How to determine the range of values of the population (with some confidence, e.g. 95%) based on a sample?

I thought about one approach to follow the 68-95-99 rule, based on population mean estimate and population standard deviation estimate calculated from a sample. But those are only estimates, we don't know the real population mean, we have only 95% confidence interval for it. Thus, I am not sure how to correctly apply this rule. Let's assume normal distribution for simplicity (but it would also be interesting to know if there is a simple answer without assuming normality.)

EDIT: based on one of the answers below, I need to clarify that I am not interested in knowing the range that contains 100% of population values (which is infinity, and is not useful), but the one that contains 95% (or 99% etc.) of population values.

  • $\begingroup$ In order to estimate the range of the population you will need its mean and standard deviation and, since you don't know the true values, you will need to use the estimated values. There's nothing wrong with using other estimates when estimating something else, so long as you make the appropriate adjustments when finding variance etc. I should note though that there may be a formal method for this which I'm not aware of, hence I'm only commenting. $\endgroup$
    – Patty
    Sep 5, 2016 at 1:45
  • $\begingroup$ en.wikipedia.org/wiki/German_tank_problem $\endgroup$ Sep 5, 2016 at 5:21
  • $\begingroup$ This question is possibly related. $\endgroup$
    – GeoMatt22
    Sep 6, 2016 at 13:57

1 Answer 1


The assumption of a normal distribution makes this problem trivial. All normal distributions are supported on the whole real line, so the population range is the whole real line, too. In other words, there is no minimum or maximum.

  • $\begingroup$ This is a good point, and makes me realize I should clarify my original question: from the more practical standpoint, I would like to know something like: "95% of population values should fall within ??? of the sample mean". Knowing the range that contains 100% of population values is meaningless. I will edit the original question. $\endgroup$
    – Viktor
    Sep 6, 2016 at 13:45
  • 1
    $\begingroup$ @Viktor In that case, what you want are called quantiles. There are a number of ways to estimate population quantiles, the simplest way being to just use the sample quantiles unaltered. Under the assumption of a normal distribution, you can estimate a population quantile by computing the same quantile of the normal distribution with mean and SD equal to your sample mean and sample SD. $\endgroup$ Sep 6, 2016 at 15:07
  • $\begingroup$ OK, got it. I wasn't sure if that would be correct to use the sample mean and SD, just because intuitively I felt that they underestimate the width of the interquantile range - after all, there is some uncertainty regarding the population mean (it's somewhere within sample mean +/- z*s.e.m.), and then we can find the quantile based on the extreme possible value of true population mean and an estimate of population SD. But that was just my intuition. If using sample mean and SD is the formally statistically appropriate way to do that, then that's the way to go. $\endgroup$
    – Viktor
    Sep 6, 2016 at 16:35

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