Measures of Central Tendency

The arithmetic mean is the sum of the observations divided by the number of observations.

The median is the value of the middle item of a set of items that has been sorted into ascending or descending order.

In an odd-numbered sample of n items, the median occupies the (n + 1)/2 position. In an even-numbered sample, we define the median as the mean of the values of items occupying the n/2 and (n + 2)/2 positions (the two middle items). Advantage: extreme values do not affect it.

The mode is the most frequently occurring value in a distribution. The only measure of central tendency for nominal data.

Unless all the observations in a data set have the same value, harmonic mean < geometric mean< arithmetic mean.

[Practice Problems] The annual returns for three portfolios are shown in the following table. Portfolios P and R were created in 2009, Portfolio Q in 2010. The median annual return from portfolio creation to 2013 for:

[Solutions] C

The median of Portfolio R is 3.0%, 0.8% higher than the mean for Portfolio R.