I am having some existential doubts on statistics; maybe someone here can help me clear out the dust. I've trying get answers from literature, but I found most reads quite vague, which still leaves me we doubts, because (from those I've read) they don't go into detail on how to organize the data and how its really processed.

So, for sake of simplicity consider a t.test: my doubts are related with the n (the number of observations); it's not clear to me how to count the number of observations. I'll try to explain with one example.

Image this hypothetical experiment were we want to compare the expression of genes between two conditions: control and treatment group of mice; each groups has 3 mice. When we do the qPCR we do it in triplicate --- thus, the total raw number of observations is $3\times3\times2=18$ --- what we should do next is to average the Cq for each mice for each target gene, which leaves us with a total of 6 observations (because we have six mice). Now, my doubts is what numbers (observations) to use to calculate the standard deviation --- should we use the triplicate values or their average? If we use the MS Excel function to calculate this, since it divides by n, and since it corresponds to the selected spreadsheet cells the result is going to be different. So, to compare these groups with a t.test should we use the 18 values or the 6 values? Conversely, if we as part of this same experiment measured the mices' weight, we might just do it once, and thus we just have 6 observations regarding the weight. Therefore, in this case we don't need to calculated the weight technical averages (because it's the value it self), meaning that we calculated the average of the biological group from one measurement only, leading to a standard deviation coming from 3 measurements only.

Another example that makes me think is the quantification of speckles with microscopy. Imagine we are quantifying the number of bright spots in cells using confocal; in this case would be correct to consider the n as each cell that we evaluate/count the number of speckles? So, when doing the statistical test can we join the cells from independent experiment that are from the same condition? e.g for each independent experiment 100 cells were analyze, which leaves us 300 cells per condition. Now, again, should we average for each independent experiment and have just 3 values that are then used for the statistical analysis? Or consider the 300 observations?

In sum, my questions are:

  1. is there a standard notation/syntax to refer to the number of observations in terms of technical replicates vs biological replicates? maybe 'k' and 'n', respectively.
  2. before doing a statistical test, should we use total number of observations including the technical replicates, or average for each biological individual/biological replicate?
  3. what counts as a biological replicate? Is it each biological individual that can give a response to a given condition (can be a mouse or can it be a cell)? (I guess that some techniques like qPCR would require a group of cells instead, due technical reasons)
  4. where to draw the line to know if an observations needs/has to be measured in replicates or not?
  5. if we are comparing means with t.test, when can and cannot we used normalized values? (e.g. qPCR, ChIP-enrichment, and relative quantification in western blot)

Thank you for your help.


  • 1
    $\begingroup$ Can you think about a better, more focused title of the question? I can't see how "existential" ever fits here. $\endgroup$
    – ttnphns
    Commented Dec 27, 2020 at 12:30
  • $\begingroup$ @ttnphns "existential" because it feels like it is something that should not cause doubts, but yet I have doubts $\endgroup$ Commented Dec 27, 2020 at 13:01

1 Answer 1


My biologist colleagues do similar research, but no-one so far could explain to me why they do it according to this schema: 3 'biological replications' $\times$ 3 'technical replications'. It seems to be the standard procedure that no-one questions, but also no-one really understands.

So, without knowing the reasons, I can only offer my more-or-less 'educated' guess: The biological triplicates are to account for variability in the subjects, while the technical triplicates account for your 'technical' variability (noise, measurement errors etc.). You should compare the variability in technical triplicates between the biological triplicates to ensure that all experiments were valid, i.e. that they really measured what you intended to measure. If some of them significantly deviate, it would point towards some problem with your experimental setup.

Assuming the variability in technical replicates does not differ between the biological replicates, you should take all 18 measurements (2 groups $\times$ 3 mice $\times$ three measurements) for further calculations, without computing any averages over technical replications.

  • $\begingroup$ > It seems to be the standard procedure that no-one questions, but also no-one really understands That's my current situation as well, and honestly I don't know if I should cry or laugh, or both. > If some of them significantly deviate Are you suggesting to make a normality test with all the observed values? like Shapiro-Wilk's test? P.S thanks for fixing the "x" in the calculation that were lost with copy+paste $\endgroup$ Commented Dec 27, 2020 at 17:32
  • $\begingroup$ I was actually thinking of ANOVA, to see whether the variances are equal. $\endgroup$
    – Igor F.
    Commented Dec 27, 2020 at 18:50
  • $\begingroup$ I see, but ANOVA would be an overkill test to compare means just between two groups? $\endgroup$ Commented Dec 28, 2020 at 10:17
  • $\begingroup$ I meant comparing the biological replications. You have three of them in each 'arm'. $\endgroup$
    – Igor F.
    Commented Dec 28, 2020 at 17:52

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