Should I be using the SD or SE? Standard Deviation or Standard Error?
I’m surprised that there are so many poor answers to this question out there, so I’ve decided to write my own, just in case my poor answer helps you any better.
The Standard Deviation
The Standard Deviation is a description of how much spread there is around the mean. You can find the formula all over, so I won’t repeat it here. A small SD indicates that your data points are closely clustered around the mean, and the larger your SD gets the more the data is spread, in a normal distribution. However, there is some intrinsically useful information that the SD carries which may help you decide whether or not you should be using it. If your distribution is normal (and yes, you should have already tested for this), then the SD tells the reader that around two thirds of the data points in your data set fall within one SD of the mean: that is the mean plus 1 SD and the mean minus 1 SD. In addition, the units of SD are practically the same as the units that you have used to collect your data and display your mean. Thus, the SD is providing your reader with a descriptor of the mean that is quantifiable and can be easily interpreted.
You can report the SD in text as follows:
Our data show that males (n = 353) were larger (387.5 ± 49.37 mm) than females (n = 321; 245.4 ± 27.61 mm).
Note that I’ve not used the terms mean or standard deviation in the text above, this is because you should set this up in your material and methods section. If you need to change between the SD and SE in your results, then you will need to indicate which is which when you report it. I have used different levels of accuracy (decimal places) for means and SD. This should be relative to how you actually measured your data. In this case, I measured animals to the nearest mm, so report the mean to one decimal place and the SD to two. Although there is no strict rule, you should not be reporting a mean with far greater accuracy that you actually measured it (e.g. to four decimal places). Note also that the sample size is given for each set. It is very important to provide this information somewhere. There may be instances (such as an experiment) where these numbers are set throughout the document, and so don’t need to be repeated in the text each time you report a result.
Standard Error
The purpose of the Standard Error is to inform the reader on the likelihood of the mean. In many biological studies, we take the sample from a population that is much larger because we can’t measure all individuals. In our example above, we see that males were larger than females, but how likely is it that the mean we obtained reflects the true mean of the entire population, both sampled and unsampled?
By now, you should have come to the conclusion that the SE is useful when comparing means. In a graph or table where you are interested in demonstrating whether or not the means are different, you can use the SE as error bars around the mean. Note that the convention is that you have plus or minus two SE in error bars (two SE above and two SE below the mean). This is because 2 SE is equivalent to a 95% probability that the mean falls within this range, meaning you’d be very sure. Showing 1 SE either side only shows that you are 68% sure, which doesn’t help much if your test statistic is 0.05. Either way, just make sure that you clearly indicate what you have done in the figure legend. In addition, you will probably want to conduct some statistical test to show that the data are indeed different. Interestingly, statisticians are moving away from many of these tests (or at least the test statistic). If you have used the SE correctly, your graph should speak for itself and there is no real need to carry out a test. In cases where it appears marginal, a test can be useful. Alternatively, you should go back and measure more animals!
In summary
In summary, the SE is telling us about the variability of the population mean (in relation to the one we measured), while the SD is giving information on the variability of the data points that we collected around their mean.
Standard Deviation or Standard Error?
To try to answer the original question posed above. You are most likely to report the SD in your text when describing the data you collected, and the SE on a graph demonstrating how likely it is that the means you obtained represent the entire (only partially sampled) population.
Read on...
If you didn’t understand the above, or want to read more on the (relatively) simple logic of how and why the SD and SE are calculated and derived, I suggest that you look for:
Streiner DL (1996) Maintaining Standards: Differences between the Standard Deviation and Standard Error, and when to use each. Can J Psychiatry 41:498–502.
I am indebted to Don Kramer for pointing me toward the above paper.
