# Confidence Interval For Mean By Bootstrapping

The standard deviation in my original sample is very large, about 100 or so. I took many bootstrap samples, found the mean of each bootstrap sample and then took the mean of these means. I found the standard deviation of the means of my bootstrap samples from this 'mean of means' to be approximately 10.

My question is: when finding a confidence interval for the population mean, do I need to divide this value of '10' by sqrt(n-1) where n is my sample size?

  mean(original_sample)+-1.96*10/sqrt(n-1)


It's just that my standard deviation has dropped so much, from 100 to 10, that maybe the factor of sqrt(n-1) has somehow been accounted for already?

## 1 Answer

Use the empirical quantiles or percentiles to construct the confidence interval. Check the wiki Bootstrap

• Empirical quantiles, referring to my value of 1.96? From the look of my bootstrapped means I've assumed normality. I'm just wondering if my 'bootstrap standard deviation of the mean' needs to be divided by the factor sqrt(n-1), since it already seems to be an sd of the mean. Apr 6, 2015 at 15:12
• To use the empirical quantiles to construct CI, I meant it is a standard method, especially when you are not sure about SD. For your current question, SD does depend on the sample size. For SD of bootstrap means, the sample size is R, bootstrapping resampling number, not n, the original sample size.
– xjf
Apr 6, 2015 at 17:50
• Yep so my boostrap sd of the mean is sqrt(sum(mean-mean of means)^2/(#bootstrap samples-1)) But, in normal practice, when produce a CI for the mean you would do mean+-1.96(s/sqrt{n-1}), so here I'm relating my bootstrap sd to 's', meaning that I would need to further divide by sqrt(n-1), right? Apr 6, 2015 at 18:16
• I'm with purewater. A pseudo-sigma is a robust measure of variation that is analogous to the standard deviation. I like to use the robust with the non-robust, ie mean with median, stdev with pseudo-sigma, to get a sense of how skewed my data is, or to detect the presence of strong outliers. If I want to bound my bootstrap sampled mean, I would make sure samples per mean are reasonable and that total count of bootstrap means are reasonable. I would then use percentiles (like 5% and 95%) on the bootstrap values to get bounds for the mean. If you want 95% CI CL this isn't a bad way to go. Apr 6, 2015 at 18:17