A colleague does replication on a quite coslty experiments. There are four different conditions, each one duplicated. The outcome with $4\times 2 = 8$ points is illustrated below:
The analysis is standardized: each pair of colored points yields an average $\mu$ associated with its dispersion $d$, to assess the variability in the given condition. Black diamonds apparently exhibit higher variation, for instance. The colleague directly computes the standard deviation on each pair of points, and uses $d=2\sigma$. I know statistics on such small dataset can be touchy. I would like to provide advices to keep numbers as clean as possible on this condition.
My questions are the following (I am not a trained statistician):
- The variance is computed directly, and not corrected by the $\frac{1}{n-1}$ factor (Bessel's correction), which I believe would be the standard with sample estimation (see here). In fact, this is the same here, since $n=2$, for each pair, so $n-1 = \sqrt{n-1} = 1$. Am I correct in insisting on mentioning this factor in an explicit manner, although it does not change the results? By "explicit manner", I mean writing in documents, or even in Excel macros (in case somebody use them in a different context), that $d=2\sqrt{\frac{(x_1-\mu)^2+(x_2-\mu)^2}{n-1}}$, instead of $d=2\sqrt{(x_1-\mu)^2+(x_2-\mu)^2}$, which the colleague uses, even if it does not make a difference. Some people are afraid of formulas in some fields.
- Sometimes, the colleague gathers the red and black dots. Am I correct that in this case the correction on $\sigma$ is mandatory, as now $n =4$ for one quadruplet, and $n=2$ for each of the two remainging couples?
- I feel that if the experiments could be triplicated, one could expect a "huge" gain in estimating sample dispersion, since one morally would expect to divide the estimate by $\sqrt{3-1}$ instead of $\sqrt{2-1}$. Does that sounds sound enough?
I know two points is not much, and we are close to the singularity limit of one-sample statistics. I have heard a colleague say: "I do not replicate, to avoid dispersion".