[Math] How to calculate the standard deviation of numbers with standard deviations

error-propagationstandard deviation

I have essentially a propagation-of-error problem I run into frequently with my scientific data. For example, I have three samples, each of which I take two measurements of. So, for each sample, I can calculate a mean and a standard deviation. However, I can then calculate the mean of the three samples together, and a standard deviation for this mean. However, this feels like it underestimates the deviation, as we have not factored in the uncertainty in the mean of each. To be specific with an example:

I have three samples (which are supposedly identical), called A, B, and C. Each sample is measured twice: for instance, A is 1.10 and 1.15, B is 1.02 and 1.05, and C is 1.11 and 1.09. Using Excel, I quickly calculate means and standard deviations for each (A: mean 1.125, stdev 0.0353…; B: mean 1.035, stdev 0.0212; C: mean 1.10, stdev 0.0141). But then I want to know the mean and standard deviation of the total. The mean is easy: 1.09; I can also calculate the standard deviation for that calculation: 0.05. But this seems to not take into account the error found in the numbers I am averaging.

Any ideas?

Best Answer

Standard deviation is only a measurement of dispersion of your data in your 3 samples. All three samples will have the same standard deviation if they are supposed identical.

In order to take precision of measurement into consideration, you have to calculate the standard error, which is basically the standard deviation divided by $\sqrt(n)$ where n is the number of measurements youjust used to calculate the mean. Adding more measurements will then involve a decrease in the standard error. This standard error (SE) can then be used to calculate a confidence interval, usually using a normal approximation saying that the "true" mean in the overall sample has a probability of 95% being in the interval [mean - 1.96*SE ; mean + 1.96*SE]

EDIT: here is a great course about the distinction between standard error and standard deviation, by Douglas Altman in the BMJ Statistics Notes: http://www.bmj.com/content/331/7521/903.full.pdf+html

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