Combining Uncertainties - Mean

[crankine]

Homework Statement

How do I combine the uncertainties to get a mean value±error?

Okay I know I have huge errors and if anyone could suggest what's wrong (I have huge errors in strain because the changes in length were tiny and the calibration errors were of the same order) that's great but my main problem is:

These are my results in GPa from a youngs modulus experiment. How do I combine the uncertainties to get a mean value±error?

97.6±27.1
95.2±67.1
125.5±41.4
192.1 (Negligible error)
154.1 (Negligible error)
95.8±30.9
95.8±25.5
110.5±24.8

Homework Equations

The Attempt at a Solution

I've taken the mean of the first part = 120.8

On the internet people seem to be saying things like
"If more than 10 but fewer than 50 trials have been taken, report the uncertainty as the
average of the absolute values of the deviations of the individual values."
In that case its ±36.1

But someone told me to use the Pythagorean combining formula which I thought might be wrong...
±95.8 nearly 3/4 of my actual value.P.S How many significant figures do you round off to if the uncertainty is greater than 1?!
if it was ±0.0002556 then it would be ±0.0003, fine, but does ±27.1 become ±30?
by the way that quote was from http://cc.ysu.edu/~jeclymer/Uncertainty%20Tutorial.pdf
page 4

 

[jbsteele]

(in case you wanted to take a look)The Pythagorean Combining Formula is the correct approach for combining uncertainties in this case. You would use this formula to find the combined uncertainty of all of the individual results. The formula is as follows: σ_combined = √(σ_1^2 + σ_2^2 + ... + σ_n^2) Where σ_combined is the combined uncertainty and σ_1, σ_2, ..., σ_n are the individual uncertainties. Using this formula, you would calculate the combined uncertainty as follows: σ_combined = √(27.1^2 + 67.1^2 + 41.4^2 + 30.9^2 + 25.5^2 + 24.8^2) = 95.8 Therefore, the mean value ± error is 120.8 ± 95.8. As for the significant figures, when the uncertainty is greater than 1, you would round off to one significant figure. So, in this case, you would round off 27.1 to 30.

 

[nlightner]

As a scientist, your main goal is to accurately and precisely measure and report your results. In order to do this, you must take into account the uncertainties associated with your measurements. In this case, you have multiple measurements with their own respective uncertainties, and you want to combine them in order to calculate a mean value with an associated error.

There are several ways to combine uncertainties, and the method you choose will depend on the specific circumstances and assumptions of your experiment. One common method is to use the Pythagorean theorem, which involves taking the square root of the sum of the squares of the individual uncertainties. This method is appropriate when the uncertainties are independent of each other.

In your case, you have some measurements with negligible errors, so you can simply take the average of the other measurements as your mean value. However, for the measurements with uncertainties, you can use the Pythagorean theorem to calculate the combined uncertainty. This would result in a mean value of 120.8±36.1 GPa.

Regarding significant figures, the general rule is to round your final result to the same number of significant figures as the measurement with the least number of significant figures. In this case, since your uncertainties are all given to one decimal place, you can report your mean value with one decimal place as well (120.8±36.1 GPa).

In conclusion, when combining uncertainties to calculate a mean value with an associated error, it is important to use a method that is appropriate for your specific situation and to report your result with the appropriate number of significant figures.