What is an ideal measure of central tendency?

Answer 1

It depends primarily on the nature of the data.

If the data are qualitative data then the only option is the mode.

Sometimes data can be ordered but the interval between adjacent categories is not always the same. An example of such data might be answers to a questionnaire where the answers are "strongly disagree", "disagree", "agree" and "strongly agree". The difference between strongly disagree and disagree may not be comparable to the difference between disagree and agree. In such cases, the median is readily available but the mean depends on arbitrarily assigned weights (for the categories).

You can have interval data, in which the values of the variable are known and the differences between the values are also quantified. In such cases both the mean and median may be used.

The median is generally preferred for skewed data since it is not greatly affected by outliers. For more symmetric data sets there is little to choose between the median and the mean since they will be close together.

However, by the Central Limit Theorem, the mean result from repeated trials will tend towards the population mean. The sample mean is a maximum likelihood unbiased estimate of the population mean. Also, the means are often one of the parameters of parametric statistical distributions.

The distribution of the mean of repeated trials has been extensively studied and there are many efficient tests for testing hypotheses concerning the mean.