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If you create a graph with error bars, or create a table with plus/minus values, you need to decide whether to show the SD, the SEM, or something else.

Often, there are better alternatives to graphing the mean with SD or SEM.

If you want to show the variation in your data

If each value represents a different individual, you probably want to show the variation among values. Even if each value represents a different lab experiment, it often makes sense to show the variation.

If you are plotting a column graph fewer than 100 or so values per data set, create a scatter plot that shows every value. What better way to show the variation among values than to show every value? If your data set has more than 100 or so values, a scatter plot becomes messy. Alternatives are to show a box-and-whiskers plot, a frequency distribution (histogram), or a cumulative frequency distribution.

If you are plotting XY data, especially with multiple treatment groups, plotting every replicate can lead to a messy graph. It can be a good first step, so you see your data fully. But then change to mean and error bar when presenting the data.

If you want to plot mean and error bar, the SD  quantifies variability among replicates. So does a graph of median with interquartile range or full range. When plotting a graph with error bars, be sure to explain how the error bars were computed in the figure itself or in its legend.

If you want to show how precisely you have determined the mean

If your goal is to compare means with a t test or ANOVA, or to show how closely our data come to the predictions of a model,  you may be more interested in showing how precisely the data define the mean than in showing the variability. In this case, the best approach is to plot the 95% confidence interval of the mean (or perhaps a 90% or 99% confidence interval).

What about the standard error of the mean (SEM)? Graphing the mean with an SEM error bars is a commonly used method to show how well you know the mean,  The only advantage of SEM error bars are that they are shorter, but SEM error bars are harder to interpret than a  confidence interval. Nonetheless, SEM error bars are the standard in many fields.

Whatever error bars you choose to show, be sure to state your choice. Noticing whether or not the error bars overlap tells you less than you might guess.

If you want to create persuasive propaganda

If your goal is to emphasize small and unimportant differences in your data, show your error bars as SEM,  and hope that your readers think they are SD

If our goal is 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.

This approach was advocated by Steve Simon in his excellent weblog. Of course he meant it as a joke. If you don't understand the joke, review  the differences between SD and SEM.

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