# Mean of a distribution

In an asymmetrical distribution the mode and median are respectively 12.30 and 16.42. Find the mean of the distribution.

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Without additional information about the distribution, there is not much that we can assert for certain. However, if we are willing to make reasonable guesses, there is often a rough empirical relationship between mode, median, and mean, for fairly well-behaved unimodal distributions. The relationship is $$\text{mean}-\text{mode}\approx 3(\text{mean}-\text{median}).$$ We can use this approximate relationship, combined with a crossing of the fingers, to estimate the mean, though an answer to $2$ decimal places would certainly be inappropriate!. There are even proofs that under suitable restrictions the relationship roughly holds.

For a detailed discussion at a not difficult level, please look here.

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Without knowing more about the distribution, it could be anything.

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Alright, here are four numbers $$12.30,\quad 12.30,\quad 16.42,\quad 20, \quad 30$$ The mode is 12.30, the median is 16.42, and the mean is 18.204.

Here are four numbers: $$12.30,\quad 12.30,\quad 16.42,\quad 20, \quad 40$$ The mode is 12.30, the median is 16.42, and the mean is 20.204.

So the given information is consistent with any of various different numbers in the role of the mean.

That is how you know that not enough information is given to answer the question.

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