Assuming that you have the following values for a data set:
- Median
- Mean
- First quartile
- Third quartile
- Standard deviation
- Number of elements
- Minimum value
, would it be possible to somewhat accurately calculate / estimate the maximum value?
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5$\begingroup$ Accurately, no. Maximum is a random variable. If you know the family of your distribution (Gaussian for instance) you should be able to calculate the density of probability of your maximum value. But, even in that case, I think it would not be very accurate. $\endgroup$– ArnaudCommented Aug 31, 2018 at 7:52
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$\begingroup$ Without making some specific and strong assumptions, all you can do is obtain bounds for the maximum. Both will be finite (the lower bound obviously is not below the third quartile and the upper bound obviously is limited by a certain number of standard deviations above the mean). $\endgroup$– whuber ♦Commented Aug 31, 2018 at 13:28
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3 Answers
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"Somewhat accurately", not. The maximum (the order statistic $X_{(n)}$ where $X_{(1)} \le \dotsm \le X_{(n)}$) will obviusly be larger than the third quartile. To say more, you will need some assumptions. I will assume iid data, else not very much can be said.
Is the distribution symmetric (can be judges by seeing if mean and median are both about midway between first and third quartiles)? Could it be close to normal (could be judged by comparing standard deviation with interquartile range)? Then you could get something by using joint distribution of order statistics. If tails could be much heavier than normal, you should look into extreme value theory.
If you could tell us something about which assumptions you are willing to assume, maybe something more concrete could be said.
answered Oct 17, 2018 at 10:16
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The method that comes most readily to mind is the German Tank Problem, which allows you to solve $E(\max\{X_i\})$. This assumes a Uniform(a, b) distribution, though. The general solution is $E(\max\{X_i\}) = \frac{a + bn}{n + 1}$. But again, that is for a specific distribution.
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Conditionally you could, yes. The condition would be that the underlying distribution is Guassian. If it were Gaussian then:
Mean = Median
Standard deviation = (3rd quartile - 1st quartile) / 1.349
You can use the number of samples to calculate the likelihood of your maximum value to confirm that it is not crazily unlikely.
Once you know it is Gaussian, then you've got a fighting chance of figuring out the most likely maximum for a given sample size using order statistics:
https://en.wikipedia.org/wiki/Order_statistic
PS: I suspect a few other distributions would be amenable to this sort of approach.
