# Combining experimental error in a mean

I understand the rules for combining experimental errors in sums, differences and ratios (as explained here), but what happens to an experimental error when you average it?

Say, ruler measurements of the length of beetles that are all values like 12.3 cm $\pm$ 2mm.
I can find plenty of explanation on the web about how to add and multiply values like that together, but what happens to the error when you take an average of (say) 10 measurements?

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Are the measurement errors independent? That's a big deal. Prima fascie, I would guess that measurements of beetle lengths w/ a ruler would be independent, but there could be effects of the order of measurement (1st few tend to be underestimated, last few overestimated, eg), also there could be a clustered structure if several measurements were taken by each of several different people. –  gung Mar 20 '12 at 14:37
@gung: right, so you're saying that the approach would be affected by the specific situation. But lets say the measurement errors are independent for the sake of argument. –  codeulike Mar 20 '12 at 15:00

The formulas on the link you quote assume that the measurements are

• independent and

• come from a Gaussian distribution around the true value (which one wants to measure).

They're essentially relying on the fact that the sum of two (or more) Gaussian distributed random variables are again distributed as a Gaussian random variable whose mean is the sum of the means and the variance ('square of the error') is the sum of the variances of the two summed distributions. See also this section on Wikipedia: http://en.wikipedia.org/wiki/Normal_distribution#Combination_of_two_independent_random_variables

You'll notice that the product of two Gaussian random variables is not a Gaussian variable any more, so the formulas assume an approximation by a Gaussian distribution.

For the ratio, the resulting distribution is a Cauchy distribution whose variance does not even exist (because it does not go quickly enough to zero when going to +/- infinity), so this is definitively an approximation for this case.

When you average $n$ measurements, $n$ has no uncertainty assigned, so (assuming all your measurements have the same uncertainty) the uncertainty of the average is the uncertainty of a single measurement divided by $\sqrt{n}$ (follows from the formula for the sum of two measurements).

While this looks like one could achieve an arbitrarily good precision by just doing an appropriately high number of measurements, keep in mind that this assumes that the deviation of the measurements from the true value are purely 'statistical' (randomly distributed).

In practice, you'll find sources of measurement bias (which are common to all measurements, thus the assumption of is not justified any more), such as the fact that you'll always take the value of the closest ruler mark.

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There are other CV contributors who have backgrounds in measurement who may be able to give you a better answer, but I can give you some initial thoughts. Later, I could edit or delete this if need be. The page you like to seems to me to be decent, but heuristic. The independence, or lack thereof, is a really big deal, and it simplifies our tack immensely that we can assume independence.

The second big issue to worry about is bias. Essentially, you have a probability distribution over potential measurements. What you want to know (be the case) is that the expected value of that distribution is the true value of that property of the object in question. For example, imagine that the ruler you use was mis-marked at the factory when it was manufactured such that all measurements are a little too small relative to the actual length. On top of that, there is still a small amount of error in the use of the ruler. As a result, the probability distribution of the resulting measurements would not be centered on the true value. Let us assume that there is no bias in your case, which again simplifies our task.

The last issue, I should think, is the shape of the probability distribution of the measurement errors. I suspect it is reasonable to assume that your measurement errors are Gaussian. That is, it is more likely that your errors are smaller, and less likely that they are larger; furthermore, that they are symmetrical about the true value. It isn't necessarily true that they are perfectly Gaussian, but I bet they are Gaussian enough for our purposes.

All of this is background, but it allows us to use well-understood theory to address your situation. Now, what happens if you have measurement errors that are independent, unbiased, and distributed as a Gaussian? It depends on what we're doing:

• Your estimated mean should be unbiased. That is, it is centered on the true value of the population mean, as your errors will tend to eventually cancel each other out.
• The standard error of your mean will be larger than it otherwise would be from sampling error alone. For a simple simulation, draw random samples from a population distribution, then perturb each one a little, and calculate the mean. Do this over and over, and examine the empirical distribution of the means; they will vary more widely than would occur from sampling error alone.
• If, instead of calculating a simple mean, you were to correlate your measurements with another variable (and assuming the population correlation $\rho\ne0$), then the resulting correlation would be attenuated. That is, it would be closer to 0 than the true value.
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