Zero is not the same as no value

Before computing any average, there is a question to settle: which values belong in the calculation at all? Getting this wrong quietly spoils the answer.

Suppose a player scores 57, 13, 0, 84, —, 51, 27 in a series. Look carefully at two of those entries. In match 3 the player PLAYED and scored 0 runs. In match 5 the player did not play at all.

Those are different things. The 0 is real data — a performance that happened. The dash is not data; there was no performance to record.

So we count the number of matches as 6, not 7, and the mean is (57 + 13 + 0 + 84 + 51 + 27) ÷ 6 = 232 ÷ 6, about 38.67 runs per match. Dividing by 7 would be wrong, because it would treat an absence as a zero score.

Here is a striking case. In a cricket match, can the MEDIAN runs scored by a team's players be 0 while the team scores 407 for the loss of 10 wickets? It can — if most of the eleven players scored very little and a few scored heavily, the middle value can easily be 0. Accounting for 19 extras, the average runs per player is (407 − 19) ÷ 11 = 35.27, wildly different from a median of 0.

Sometimes the right choice is to leave values out altogether. Sita's mango tree yielded 0, 0, 8, 24, 41, 16, 5, 0, 0, 0, 0, 0 mangoes from January to December.

Averaging over all twelve months gives about 7.8 mangoes a month — a number that describes no month at all, since the tree either gives nothing or gives a lot. If we want the mean or median per month, it is more appropriate to consider only the summer months when mangoes are expected to grow. Deciding what to include is part of the analysis, not something that comes before it.