# What is standard error used for?

I'm using a tutorial I found and plotting mean values along with the standard errors to show my data. But I'm having a problem discussing the results. My plot is as shown below: some of the standard errors (shown as a error bar) vary much and some of them are very close to zero.

• A side-issue here is that using bars is likely to prove confusing. Trivially, downward bars are a bit harder work than upward bars. More fundamentally, bars starting at 1e-3 is arbitrary. More positively, showing point estimates by point symbols and adding error bars would be a lot simpler than showing bars plus error bars. Google "dynamite plot" for more. – Nick Cox Aug 16 '13 at 18:23
• I'm not sure what the question is. Based on the answer you marked correct and title it might be to just know what a standard error is. But based on what you have here it seems you need help describing the data. Could you please clarify in the question? Also, if you want help describing the data then please relate more about the data, not just the figure. The N's in each group and what the values mean would be helpful. Any transforms done would be helpful as well. – John Aug 16 '13 at 19:06

Error bars in general are to convince the plot reader that the differences she/he sees on the plot are statistically significant. In an approximation, you may imagine a small gaussian which $\pm1\sigma$ range is shown as this error bar -- "visual integration" of a product of two such gaussians is more-less a chance that the two values are really equal.

In this particular case, one can see that both the difference between red and violet bar as well as gray and green ones are not too significant.

• what about standard error in this case? as plotted error bars. – berkay Mar 23 '11 at 1:14
• It's a poor error bar if that's the purpose. Non-overlap of the bars is not sufficient for statistical significance and the amount of non-overlap needed to actually be significantly different at 0.05 varies with N. And what the heck does "not too significant" mean? Both of those marginal conditions you point out would fail a t-test. – John Aug 16 '13 at 19:00
• @John As I wrote, error bars are a visual clue to help make ad hoc assessments while investigating the plot; actual testing needs some hypothesis to be tested, thus obviously should happen in the text. – user88 Aug 17 '13 at 21:39

In general, the standard error tells you how uncertain you are that true value of the top of the bar is where the bar says it is. When there are multiple bars, it can also enable comparisons between bars, in the sense of a statistical test. However, interpreting them in this way requires some assumptions, shown graphically below. If you are really interested in comparing the bars to see if the differences are statistically significant, then you should run tests on the data and display which tests were significant,like this.

In addition, I would suggest using confidence intervals rather than standard errors.

This paper is well worth the read:

Cumming and Finch. "Inference by Eye: Confidence Intervals and How to Read Pictures of Data." Am Psych. Vol. 60, No. 2, 170–180.

Their overall conclusion is: "Seek bars that relate directly to effects of interest, be sensitive to experimental design, and interpret the intervals."

For independent samples, using confidence intervals, half overlap of the CIs means the difference is statistically significant.

For independent samples using standard error bars instead, the following graph shows you how to figure out statistical significance:

• This isn't really an answer (yet). Would you mind augmenting this citation w/ some info about how it helps to answer the OP's question? (btw, I'm not the downvoter) – gung - Reinstate Monica Aug 16 '13 at 18:41
• @gung Real life intervened so I posted a partial answer. Updated. – Ari B. Friedman Aug 16 '13 at 18:53

As mbq says, error bars are a way of letting your readers to get a feel if the differences between two groups are significant - i.e. if the variation within each of your groups is small enough to believe that the difference you've found for the mean between your groups.

All else being equal, larger error bars mean more within-group difference, but it looks like the y-axis of your plot is log-transformed, so the lower groups aren't quite on the same scale as the higher ones.

You should be aware, many of your readers won't understand what error bars represent, even if you explicitly explain it! Often you can achieve the same goal with a jittered dot-plot or a boxplot (or both together) to achieve the same effect.

• regarding the article you mentioned, it's an interesting observation, however it doesn't come as a surprise to me. I find a significant portion of statistical concepts and common practices as confusing and convoluted (even though I have a strong background in mathematics, and have taken a number of courses in mathematical statistics). I personally feel a lot of the concepts would have been much easier to understand if they were taught visually and using examples, instead of long and convoluted verbal explanations. – posdef Mar 23 '11 at 10:23

Plenty of researchers have trouble interpreting these graphs. See http://scienceblogs.com/cognitivedaily/2008/07/31/most-researchers-dont-understa-1/ for a more detailed elaboration.