There are situations where none of the three averages is fully satisfactory. For example, if the number of items in a series is very small, none of these averages has any meaning as a measure of central tendency. Moreover, if the items are what has been called "freakishly deployed" (as when concentrated at one end), an average for the series can be found, but it is not descriptive of the series.
The data available may exclude the use of certain averages. For instance, in open-end distributions, the mean cannot be accurately found. But the median or mode can be used unless the median or modal class happens to be open-ended. In a situation of bimodality, the mode makes no sense as a measure of central tendency.
The characteristics of the averages and the demands of the problem to be solved usually determine which one is to be used. The data may call for a certain average.
In such problems as the average score on personnel tests or the average productivity rate of workers, where the values in the series are really ranks, the median is the average to use. These scores and rates are not additive, that is, they indicate not unit quantities, but rather the position of an individual with respect to other individuals. Therefore, a positional average namely the median is appropriate.
The arithmetic mean is the most commonly used and the best known of the averages, and is preferred unless precluding circumstances are present, such as extreme values at either end of the series or open-end classes, or varying class intervals, or unless we definitely wish to establish the most frequent value or some other positional average. Where further computational techniques are involved in the investigation, the mean is the average to be used.
Great care should be used in choosing an average. On occasions, it may even be advisable to work-out all three averages and present them. The added burden is preferable to the use of a single average that may be an incomplete description.
But considering all these points, the statistician, guided by the desire to present an accurate picture of the data and to command respect, is the final judge of which average is the most appropriate.