# Why is standard error sometimes used for “error bands” in plots?

It seems that often what someone really wants to plot is a confidence interval of some kind, but using SE for this purpose I think only ends up comprising something like a 68% confidence band. Therefore, plotting SE for error bars instead of a wider band more representative of the significance level of your analysis visually suggests significance in your data that may not actually be there.

Consider the following concrete example:

set.seed(123)
X <- rnorm(100, 0, 1)
Y <- rnorm(100,1.7,5)
df = data.frame(X,Y)

boxplot(df)

se.x = sd(X)/sqrt(length(X))
se.y = sd(Y)/sqrt(length(Y))

X.err.CI = 1.96*se.x
Y.err.CI = 1.96*se.y

plot(1:2, colMeans(df), ylim=c(-1,3), xlim = c(0.5,4.5), col="dark green"
, main="Comparison of SE bars vs 95% CI")
lines(c(1,1), c(mean(X) + X.err.CI, mean(X) - X.err.CI), col="dark green")
lines(c(2,2), c(mean(Y) + Y.err.CI, mean(Y) - Y.err.CI), col="dark green")
text(1:2 + .2, colMeans(df), c("X","Y"))

points(3:4, colMeans(df), col="blue")
lines(c(3,3), c(mean(X) + se.x, mean(X) - se.x), col="blue")
lines(c(4,4), c(mean(Y) + se.y, mean(Y) - se.y), col="blue")
text(3:4 + .2, colMeans(df), c("X","Y"))

abline(v=2.5, lty=2)

legend("topright"
,c("95% CI", "+/- SE")
,lty=c(1,1)
,pch=c(1,1)
,col=c("dark green", "blue")
)


If we just base our analysis on SE (the image on the right), visually it appears that there is significance between the means of X and Y because we don't have overlap in our error bars. But if we're testing at a 5% significance level, plotting the 95% confidence bands shows that this is clearly not the case.

Since we can expect that a test at the 32% level will never be appropriate, why even show the SE bars since they will probably be interpreted as though they represent a confidence interval? Do people use SE bars instead of more meaningful CIs because it's moderately easier to calculate (e.g. using a built-in function in Excel)? It seems that we're paying a pretty high cost in terms of the interpretability of our graphic in exchange for a few minutes' less work. Is there some value/utility in SE bars that I'm missing?

For context, I was prompted to write this after skimming this article. I was frustrated by the lack of confidence intervals in the plots provided by the authors, and then when they did finally provide them, it turned out they were just SE bars.

• Without trying to be ultra-cynical, there seem to me to be two overwhelming reasons why people draw bars +/- one SE: "honest" (e.g. 95% or 99%) confidence intervals would look horribly large in some problems; many people just follow what they believe to be standard practice. I set on one side that using CIs for graphical significance tests is fraught with problems. – Nick Cox Oct 25 '13 at 16:47
• On the latter see e.g. ncbi.nlm.nih.gov/pmc/articles/PMC2064100 and other papers with the same first author; tandfonline.com/doi/abs/10.1198/000313001317097960 – Nick Cox Oct 25 '13 at 16:54
• When you base your analysis on overlapping error bars, you will be far more conservative than you think. See stats.stackexchange.com/questions/18215/… for an analysis. It demonstrates that typically (e.g.) a test based on non-overlap of two independent 95% confidence intervals actually is (approximately) a 99.5% test. Roughly, looking for overlap among two-sided 1 SE error bars is about a 95% test--just what you might like to do! – whuber Oct 25 '13 at 17:54
• People who are using 1 s.e. bars are almost certainly not using it to 'test for a difference'; to do so would have the problems you note. – Glen_b Oct 25 '13 at 23:28