# How to understand confidence interval?

I'm learning confidence interval/level, and I was taught that confidence interval can measure how representative our sample is. But I don't know why.

For example, assuming that we are given a sample of size N from population, and after calculating, we get the sample mean $\overline x$ and the standard deviation of the mean $\sigma_\overline x$. Given the confidence level 95%, we get the confidence interval.

$$\overline x \pm1.96\sigma_\overline x$$

It means that, were this procedure to be repeated on multiple samples, the calculated confidence interval ($\overline x \pm1.96\sigma_\overline x$) would encompass the true population parameter 95% of the time.

What confuses me is that $\overline x$ is different for each sample. So the confidence interval is different for each sample. If we draw a sample, and get its parameters from calculation

$$\overline x = 5\\ \sigma_\overline x = 1$$

Can we say were this procedure to be repeated on multiple samples, the calculated confidence interval ($[5-1.96, 5+1.96]$) would encompass the true population parameter 95% of the time? I think the answer is negative, because for each sample, $\overline x$ is a different value, so the confidence interval for each sample is different. If each sample's confidence interval is different, why should we use it to denote how representative our sample is?

And if we have a confidence interval $[5-1.96, 5+1.96]$, how should I interpret the interval without using $\overline x$(because $\overline x$ varies from sample to sample)?

• You are constructing a random interval (upper and lower confidence limits) to cover a fixed true parameter. Before you obtain the realization of the limits, it has a probability of, say $95\%$ to cover the true parameter. So if you independently, repeatedly to observe realizations of these limits for many times, you expect you have close to $95\%$ of them covering the true parameter. In reality, you may not be able to repeat this and you can only observe 1 realization. You may read this first: en.wikipedia.org/wiki/… – BGM Sep 13 '16 at 5:14
• So as you said "a given realised interval", is different from "a random interval before you obtain the realization" – BGM Sep 13 '16 at 5:27
• @BGM My struggle is that, every sample has different interval. Given a confidence level of 95%, if the CI of my sample is $[5-1.96, 5+1.96]$, I cannot say "if we get more samples of the same size in the future, population mean will lie in $[5-1.96,5+1.96]$ for 95% of them(because $\overline x$ is different for each sample)". Nor can I say "there's 95% probability that the population mean will lie in the interval $[5-1.96,5+1.96]$(This is definitely not true according to the definition of CI)". If so, how can confidence interval give us some confidence about the sample? – Searene Sep 13 '16 at 5:32