My first question here and i hope that its not to obvious or stupid.
My problem: I have a number of $x$ values ( $x$ can vary) that follow a normal distribution. I can calculate all normal descriptive statistics (minimum, mean, quartiles and so on), but those $x$ values only describe a small fraction of a potentially much bigger range of values (which still follows the same distribution and the average should not vary much).

So what i really want is to estimate the minimal value of the unknown full distribution. Something like an estimation of the minimal x-intercept of a known distribution with given slope and tip value.

Imagine for instance the standard trees dataset in R (see picture below). I don't want to know what is the minimal value of the given height values (60), but instead estimate the possible true minimal value in a most accurate way (might be $height<=50$).

enter image description here

Is this possible or just stupid? Or am i missing a basic statistic principle which already allows me to estimate those values?

  • 1
    $\begingroup$ There is no minimum possible value of a normal distribution. Instead, you can talk about the range that contains 95% of the data (within two standard deviations from the mean) or contains 99.7% of the data (within three standard deviations from the mean): see here $\endgroup$ – David Robinson Nov 18 '13 at 20:34
  • $\begingroup$ You assert normality right in your title and your opening sentence. If the minimum possible value is not $-\infty$, then your assertion is false. (i) How do you know with such certainty that you have normality? (ii) why do you need the population minimum? $\endgroup$ – Glen_b Nov 18 '13 at 20:37
  • 1
    $\begingroup$ Related, though perhaps pedantic, is the acknowledgement that the normal distribution, because it has support on the whole real line, is not a great representation of something like "height of trees," since heights of trees are bounded below by 0. A distribution with support $x \in [0, \infty)$ might be more appropriate. $\endgroup$ – Sycorax Nov 18 '13 at 20:38
  • $\begingroup$ The three-sigma rule might be what i need. As user777 correctly noted the data can not go to infinity (both on the lower and the upper tail). Do i need to assume another value-distribution if not normal? @Glen_b I know that the distribution of values will always approximate a normal distribution. I need the minimum for my analysis, but my sample is definitely incomplete. Therefore the minimum of the sample values is very rough. $\endgroup$ – Curlew Nov 19 '13 at 11:20
  • $\begingroup$ If you can accept probabilistic bounds, you would probably want some form of model; extreme-value distributions are mostly used for upper tails, but the distributions carry across to lower tails with a few appropriate adjustments; I think that the implication we should be able to draw is that the distribution of the minimum for an observed distribution with a lower bound might be approximated by a Weibull, for example. This is not really my area however. $\endgroup$ – Glen_b Nov 19 '13 at 19:22

There is no minimal value or "x intercept" since the tails go off to infinity. What you may want to look at is tail probabilities. For instance, for a N(0,1), try finding out P(X>4). Very small probability.

  • $\begingroup$ Also note that the statement of Chebyshev's Inequality gives you those probabilities, in general. $\endgroup$ – Marc-Andre Seguin Nov 18 '13 at 20:36
  • $\begingroup$ Mhh, so i just need to calculate the $k=10$ standard deviation from the given mean to approximate the potential minimal value with 99% probability. Do i get this right? $\endgroup$ – Curlew Nov 19 '13 at 11:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.