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Suppose we draw a histogram of the distribution of all the possible sample averages drawn with replacement from the set $\{1,2,7,8,14,20 \}$. I know that $\frac{1+2+7+8+14+20}{6}$ has the highest chance of occurring.

So this average would be the peak? Can we say that this distribution is unimodal? Do we know which averages are least likely to occur? I intuitively think that numbers that are all the same have the least probability of occurring. This seems to be similar to Zipf's Law or Benford's Law.

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The value of a random variable that has maximum probability is called its mode and should not be confused with its mean. In particular, the mean is not necessarily a value that the random variable takes on. Example: $X$ takes on values $0$ and $1$ with probabilities $\frac{1}{4}$ and $\frac{3}{4}$ respectively. The mode is $1$ while the mean is $\frac{3}{4}$. –  Dilip Sarwate Dec 29 '11 at 1:27
In this case, though, the average of all the points is indeed the most likely average (though not by much). I'm trying to construct an example where this is not the case, but failing, and I can't seem to prove that the average of the most likely set is always the most likely average (which seems unlikely in general). –  Lopsy Dec 29 '11 at 1:41
When you say "sample averages drawn with replacement from {1,2,7,8,14,20}" how many samples are you considering? Your set has six elements, but you could form a sample average from any number of draws greater than zero. In particular, if you only have one sample then there are six equally likely values for the sample average. –  Chris Taylor Dec 29 '11 at 8:18

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