Example:A stock you and your research partner are analyzing has 12 years of annualized return data. The returns are 12%, 25%, 34%, 15%, 19%, 44%, 54%, 33%, 22%, 28%, 17%, and 24%. Your research partner is exceedingly lazy and has decided to collect data based on only five years of returns. Given this data, calculate the population mean and calculate the sample mean. (Your partner's data set is shown above as bold).
Population mean = 12 + 25 + 34 + 15 + 19 + 44 + 54 + 33 + 22 + 28 + 17 + 24 / 12 = 27.25%
Sample mean = 25 + 34 + 19 + 54 + 17 / 5 = 29.8%
Arithmetic meanis the sum of the observation values divided by the number of observations. It is the most widely used measure of central tendency, and is the only measure where the sum of the deviations of each value from the mean is always zero.
Example:A data set contains the following numbers: 5, 9, 4, and 10. The mean of these numbers is: ( 5 + 9 + 4 + 10) / 4 = 7. The sum of the deviations from the mean is: (5 - 7) + (9 - 7) + (4 - 7) + (10 - 7) = -2 + 2 - 3 + 3 = 0.
Geometric meanis often used when calculating investment returns over multiple periods, or to find a compound growth rate.