Mean

The arithmetic mean is the central value of a set of results: you add up all the observed values, then divide that sum by the number of values. It summarizes an entire series with a single representative number, around which the results are distributed.

The mental image is that of a fair sharing. If all the results were pooled and then redistributed in strictly equal parts, everyone would receive the mean. It is also the balance point of the series, the one where the deviations upward exactly offset the deviations downward.

The calculation is direct. For five die rolls giving 2, 4, 4, 6 and 4, you add them up: 2 plus 4 plus 4 plus 6 plus 4 make 20. You then divide by the number of rolls, five, which gives 20 divided by 5, that is 4. The mean of this series is therefore 4.

The mean should not be confused with expectation. The mean concerns results that have actually been observed, after the draws; the expectation is the expected theoretical average, weighted by the probabilities, which can be computed even before playing. The mean is observed, the expectation is predicted.

A common trap is to take the mean for a value that would come up often. With a fair die, the expectation is 3.5, a number that no face can display. The mean is not necessarily a possible result; it is a marker of tendency.

The law of large numbers links the two notions: the more draws you accumulate, the closer the observed mean gets to the expectation. This is why, by rolling a die or the site's number generator over a long stretch, the mean of the results ends up gravitating around the expected theoretical value.