# Which error bars to use when comparing only two subjects when many replicates?

I want to plot discrete time measurements made for two subjects, but where many cells were observed for each subject in triplicate.

I think the experimental unit is the subject, so I should just get a mean for each subject and plot that. Another option is to get a mean of each cell's tripicates and plot those.

In any case, I can't tell if I should use standard error or standard deviation for the error bars. As I understand it, standard deviation is meant for different samples, and it's not clear to me that each cell should count as a sample. On the other hand, I keep reading that most of the time you should use standard deviation for error bars.

Edit:

I was computing the standard deviation and standard error for each subject at each timepoint. I wasn't taking into account the fact that the observations were triplicates. So basically I calculated the means, SD, and SE and simply plotted those values.

Should I instead calculate the mean, SD, and SE for each triplicate and plot those? Maybe plot a smooth line running through them using loess?

They serve different purposes. Standard deviation describes variability in a population. Standard error describes variability in an estimated parameter. Do you want to make a statement about how precisely your experiment quantified the mean among replicates? Or do you want to make a statement about how much variability there was among the replicates?

Suppose you're selling some new technology that quantifies some value. You'd probably want to report the standard deviation among replicates to give people a sense how much variability they can expect when applying your tool. Reporting the standard error wouldn't mean much, as you can always make the standard error go down simply by collecting more data from the same population. A low standard error may not imply a highly precise tool, but a low standard deviation would.

On the other hand, if you're applying the tool in practice, you may want to report standard error to give a sense of how well you're quantifying different populations under study. Standard error is more closely linked to whether populations are "significantly different" or not, as that comes down to how well you've estimated the population parameter. Oftentimes the goal of applying such quantification tools is to describe whether populations are on average different in both a statistical and meaningful sense, even if those populations are quite variable.

There is no fixed rule what to use. In some specific situations there may be arguments that one or the other is more appropriate, but this will depend on your subject area, aims of analysis, and also on your audience. The most important thing is that you state clearly and unambiguously what you use, whichever it is. You may in fact find out what works better when interpreting results, namely if you make reference to the uncertainty of the estimates (which is quantified by standard errors), or rather to how much data one could expect close to your fits (which is quantified by standard deviations).

Also regarding what means to plot this is in the first place dependent on the focus of interest and interpretation in that study. For your own private use nothing should stop you from plotting both variants and learn from them whatever you can, and then you can present what was more useful for you to understand your results (or even both of they show different things of interest). Note however that if you plot triplicate means make sure you do it in a way that shows which belong together to the same subject.

In general I use standard deviations or quantiles for descriptive purposes, i.e. describing my sample, my sampling procedure, etc. - basically whenever I would use boxplots to visualize my data.

When my goal is to do any type of inferential or frequentist statistics, and I need an estimation of the error of a statistic, e.g. when I'm interested in the sampling distribution of the sample mean to make generalizing statements about the underlying population parameter, I would use the standard error of the mean (or confidence intervals).

On the contrary, when I'm doing Bayesian analyses I would only use standard deviations or quantiles since you are working with a full posterior distribution of plausible parameter values, hence the concept of a standard error (and sampling distributions) doesn't exist.

• in this case my "population" is only one person, shouldI treat it as descriptive then? Commented Mar 15 at 14:09
• @maglorismyspiritanimal ultimately it depends what your underlying scientific question is and the goal of your sampling procedure. Given from what you said and without knowing more, I would probably take a mean of the triplets and then do boxplots for visualizing the distribution of those measurements. But again you need to me more specific with your question. Commented Mar 15 at 14:29