Error propagation through a mean

[Z21Jazz]

I have taken 10 measurements and each measurement has it's own associated uncertainty. If I take the mean of these 10 values and I want to know the standard error of the mean, including the uncertainties in the individual measurements, can this be done as follows.

For each measurement (n) I would either add the individual error (if n>mean value) or subtract the individual error (if n<mean value) to have a set of worst case values. I would then calculate the standard error of the mean of the new worst case values. I think this would then account for the error in individual analyses as well as the spread across the 10 measurements. Is this correct? If so, is there a name for this method? If not, why not and what is the correct way to propagate these errors? Thanks ahead of time!

 

[Miner]

I think we need more information on what you intend. Your 10 measurements already include the measurement variation. Do you somehow know the true (actual) values?

The following is the equation relating these: Variance (Observed) = Variance (Actual) + Variance (Measurement error) where Variance is the square of the standard deviation.

 

[katxt]

Assuming you know the variance of each measurement, and the measurements are independent, then
Var(sum) = sum of the individual variances.
Var(mean) = Var(sum/N) = Var(sum)/N^2 = sum of the individual variances/N^2
so se(mean) = sqrt(sum of the individual variances)/N
I think!

 

[Z21Jazz]

Below is a picture of some fake example data. @Miner, I do not know the true values but I am measuring isotope ratios on an analytical instrument (Secondary Ion Mass Spectrometer). Each measurement has an associated standard error (Error column). I used the equation =IF(Measurement > Mean, Measurement+Error, Measurement-Error) to calculate a row of worst case data given the uncertainties in each measurement. I then took the standard error of the mean of this column instead of the Measurement column thinking that this column incorporates the spread from measurement to measurement and also the individual uncertainties in a given measurement. I hope this clarifies my question.

 

[katxt]

It's ingenious but seems somewhat ad hoc. Here is another idea - a bootstrap.
Resample the measurements and errors size with replacement.
For each resampled measurement and error pair in the sample generate a random normal with the appropriate mean and sd.
Find the mean of the sample. Record. Repeat several thousand times.
The SE is the the SD of all the sample means.

 

[katxt]

It also depends on the source of the measurement. Is this a random sample from a large number of possibilities and you want the SE of the population mean? Or is this the full data and you want to know how reliable the calculated mean is?

 

[Miner]

Below is a picture of some fake example data. @Miner, I do not know the true values but I am measuring isotope ratios on an analytical instrument (Secondary Ion Mass Spectrometer). Each measurement has an associated standard error (Error column). I used the equation =IF(Measurement > Mean, Measurement+Error, Measurement-Error) to calculate a row of worst case data given the uncertainties in each measurement. I then took the standard error of the mean of this column instead of the Measurement column thinking that this column incorporates the spread from measurement to measurement and also the individual uncertainties in a given measurement. I hope this clarifies my question.

View attachment 3306

Is the error column truly the "Standard Error" or a stated uncertainty? As katxt pointed out Standard error is the standard deviation of multiple means. That means that each measurement (e.g., 10.1) is the average of a group of averages of multiple individual measurements. Is this what your SIMS is reporting?

 

[Z21Jazz]

Thank you both for your questions and input.

@Miner, Yes every "measurement" value above is the average of a group of averages. Every measurement runs for several cycles in order to get enough ion counts to get a low standard error for the set of cycles. Each cycle is say 15 seconds and the software computes a mean for that cycle. The average and standard error for all cycles are computed and are shown as the measurement and error columns above. After a measurement is complete we will move to a new spot on the solid samples we are analyzing and start another measurement. We can assume the materials we are analyzing are homogenous and variations in measurements and their associated errors are related to instrumental instability.

@katxt, I am randomly sampling a homogenous material that should give me the same value for every measurement. Variations are due to instrumental instability. So, I want to know the SE of the population mean. As far as resampling, SIMS is a destructive technique and so I cannot resample the exact same material.

 

[katxt]

As far as resampling, SIMS is a destructive technique and so I cannot resample the exact same material.

OK. Fortunately "resampling" in a statistical context is completely non destructive. It means resampling the data. The picture shows the original data put into a simple Excel program. Columns F:G hold your original data, I:J hold a resampled version. Note that the data is resampled in pairs, not all pairs are in the resample, and pairs may occur more than once. Then col K holds a simulated true value using =NORMINV(RAND(),mean,sd) and K12 holds the mean of that particular resample. This is one possible mean. This mean is transferred to the list in col D. Finally, after 2500 repeats (in this case), the SE mean is found by the SD of all the possible means.
If this interests you, I can post the program and an associated paper. kat

 

[Z21Jazz]

@katxt, Yes, this Monte Carlo simulation does interest me. Please do share the program and the paper. Thanks!

 

[katxt]

Don't worry about the coding. Section 5.4 is the closest with a simulation added. Cheers, kat

 

[hlsmith]

Below is a picture of some fake example data. @Miner, I do not know the true values but I am measuring isotope ratios on an analytical instrument (Secondary Ion Mass Spectrometer). Each measurement has an associated standard error (Error column). I used the equation =IF(Measurement > Mean, Measurement+Error, Measurement-Error) to calculate a row of worst case data given the uncertainties in each measurement. I then took the standard error of the mean of this column instead of the Measurement column thinking that this column incorporates the spread from measurement to measurement and also the individual uncertainties in a given measurement. I hope this clarifies my question.

View attachment 3306

I haven't read all the posts, but this example makes the errors look non-systematic? What is the source or cause. Can you provide the actual context you are thinking about. There are measurement error corrections if they are systematic. Examine, quantitative bias analysis. It typically helps if you can have a validation sample collected with truth (criterion standard).

 

[Z21Jazz]

@hlsmith, So for this example I am measuring a standard that has a known lithium isotope ratio (7Li/6Li). The value in the measurement column is the SIMS instrumental mass fractionation (IMF) in per mille (‰) deviations from the known standard. We measure the standard multiple times before and after measurements of unknown samples so that we can correct for the IMF. It is typical to see variations of 5-10 ‰ over the course of a day with the SIMS instrument we use. These instruments have multiple lenses that have very specific voltages applied to them as well electrostatic and magnetic sectors. Variations in the voltages applied to the different parts of the instrument can result in slight deviations in the isotope ratios that can be picked up in a per mille scale.