If you have two data points (N=2) for two individual analyses of a sample then is it logical to calculate standard error (SE)?
No it does not make sense; the equation may work but not the statistics background.
What is advised in statistics handbooks is, if you have a SERIES of duplicates, for which you expect the precision to be the same, is to calculate the pooled standard deviation, which will be a much better estimate of the analytical precision. I use Box, G. E. P., Hunter, W. G. & Hunter, J. S. (1978). Statistics for Experimenters, an Introduction to Design, Data Analysis and Model Building. New-York, Wiley and Sons, 352 pp., but I am sure there are more recent books...
Pooled standard deviation is calculated as the square root of individual variances pondered by individual degrees of freedom or, more simply, as the square root of (the sum of (squared difference to the mean) divided by the sum of degrees of freedom).
It does not work when the values cover different orders of magnitude, as then you can NOT assume that the precision is the same.
Yes, it does. The equation for calculating the SE work for two values. But the ideal situation would be to use the raw data in your graph or table (namely the two values that you have) than representing the mean ± SE.
No it does not make sense; the equation may work but not the statistics background.
What is advised in statistics handbooks is, if you have a SERIES of duplicates, for which you expect the precision to be the same, is to calculate the pooled standard deviation, which will be a much better estimate of the analytical precision. I use Box, G. E. P., Hunter, W. G. & Hunter, J. S. (1978). Statistics for Experimenters, an Introduction to Design, Data Analysis and Model Building. New-York, Wiley and Sons, 352 pp., but I am sure there are more recent books...
Pooled standard deviation is calculated as the square root of individual variances pondered by individual degrees of freedom or, more simply, as the square root of (the sum of (squared difference to the mean) divided by the sum of degrees of freedom).
It does not work when the values cover different orders of magnitude, as then you can NOT assume that the precision is the same.
@ Catherine M G C Renard: I cannot agree. Could you be more specific on what is the "statistics background" you are referring to? This term seems vague. Using N=2 (and d.o.f. = 1) does not violate the definition of standard deviation.
@ Jeannie Horak: I cannot agree either - naming one arbitrary number (e.g. N=3) as the minimum is irrelevant, as it may very much depended on the particular field. In case of retrospective studies, a data analyst deals with what is available. There always exists a need for giving an estimate of data spread, and standard deviation (from N=2) may be as informative as reporting the two measurements or their range, it is only a matter of particular application and convenience.
Besides the fact that having more data increases the confidence estimates and reduces the error estimates in general, there is no fundamental reason why statistics such as average or standard deviation cannot be given for two measurements. In principle (e.g. in terms of the assumed distribution of the population from which the data is obtained), they are as valid estimates as any other coming from more data.
In any case, I am open to discussion on this topic.
Dr. Wyrzykowski, with all due respect, I'm not sure if you are being serious or approaching this purely as a theoretical exercise (again no offense). A range or spread intuitively implies a practical set of data. If there are only two numbers, inferring these as your range would be highly speculative to the point of absurdity. They are simply the only two observations you have. Now of course one could bring up all sorts of caveats claiming that the estimate is more-or-less sound, due to expectations based on knowledge of the population distribution, etc. That doesn't mean anyone should conceptualize or report an SE from two observations or that it "makes sense" as Arshad Ali asked. Neither you nor I would be satisfied wrapping up a quantitative analysis having only two measurements. Note that Arshad asked about SE (standard error), which is derived from sample SD but conceptually very different. This is regardless of the merits of SD for an n of 2.
Unfortunately, I have been seeing plenty of papers recently with inferential statistics performed on n=3 samples and graphs peppered with lots of meaningless and undefined SEM bars. The problem is not whether the statistics are wrong due to some kind of pedantic classroom rule. Rather, it's that this quantitative frosting tends to imply an unwarranted level of confidence (literally) in an experimental outcome, both to the audience and even to oneself. Whether a journal will publish without a 'sufficient' n just adds to the arbitrary nature of the issue; many will and those that won't aren't necessarily any sounder in their judgment.
The problem as I see it is that researchers need to accept that there is nothing inherently wrong with presenting qualitative, non-interpreted data. Some interesting 'anecdotal' data as in, for example, a clinical case report could be far more relevant than a page full of bar graphs showing minuscule but 'significant' differences. Calling two measurements n1 and n2 does not make them any more meaningful.
Since a key-term in this question in 'analytical chemistry', lets think back to Humphry Davy, who managed to discover so much before statistics was even invented! An outdated, romantic, and naive example? Well if the only things left to discover in nature are to be found exclusively through arbitrary probabilistic analysis, then we have a problem.
Dr. Horak, my understanding is that the n=2 one sees for ELISA tests is referring to the number of replicate dilutions used for test validation. The regression in an ELISA is fit over more than 2 data points (wells). The n referred to by the original poster is different than the n in an ELISA validation.
Below is my explanation of the issue for Arshad Ali:
Are you trying to construct a sampling distribution based on two observations/subjects (n=2)?
If so, then conceptually you are trying to estimate the population distribution based on a kind of 'midpoint' of two numbers. I say midpoint because the idea of a mean, median, or other measure of central tendency intuitively makes no sense to me considering you only know two numbers.
Whether the formula works or not is beside the point. Common sense would dictate that one would need a certain number of observations to generalize about anything in life.
Wow, it’s been quite a learning to read diverse arguments from all you. I really appreciate your input. Thanks all!
n=3 is arbitrary and folckloric:
https://www.graphpad.com/support/faqid/591/
Regards!
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