Is the arithmetic mean an average for every purpose?

The claim. The statement accepts that averages differ in purpose but then credits the arithmetic mean with being ideal for all of them. The first half is right; the second half goes too far.

Why the arithmetic mean is usually preferred. No other average meets so many of the tests a good average is judged by. Every single item of the data enters its value, and it can be carried into further algebra: the means of two groups can be combined into one, for instance. A formula fixes it, so two people working on the same data cannot reach different answers, and it needs nothing more than adding up and dividing, so it is quick to work out and readily grasped. That is why it is the average used most often, and why later measures such as the standard deviation and correlation are built on it.

Where it fails.

1. Extreme values. One very large or very small item pulls the mean towards it; for an income or wage series with a few very high earners, the median describes the typical worker better.
2. Open-end classes. When the first or last class has no limit, mid-points cannot be fixed and the mean cannot be found, while the median still can.
3. Qualitative data, or a search for the most common value. A dairy deciding which milk-pouch size to put on shelves wants the size bought most often; the mean pouch size may match no pouch it sells. An attribute such as intelligence or honesty can only be ranked, not measured. In such cases the mode or the median serves and the mean does not.
4. A value that may not exist. The mean of a discrete series can be a value no observation takes (2.4 children a family).

Conclusion. The arithmetic mean is the best general-purpose average, but not an ideal one for every purpose. The choice of average depends on the purpose of the study and on the nature of the data: the mean for symmetric quantitative data, the median for skewed or open-ended data and the mode for the most typical or most frequent value.