What are the different types of averages?

1. I know there are three types of averaging methods in statistics: Mean, Mode and Median. Are there any other type of averaging methods that statisticians use?

2. How do I know which method is best for a particular set of data?

1. One popular type of average that you have not mentioned is the trimmed mean (recommended by, for example, Wilcox, 2010) which I think of as a middle road between the mean and the median. You get the trimmed mean by first discarding $n$ % of the lower and upper part of your sample and then taking the mean of the resulting subset, where $n$ can be , for example, 10. The resulting average is generally more robust to outliers than the mean.

2. If your data looks normally distributed (or generally heap shaped) the mean is a good description of the general tendency of the data. If the data is skewed then often the median or a trimmed mean can be a better description of the general tendency.

References

Wilcox, R. R. (2010). Fundamentals of Modern Statistical Methods: Substantially Improving Power and Accuracy, Springer, 2nd Ed.

For the geometrically minded, there are means based on monotonic transformations of data.

The geometric mean of a random variable is defined as $$\mbox{G.M.}(X) = \exp \left( \int_{\Omega_X} \log(x)df_x \right) .$$

This is excellent at handling measures of the things that are known to grow exponentially such as income, bacterial colonies, disease progression, etc. One of the reasons log transforms are so highly favored in biostatistics is due to their ability to estimate geometric means with regression models.

The harmonic mean of a random variable is defined as $$\mbox{H.M.}(X) = \left( \int_{\Omega_X} x^{-1}df_x \right) ^{-1}.$$

This is excellent at estimating averages of rates where you have incidents, tasks, or events in a numerator and measures of person time in the denominator. For health planning or corporate mergers, you might be interested in staffing locum tenens doctors to travel between 3 acquired community hospitals to serve specific breakouts of MRSA. Traveling between places, you need to jointly average time per task across the various hospitals and their protocols. A harmonic mean tells you that.

With respect to choosing amongst the three types of averages you list, it is generally considered that the mean is appropriate for continuous equal-interval data, the median is for ordinal data, and the mode is for nominal data. However, this scheme is quite limited. See which-mean-to-use-and-when for more sophisticated thoughts on the topic.