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I have N samples, that have mean X and standard deviation sigma. So far so good. I know that each sample has been measured with a measurement error e. Because the data is real world data the instrument error is not negligible, and thus I would like to account for it. The mean/standard deviation description assumes that the instrument errors are negligible, but in this case I think I need account for them somehow. I have looked around but this 'trivial' problem does not seem to have a standard answer (i.e. I cannot find any solid indication of what to do with non trivial instrument errors affecting the readings I use to calculate a mean). I would be obliged if anyone could shed some light on this.

PS to make the issue perfectly clear: the value I am looking at is a measure of heat resistance. Each value is obtained by combining two temperatures and one heat flow, and thus each point measurement has an associated instrument error. The data was collected in a thermal chamber, where the temperature fluctuates around the stated temperature. Hence the distribution of the observed values is affected by (1) a sizeable instrument error and (2) a real effect that is affecting the distribution. Given that I know that each point estimate has an instrument error that is of the same order of magnitude of the SD of the final distribution I would be interested if this instrument error can be at any point taken into account explicitly.

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    $\begingroup$ The mean/SD description assumes nothing at all about measurement errors. It describes the actual measurements, not the underlying numbers that are being measured. So exactly how do you want to "account" for the measurement errors? Do you want to study them separately and use that information to adjust your estimates of the mean and standard deviation of the underlying numbers? $\endgroup$ – whuber Mar 7 '15 at 16:00
  • $\begingroup$ @whuber that is an interesting point. I added details to my question and I'd like you to expand on your comment, which I think is a solid starting point for this converstion. $\endgroup$ – user1256923 Mar 10 '15 at 16:04
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Is your measurement error biased? Why or how? Accounting for this could get complicated.

If your measurement error is unbiased, just some iid noise added to your data, it will usually just make your standard errors larger and significance lower. In this case you live with it and grandstand that your results a are significant despite measurement error. Or get more data.

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