# How do I handle measurement error in sparse data?

Under-educated non-statistician seeks short-term relationship for very one-sided benefit.

System in question involves rocks and physical properties. Modelling bits of the earth typically means few real measurements compared to volume of model. Many estimates are required, and I have NO IDEA how to handle the statement of uncertainties. For illustration let's say the process flow of modelling is 'measure small sample of rocks: get property X (via mean of measurements)', then 'use X in simple model to determine Y'. E.g., Y = mX + b. If I have 20 rocks that I measure, and get a mean for property X, how do I show its uncertainty to start with, and then how do I propagate that through in my calculation of Y (assuming the uncertainty around m is insignificant in comparison)?

I've looked at my data graphically, and they seem to have mostly Gaussian shape. E.g. this is a plot of the kernel densities of property X: Some skew and bumpiness evident, possibly affected by small sample size. OK so there's a fairly large spread of values for some of the rock types, but in essence we deal with large volumes, and have to idealise our models quite a bit, so tails like you see on the yellow curve far left, while real, are not going to be explored in our numerical simulations. I was once told by an older pro that if my model can explain about 85% of observations then I should buy champagne.

This: Estimating error from repeated measurements seems to be a similar question, but I don't even really understand the accepted answer. Std dev/sqrt(n) is the 'standard error'?

• What is Y in your circumstance? Another process measured (with error?) at the same sample location in space? Off hand your description sounds more appropriate for errors in variables models than the linked question you provided. – Andy W Apr 20 '12 at 15:41
• @AndyW Yes, that's correct, Y is another measured variable, however its uncertainty is minor, insignificant compared to that of X. (thanks for putting the image back in) – a different ben Apr 21 '12 at 2:08
• I think I should delete this question - it's far too vague. Comments? – a different ben Apr 22 '12 at 12:26
• It is just a difficult topic (perhaps too large in scope to easily answer on this site). In my experience it is more common for people to ignore such error than to deal with it. Slightly different context (error is from interpolation, not admitting initial measurement error), but here are two papers I am familiar with (Anselin & Lozano-Gracia, 2009; Foster et al., 2012) that may be of interest. – Andy W Apr 22 '12 at 12:48
• As far as deleting the question, it is more typical for community norms to not delete the question. Potential options are leave it and hope someone answers it, edit to refine the question scope, and/or answer the question yourself when you find your solution. – Andy W Apr 22 '12 at 12:50

Your problem has actually got two main parts.

The first is related to the statistics. You will need to assess the data in light of your knowledge of the system and the option to match different distributions to confirm the kind of data you have is a good first step. Once you have a good model you can then begin to make calls as to how best to analyse - i.e. can you get away with normal distribution type analysis e.g. using a simple mean or do you need to use median or is the distribution indicating that there are more underlying complexities.

This brings us to the second point - probably more in my area of expertise - this is whether you have sufficient sample. I am not referring to the statistical context (sort of) of a sample but actually the geological/metallurgical assessment of a mineral representative sample. As a metallurgist/mineral processing engineer, this is typically a bigger challenge than the statistics. If you haven't managed to get the right kind of sample you might as well halt!

To confirm that you have a sample of relevance, you would need to consider things sampling practice for your commodity. For example, if you are looking to understand the particle density distribution for an orebody you would need a LOT of sample to begin to represent the whole. I suspect that since you are looking at particles, you are more likely trying to understand the density of discrete minerals, Probably in an insitu context - but this is probably not the forum to go into detail about that! I can recommend jumping on the LinkedIn forum on sampling orebodies if you want more in this area.

For those not familiar with mineralogy, the issue at heart is that particle analysis does not allow the selection of discrete populations. This means that there is a lot of confounding of the data by associated minerals and the choice about where to get the sample.

Hope this helps.

Mark

As mentioned in the comments, the question is a bit vague so it is hard to make sure I actually answer it.

If your property X is the mean of twenty measurements, then you can compute a standard deviation from that sample, say σ. If you believe that measurements are independent, the standard deviation of X is σ / √20.

Then the question is whether m is a constant or if you actually want to estimate it from your data. If it is a constant, then the standard deviation of Y is m σ / √20. If you actually have a regression problem, like trying to fit m and b and then use that model to predict Y from X, it is probably better to use all your data points (no averaging). Then the variation is much larger and will depend on the value of X. If X is Gaussian you can look up the formula from Wikipedia at the paragraph "Normality assumption".

To my knowledge there is no general method to propagate uncertainty, which means you'll have to work your way through each problem. To convince you I will use a pathological case. If X has a uniform distribution between 0 and 1 (variance 1/12), then tan(π(X - 1/2)) has a Cauchy distribution and thus an infinite variance.