# Correct measure of error for experimental data

I am recording data in real time, my signal has a stability period and then a relatively stable period (with noise, red) outlined in the image. I am require to repeat this type of measurement N times (lets say 10 times) and between each measurement the signal must be switched off and turn back on again, hence 10 stability periods.

My question is, how should the average value and associated error be reported?

I believe I should be evaluating the mean value (between the red lines) for each of the 10 repeats. Here the standard deviation would reflect the noise.

To then consider the 10 repeat measurements I take the average of these mean values, but i don't understand how the errors propagate, this obviously isn't the SD on these mean values.

If anyone can provide any insight, it would be much appreciated.

• If you're confident the different "stability periods" should represent the same state, you can take the average over the $N$ repetitions of your signal. As a general rule, save all the raw data so that you re-analyze it later on. Commented Sep 26, 2017 at 22:58

The average value could be calculated either as the average of all N averages, or the average of all n data points when the N runs are pooled $(n \gg N)$. These options will give the same answer.

The appropriate measurement of the error is the standard error in the mean (SEM). This should be the SEM of all data points, not the SEM of the N averages. SEM can be calculated from the standard deviation of the pooled data set divided by $\sqrt{n}$.

• I agree, but note: This is founded on the assumption that the noise has a Normal distribution - an assumption that preferably has some justification but frequently just seems to be overlooked. There are numerous other possible distributions. For example, if due to a single random source of error (a single dice role), the distribution would be flat. Commented Sep 26, 2017 at 23:32
• Thanks sammy, as a follow up, lets say per N repeats n=1000, doesn't this mean that N really = 1000 single point measurements? Out of interest why is SEM the correct choice for the measure of error rather than SD? Commented Sep 27, 2017 at 7:07
• To JMLCarter on the normally distributed noise you, were correct to point out that I have made this assumption :). What impact does normal noise have on the error analysis vs other distributions? Commented Sep 27, 2017 at 7:09