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?

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    $\begingroup$ 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 -Reinstate Monica Nov 18 '13 at 20:37
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    $\begingroup$ 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 says Reinstate Monica 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 -Reinstate Monica 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

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