# Why Arithmetic Average Fails to Measure Average Percentage Return over Time

## Arithmetic average: the basic tool in statistics

**Arithmetic average** is a good tool for *measuring central tendency* of data sets which represent independent values and values taken at one point of time, for example when you’re calculating average return of a number of stocks in a given time period (see arithmetic average calculation example).

## Calculating average return over multiple time periods

On the other hand, when you have a set of data which has a *time-series nature* and is *expressed in percentages*, **arithmetic average has weaknesses**. The classic example is calculating average return of a stock over several periods of time. Let’s see how and why **arithmetic average fails** here.

## A stock investment example and arithmetic average

Let’s say you have bought a stock. Your stock performed like this in the last two years:

- Year 1: +70%
- Year 2: -50%

**What is the arithmetic average of the stock’s return?** The sum of +70% and -50% is +20%, divided by 2 gives us +10%. This looks like a solid return.

## Arithmetic average fails to discover the loss

**How much money have you made in reality?** If you have invested a million dollars, at the end of year 1 your position was worth 1.7 million. In the second year, its value has halved, and you end up at 0.85 million – less than what you’ve invested. You have lost money. The **arithmetic average is misleading**.

## Why is arithmetic average not working?

When you have **stock performance expressed in percent**, the percentage advance or decline in every particular year is **measured relative to the stock’s price at the beginning of that year** – which is different for each year. You are comparing incomparable numbers. 50% of 1.7 million is greater than 70% of 1 million, though it doesn’t seem so looking at the percentages only.

## Geometric average is better for averaging performance over time

When you are investing, dollars or euros matter more than percentages in the end. Whenever you have **time series kind of data ****expressed as**** percentages** (which is a very frequent case in finance and investing), **arithmetic average is misleading**. **Geometric average** is much more suitable here, as it automatically takes the different starting values in every single year into consideration.