# Subtracting two means and convert to %. What happens to the standard deviation

I have two sets of data to track the changes of my subject of analysis. I collected at two different time points, time = 0 and time = 10 min

At time = 0 min, the raw data are [12, 9, 10]. mean is 10.3 and SD is 1.5 At time = 10 min, the raw data are [5, 7, 6]. mean is 6.0 and SD is 1.0

I would like to calculate the changes from 0 to 10 min in terms of percentage.

For mean, i took [(10.3 - 6.0)/10.3]*100 = 44.6 %

What is the correct method to obtain the standard deviation in percentage?

• Same way you calculated the percentage change in means, as long as you are comparing the same concept while doing so (using estimates with the same units only). Although this has little to do with statistics, since this is not inferential but more descriptive in nature. – user2974951 Dec 26 '18 at 8:32

## 1 Answer

To obtain some (at least approximate) statistics about a random variable, you need to assume some form . of distribution or create an (even implicitly) approximate one. When you have some samples, you can estimate the mean and std. deviation using usual formulas. They are not the true statistics, albeit data estimates, which are assumed to be close the true values. You actually did this for your samples, i.e. you estimated the mean and deviation of your data from your samples. These are point estimates and don't give much hint about the underlying distribution of your statistics. In order to get statistics about your estimated statistics, you need to obtain several of them, e.g. for getting std of your mean estimate, you need to have several mean estimates. But, your data is constant, and you have only one mean estimate here. Same problem arises when we ask about the deviation of your estimated deviation, or the percentage you ask for.

One way to create several of these estimates is to use bootstrapping. You resample your data with replacement and obtain several realizations, e.g. $$[12,9,10],[10,9,12],[10,10,9],...$$, record the mean of each of these, redo the same for the second set, calculate the percentage. After all, you'll have several percentages and you can directly estimate the deviation you ask for. But note that, this is a computational methodology.