# Arithmetic Mean and Standard Deviation

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The arithmetic mean (AM) and arithmetic standard deviation (ASD) should only be used for short-term investments that have a maturity of one year or less because it is a simple interest equation that does not account for the compounding that occurs over multiple years.  If compounding is occurring, even for an investment less than one year in maturity, then the geometric mean and standard deviation should be used instead.  Assuming that you are analyzing “P” number of periods that are less than a year in duration and utilize simple interest, the following equations can be used to derive the period and annualized rates of return and standard deviations:

 Total Periods < 1 Year, Period Output $AM=\frac{1}{P}\displaystyle\sum_{i=1}^{P}{r}_{i}$ $ASD = \left[exp\left(\frac{1}{\left(n\times P \right)}\displaystyle\sum_{i=1}^{n\times P} \ln \left(1+{r}_{i} \right)\right)-1 \right]P$
 Total Periods < 1 Year, Annualized Output $AM=\displaystyle\sum_{i=1}^{P}{r}_{i}$ $ASD = \left[exp\left(\frac{1}{\left(n\times P \right)}\displaystyle\sum_{i=1}^{n\times P} \ln \left(1+{r}_{i} \right)\right)-1 \right]P$
 Total Periods >1 Year, Period Output $AM=\frac{1}{n\times P}\displaystyle\sum_{i=1}^{n\times P}{r}_{i}$ $ASD = \left[exp\left(\frac{1}{\left(n\times P \right)}\displaystyle\sum_{i=1}^{n\times P} \ln \left(1+{r}_{i} \right)\right)-1 \right]P$
 Total Periods > 1 Year, Annualized Output $AM=\frac{P}{n\times P}\displaystyle\sum_{i=1}^{n\times P}{r}_{i}$ $ASD = \left[exp\left(\frac{1}{\left(n\times P \right)}\displaystyle\sum_{i=1}^{n\times P} \ln \left(1+{r}_{i} \right)\right)-1 \right]P$

Note that sample statistics are being applied since the 1 (one) has been subtracted from all of the “P” and “n×P” denominators for the standard deviation calculations.

Definitions:

n = number of years

P = number of periods per year

r = periodic return

Two example spreadsheets are attached which show how to calculate the Arithmetic and Geometric means and standard deviations for stocks, but these same techniques can be applied to any set of data.

Analysis of Returns

Arithmetic & Geometric Example

For investments with a maturity horizon greater than one year, the Geometric Mean and Standard Deviation should be used, and these equations can be found on another blog post here:

Geometric Mean and Standard Deviation