Uncertainty Calculation for Lab Report: Methods & Comparison

[omoplata]

I'm trying to write a lab report.

I've taken some measurements several times and have calculated the results and the systematic uncertainty for each set of measurements. i.e. I have $$x_{i}$$ s, which are the results, and I have $$\Delta x_{i}$$ s, which are the systematic uncertainties, calculated using uncertainty propagation formulas and the systematic uncertainties of the measurements.

I need to take the average of $$x_{i}$$ s. How do I calculate the error for that?

I can think of two methods. I don't know which is correct.

Method 1: Calculate it using uncertainty propagation formulas and $$\Delta x_{i}$$ s.

Method 2: Use the standard deviation of the $$x_{i}$$ s as the uncertainty.

Which is correct?

 

[SpectraCat]

I'm trying to write a lab report.

I've taken some measurements several times and have calculated the results and the systematic uncertainty for each set of measurements. i.e. I have $$x_{i}$$ s, which are the results, and I have $$\Delta x_{i}$$ s, which are the systematic uncertainties, calculated using uncertainty propagation formulas and the systematic uncertainties of the measurements.

I need to take the average of $$x_{i}$$ s. How do I calculate the error for that?

I can think of two methods. I don't know which is correct.

Method 1: Calculate it using uncertainty propagation formulas and $$\Delta x_{i}$$ s.

Method 2: Use the standard deviation of the $$x_{i}$$ s as the uncertainty.

Which is correct?

If all of your x_i's are sampled from the same distribution, then you can use the standard deviation of the mean formula ... just take your estimate for the standard deviation of the distribution (i.e. your calculated standard deviation for a single value), and divide by the square root of the number of trials.

 

[omoplata]

Thanks for the reply,

If all of your x_i's are sampled from the same distribution, then you can use the standard deviation of the mean formula ...

I don't know if it's from the same distribution. $$x_{i} = \rho$$ where I used $$V=I R$$ and $$R = \frac{\rho l}{A}$$ for the calculation of the resistivity of the same semiconductor sample at different lengths and cross sections. If it was an ideal sample and measurement I'm supposed to get the same resistivity $$\rho$$. So I guess there would be NO expected distribution?

What should I do?

 

[SpectraCat]

Thanks for the reply,
I don't know if it's from the same distribution. $$x_{i} = \rho$$ where I used $$V=I R$$ and $$R = \frac{\rho l}{A}$$ for the calculation of the resistivity of the same semiconductor sample at different lengths and cross sections. If it was an ideal sample and measurement I'm supposed to get the same resistivity $$\rho$$. So I guess there would be NO expected distribution?

What should I do?

All real measurements sample a distribution, even if the quantity being measured is completely precise (which it usually isn't), your measurement technique will always have an associated uncertainty. The term "distribution" just refers to the range and relative probabilities of possible values for the measurement. Typically we assume the values will be normally distributed around a mean value, which we then take to represent the "real" value of the quantity to be measured. The standard deviation as calculated from a finite number of measurements, gives you an estimate of how broadly the possible values are distributed around the mean value. Finally, since you normally estimate the mean of the distribution as well (by averaging a finite number of values), the standard deviation of the mean (as I described it above), gives you an estimate of by how much your calculated mean is likely to deviate from the true mean of the underlying distribution that you are sampling.

In your case, since the theory tells you to expect that the resistivity of the sample is constant, you are free to assume that all of your measurements are sampling the distribution of possible values of the resistivity. In other words, you should be able to use the standard deviation of the mean to estimate the uncertainty of your mean value.

 

[omoplata]

Great! I'll use the standard deviation as the uncertainty then. Thanks a lot for the help!

But then did I calculate the systematic uncertainties, $$\Delta x_{i}$$ s for nothing?

Should I at least use the uncertainty propagation formulas to calculate the final systematic uncertainty for the average and see which one is larger(between the standard deviation and the propagated systematic uncertainty), and use the larger one for the uncertainty? Or should I just ignore the calculated $$\Delta x_{i}$$ from here onwards?

 

[SpectraCat]

Great! I'll use the standard deviation as the uncertainty then. Thanks a lot for the help!

Note that I said standard devation of the mean .. that is different from just standard deviation, as I explained in my first post in this thread.

