Confused about how to find the mean of uncertainties for pendulum exp.

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SUMMARY

The discussion focuses on determining the mean value of gravitational acceleration (g) and its uncertainty from pendulum experiment data. The relevant equations provided are g = (4π²L/T²) and Δg = [g(ΔL/L + 2ΔT/T)]. Participants emphasize the importance of using both the standard deviation and the weighted average method for combining measurements, particularly when dealing with a limited number of data points. The consensus suggests trying both methods and selecting the one that yields a tighter uncertainty range.

PREREQUISITES
  • Understanding of pendulum mechanics and the formula for gravitational acceleration.
  • Familiarity with statistical concepts such as standard deviation and standard error of the mean.
  • Knowledge of how to calculate uncertainties in measurements.
  • Basic proficiency in data analysis and interpretation of experimental results.
NEXT STEPS
  • Research the concept of "standard error of the mean" and its application in experimental physics.
  • Learn about the method of weighted averages for combining measurements with different uncertainties.
  • Explore systematic errors in pendulum experiments and how they affect results.
  • Investigate statistical software tools for analyzing experimental data, such as Python's NumPy or R.
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Students conducting physics experiments, educators teaching measurement and uncertainty, and researchers analyzing pendulum motion data.

Zuvan
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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)
0.870.20.00110.430.292
1.280.40.0019.490.172
1.550.60.0019.860.144
1.780.80.0019.970.124
1.981.00.00110.070.112

Many thanks for your help.
 
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Zuvan said:
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)
0.870.20.00110.430.292
1.280.40.0019.490.172
1.550.60.0019.860.144
1.780.80.0019.970.124
1.981.00.00110.070.112

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.
 
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.