# “Confidence intervals” or equivalent for population distribution

I feel like this is a basic question and/or I'm missing/have forgotten something obvious.

Is there an equivalent of "confidence intervals" which one uses for the mean, but for the whole population?

E.g. given a sample distribution (which may or may not be normal), is there some way I can specify limits and say, "with y% confidence, we can say that any given x from the population will fall within these two limits)?

I suppose if the data is normally distributed, I could look at the Z-scores?

Should point out that some of the samples of interest are NOT normally distributed, although some are.

• the quantiles of the population distribution would be a confidence interval, by example if you want $95\%$ confidence interval you can take the $0.025$ and $0.975$ quantiles of the distribution. – carlos Jan 16 '18 at 17:12
• This would be called a tolerance interval for 100% of the population. If you believe data come from a normal distribution, then you don't need any data to answer this question! The only correct solution is that there are no finite limits for the population. This problematic situation suggests you might not be asking the question you intended to ask. – whuber Jan 16 '18 at 17:15
• @Whuber - good point. I hadn't thought of it that way. It did occur to me that if the data come from a normal distribution, then I can follow the standard rules of a normal distribution, based on the population mean and standard deviations. However, I also have some non-normal distributions. So if, for example, I want limits within which 95% of the population values would be expected to fall, and I have a non-normal distribution, how do I calculate this? Even for a normal distribution, do I need to worry about, e.g., the confidence intervals for the mean and stand devs? – Statsanalyst Jan 16 '18 at 17:18
• @whuber - to clarify the situation, I have various samples for different populations, which are times of day that a particular even happens. I want to be able to say, for each population, that, e.g., 95% of the events from the population would be expected to happen between 4:15pm and 4:50pm, etc. It's to do with resource scheduling. Some of the samples show a normal distribution, and some do not (measured using Shapiro-Wilk tests). – Statsanalyst Jan 16 '18 at 17:22
• That's precisely what a tolerance interval is for. Although the theory is well-developed for Normal distributions, it is not as much for non-Normal distributions. (As @carlos notes, a tolerance limit (one end of an interval) can be considered a confidence limit for a percentile of the population.) One aspect of tolerance intervals that is well-developed is the theory of non-parametric intervals. These can be obtained as pairs of order statistics of samples, especially when the samples are relatively large. – whuber Jan 16 '18 at 18:33