But then did I calculate the systematic uncertainties, $$\Delta x_{i}$$ s for nothing?

I am not sure what you mean by the systematic uncertainties ... do you mean the deviations of the individual resistivity values from your calculated mean? The term "systematic uncertainty" is typically used to refer to something else entirely.

Should I at least use the uncertainty propagation formulas to calculate the final systematic uncertainty for the average and see which one is larger(between the standard deviation and the propagated systematic uncertainty), and use the larger one for the uncertainty? Or should I just ignore the calculated $$\Delta x_{i}$$ from here onwards?

You should definitely use the propagation rules to convert the uncertainties in the measured values (which I guess are voltage, length and area?) in your experiment, into the uncertainties for the calculated values (i.e. resistivity).

 

[omoplata]

Note that I said standard devation of the mean .. that is different from just standard deviation, as I explained in my first post in this thread.

So, the standard deviation of the mean is equal to $$\frac{ \sqrt{ \frac{ \sum_{i=1}^{N} ( \overline{x} - x_{i} )^2 }{N} } } { \sqrt{N} }$$ ?

I am not sure what you mean by the systematic uncertainties ... do you mean the deviations of the individual resistivity values from your calculated mean? The term "systematic uncertainty" is typically used to refer to something else entirely.

What I meant by systematic uncertainty was, I took the least possible measurements (which depends on my measuring instruments) for $$V_{i}$$, $$I_{i}$$, $$l_{i}$$ and $$A_{i}$$ as $$\Delta V_{i}$$, $$\Delta I_{i}$$, $$\Delta l_{i}$$ and $$\Delta A_{i}$$ (I call them "systematic uncertainty of the measurements"), and then I applied the error propagation formulas to the final equation $$\rho_{i} = \frac{V_{i} A_{i}}{I_{i} l_{i}}$$ to find $$\Delta \rho_{i}$$ (I wrote $$\rho_{i}$$ as $$x_{i}$$ and $$\Delta \rho_{i}$$ as $$\Delta x_{i}$$ in my original post). Here $$\Delta \rho_{i}$$ is what I call the "propagated systematic uncertainty".

You should definitely use the propagation rules to convert the uncertainties in the measured values (which I guess are voltage, length and area?) in your experiment, into the uncertainties for the calculated values (i.e. resistivity).

So which should I use for the final uncertainty of resistivity?
1. Standard deviation of the mean?
2. Final propagated uncertainty for the mean resistivity?
3. The larger of the two?
4. Or the two combined in some way?

Sorry about my difficulty in understanding this. I don't know much statistics/error analysis.

 

[omoplata]

OK, I have found out from "Data Reduction and Error Analysis for the Physical Sciences" by Bevington and Robinson that standard deviation of the mean is,
$$\sigma_{\mu} = \frac{ \sqrt{ \frac{ \sum_{i=1}^{N} ( \overline{x} - x_{i} )^2 }{ N - 1 } } }{ \sqrt{ N } }$$

I still don't kow how to calculate the final uncertainty.

 

[SpectraCat]

OK, I have found out from "Data Reduction and Error Analysis for the Physical Sciences" by Bevington and Robinson that standard deviation of the mean is,
$$\sigma_{\mu} = \frac{ \sqrt{ \frac{ \sum_{i=1}^{N} ( \overline{x} - x_{i} )^2 }{ N - 1 } } }{ \sqrt{ N } }$$

I still don't kow how to calculate the final uncertainty.

Notice that the quantity in the numerator is just the standard deviation for a single measurement, as I said in my first post.

As you surmised, you can calculate the final uncertainty in one of two ways. You can estimate mean values and standard deviations for your measured quantities and use error propagation to convert those into a mean value and standard deviation for the resistivity. Then dividing by sqrt(N) will give you the standard deviation of the mean.

Alternatively, you could calculate the resistivity for each individual trial, and then use those values to estimate the mean, standard deviation and standard deviation of the mean for the resistivity.

I recommend doing it both ways, and then checking your results. I think the two estimates should agree with each other fairly well if you are doing everything correctly, so I don't think it matters much which one you use. Of course the most conservative way is to estimate both ways and report the largest value. If I recall correctly, I believe the first method generally yields more conservative (i.e. larger) estimates of the uncertainties, except perhaps when you are dealing with small numbers of trials, so that is the one that is typically used.