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I'm not an expert on this topic. I'm working on a poster for a conference. I need to plot SEMs, confidence intervals, or some measure of uncertainty around a measure of central tendency (e.g., median). I have small samples, n = 6, of proportion data. First, I plotted error bars with the SEMs, but I noticed that the error bars could exceed the limits of the binomial distribution (e.g., position < 0 or > 1). This is certainly awkward. So I secondly opted to work with bootstrap techniques. Maybe I could use 68% confidence intervals to substitute the first intended SEMs. I thought that this would be a good idea, given that confidence intervals of a bootstrapped distribution might be assymetric, putatively solving the limit exceeding problem. However, I read that bootstrapping is not recomended for samples < 9. Is this an unbending rule, or may I proceed with caution to estimate uncertainty through bootstrapping? How can I estimate uncertainty otherwise?

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    $\begingroup$ If you're using R, you can do this to get a 95% confidence interval: binom.test(positives, sample.size, conf.level=.95)$conf.int $\endgroup$ – user54038 Jul 20 '18 at 21:48
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When N is small the standard measures don't work well for proportions. There's actually quite a large literature on this. A good entry point is Agresti and Coull Approximate is Better than Exact for Interval Measure of Binomial Proportions. They offer several methods to deal with this problem, but the simplest is to add 2 "successes" and 2 "failures" to the data and proceed as usual.

R and SAS both offer several choices to do this estimation. Other statistical packages may do so, as well.

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