Finding final g and uncertainty from multiple pendulum measurements

[Zuvan]

Homework Statement

State the value of g and its uncertainity (Pendulum experiment)

Relevant Equations

g = (4pi^2L/T^2)
Δg = [g(ΔL/L + 2ΔT/T)]

I'm confused about how to find the final value of g and its uncertainty. I've done a bit of research and I have encountered conflicting information, some say you have to weight the measurements, some say you have to find the standard deviation then divide by two, etc. I have the following tabulated data:

T Period (t/N) (s)	L Pendulum length (m)	ΔL (m)	g = (4pi^2L/T^2) (ms^-2)	Δg = [g(ΔL/L + 2ΔT/T)] (ms^-2)
0.87	0.2	0.001	10.43	0.292
1.28	0.4	0.001	9.49	0.172
1.55	0.6	0.001	9.86	0.144
1.78	0.8	0.001	9.97	0.124
1.98	1.0	0.001	10.07	0.112

Many thanks for your help.

 

[haruspex]

Homework Statement:: State the value of g and its uncertainity (Pendulum experiment)
Relevant Equations:: g = (4pi^2L/T^2)
Δg = [g(ΔL/L + 2ΔT/T)]

I'm confused about how to find the final value of g and its uncertainty. I've done a bit of research and I have encountered conflicting information, some say you have to weight the measurements, some say you have to find the standard deviation then divide by two, etc. I have the following tabulated data:

T Period (t/N) (s)	L Pendulum length (m)	ΔL (m)	g = (4pi^2L/T^2) (ms^-2)	Δg = [g(ΔL/L + 2ΔT/T)] (ms^-2)
0.87	0.2	0.001	10.43	0.292
1.28	0.4	0.001	9.49	0.172
1.55	0.6	0.001	9.86	0.144
1.78	0.8	0.001	9.97	0.124
1.98	1.0	0.001	10.07	0.112

Many thanks for your help.

Yes, this is a common source of confusion.

For the moment, I'll set aside systematic errors... see later.

On the one hand, you have an a priori estimate of the precision of each data value (what are you using for Δt?), and on the other you have a computable variance from several calculations of what ought to be the same g value.

I believe there ought to be a mathematically justified way of combining these two, but I've never seen it done. Instead, people use the first way (and the equation you quote) when there are only a few data points and the second when there are lots. With five, I would try both and use whichever gives the tighter range.

For the variance method, what you need is the "standard error of the mean". Look this up. Bear in mind that, in principle, the more datapoints you have the more you can trust their average. This means that taking the standard deviation of the calculated values as the uncertainty must be wrong - more and more datapoints would not make that diminish.

But averaging doesn’t shrink systematic errors, like consistently overreading the period.

 

[Cutter Ketch]

There are certainly several ways to do it, and you shouldn’t think of any of them as wrong. However some are definitely better than others.

Because this is the homework forum, I’ll throw it back on you and ask you to show us some of the things you’ve tried.