Standard deviation versus standard error (May 16, 2005)

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Someone asked me about when you should report the standard deviation and when you should report the standard error. This is often done on graphs using a vile and disgusting approach known as error bars. People get these confused easily, and since the standard error is always smaller, here is a good strategy.

• When you are trying to emphasize small and unimportant differences in your data, show your error bars as the standard errors for the groups and hope that your readers think they are standard deviations.
• When you are trying to cover-up large differences, show the error bars as the standard deviations for the groups, and hope that your readers think they are a standard errors.

Actually, I always prefer the standard deviation, because that is a measure that tells you something about the data itself. Here's an example from an article published in an open source journal.

• Can pulsed ultrasound increase tissue damage during ischemia? A study of the effects of ultrasound on infarcted and non-infarcted myocardium in anesthetized pigs. Olivecrona GK, Hardig BM, Roijer A, Block M, Grins E, Persson HW, Johansson L, Olsson B. BMC Cardiovasc Disord 2005: 5(1); 8. [Medline] [Abstract] [Full text] [PDF]

Here is the abstract of this article:

Background The same mechanisms by which ultrasound enhances thrombolysis are described in connection with non-beneficial effects of ultrasound. The present safety study was therefore designed to explore effects of beneficial ultrasound characteristics on the infarcted and non-infarcted myocardium. Methods In an open chest porcine model (n = 17), myocardial infarction was induced by ligating a coronary diagonal branch. Pulsed ultrasound of frequency 1 MHz and intensity 0.1 W/cm2 (ISATA) was applied during one hour to both infarcted and non-infarcted myocardial tissue. These ultrasound characteristics are similar to those used in studies of ultrasound enhanced thrombolysis. Using blinded assessment technique, myocardial damage was rated according to histopathological criteria. Results Infarcted myocardium exhibited a significant increase in damage score compared to non-infarcted myocardium: 6.2 ± 2.0 vs. 4.3 ± 1.5 (mean ± standard deviation), (p = 0.004). In the infarcted myocardium, ultrasound exposure yielded a further significant increase of damage scores: 8.1 ± 1.7 vs. 6.2 ± 2.0 (p = 0.027). Conclusion Our results suggest an instantaneous additive effect on the ischemic damage in myocardial tissue when exposed to ultrasound of stated characteristics. The ultimate damage degree remains to be clarified.

I like seeing the standard deviation, because then I can apply the rough rule of thumb that says that most of the data will be between plus/minus two standard deviations. So the non-infected myocardium had most of the damage scores somewhere between 1.3 and 7.3. The infarcted myocardium had most of the damage scores between 2.2 and 10.2. The ultrasound exposure had most of the damage scores between 4.7 and 11.5. This tells me that even though the p-values are very small, there is still a fair amount of overlap in the individual damage scores.

Another advantage of reporting the standard deviation is that you often see interesting relationships between the means and standard deviations. In particular, groups with large means often have larger standard deviations as well. This sort of relationship might be missed if you reported standard errors, especially if the sample sizes in each group are not all the same.