Is it appropriate to put "error bars" on data when you have the full population? I have data on the large number of people who were applying to receive an award.  Everyone who applied must fill out a survey and specify their gender (although they can choose "prefer not to say").  My question: is it appropriate to put "error bars" on this data given that the survey was not a sample of the population, but rather the whole population?  (Any citations would be helpful.)
 A: @Ben is of course right that there is no "error" if you have the full population. But at least colloquially, "error-bars" are not always referring to uncertainties comming from a incomplete sample. And just a single number is seldomly useful, even if it is "exact". Two examples:

*

*If you say "$34.7$ percent of applicants were male", then that number is exact and should not have an error-bar. BUT: If your goal is to say something about gender-bias for example, the exact number itself is not sufficient. You need to add a line like "given the total number of applicants that we have, a deviation from a 'perfect' 50% by $\pm X$ is statistically expected without any bias existing". This number $X$ is not an error-bar of your data. But it can serve the same purpose: It informs the reader to what degree the first number you quote is meaningful.


*If you say "on average, the applicants are $34.7$ years old", it might again be less helpful to the reader as it could be. Saying "the mean age of applicants is $34.7$ years with a standard deviation of $4.6$" is much more useful. This second number is of course not an error-bar, but it serves the same purpose because it tells the reader a range of expected results, which is generally more helpful than one central result alone.
(After thinking about it, case (1) is essentially equivalent to the thing @Ben was referring to as "superpopulation")
A: Error bars show intervals; these intervals must represent something
Error bars in a plot show an interval for a particular quantity, and like any element of a plot, these intervals must actually represent something.
Intervals in a plot are generally useful in two situations: when you want to show an interval estimate for an unknown quantity, or when you want to show an interval that represents some descriptive aspect of the sample data.
The most common use of error bars is when you have an unknown quantity that you are estimating with data.  Typically this will occur when we have an estimate of some unknown quantity (which is a single point) and then the error bars are used to give an interval estimate for the quantity which encompasses the point estimate.  In your case, you have a full population of values, so there is no unknown quantity of interest to you.  Sometimes people may wish to use full population data to make an inference about some unknown aspects of a hypothetical "superpopulation" (and infinite extension of the population), in which case there may be some unknown quantity of interest.
The other case where error bars are sometimes used is when you want to show an interval range pertaining to a descriptive quantity that is known, rather than as an interval estimate for an unknown quantity.  For example, a graphical element that is somewhat akin to an "error bar" is used in a box-and-whiskers plot.  More generally, if you have a set of sample data with continuous measurements divided into categories, it is not unusual to give a barplot of means with bars showing some descriptive range for the subsample of continuous values in each category (e.g., interquartile range, or a certain number of sample standard deviations from the sample mean, etc.).  In this case the bars are used to indicate some kind of descriptive range pertaining to a sample.  Strictly speaking, these bars don't represent "errors" but they are nevertheless graphically identical to bars indicating interval estimates accounting for "errors".
Often there is a crossover between these two cases, when a descriptive interval pertaining to the data also functions as an interval estimate for some unknown quantity for an object outside the data.  In any case, whenever you produce a plot with error bars, you must ensure that the meaning of your bars is clear.  For certain plots like the box-and-whiskers, the meaning of the bars is fixed by convention (so no explanation is required), but in other cases you should tell your reader what the error bars on your plot represent.
If you think you might need to use error bars, or other bars that are graphically identical to error bars, you need to step back and ask yourself: What is the purpose of the interval I am proposing to show with these bars?  It it an interval estimate of an unknown quantity?  (If so, what is the unknown quantity?  What type of interval estimator are you using?)  Is it an interval representing some aspect of the data?  (If so, what aspect of the data does it represent?)  If you do not have satisfactory answers to these questions, then you do not have a need to use error bars in your plot of the data.
