distribution before is gaussian (or anything other that is reasonably well-behaved) it is probably not gaussian any more afterwards and can have funny effects. If your relative uncertainties are small this does not matter much, but with large uncertainties the second approach can fail completely.

Unrelated: You should not measure a single period. Measure 10 or even more in a single time measurement. That way your uncertainty goes down linearly with periods.