Arithmetic Average: When to Use It and When Not

Arithmetic average is the most popular measure of central tendency and (the reason for its popularity is that) it is the easiest one to calculate. However, like every other statistical measure, arithmetic average has strengths and weaknesses and it is more suitable in some situations than in others. The following is a summary of situations when using arithmetic average is appropriate and those when arithmetic average should be replaced or complemented by other measures.

When to Use Arithmetic Average

  • When you work with independent data, for example performance of multiple stocks or investments in a single period of time (otherwise geometric average may be better).
  • When all items in the data set are equally important (otherwise use weighted average).
  • When you need a quick, easy, and rough information about the overall level. Arithmetic average is the easiest one to calculate.
  • When you don't have a computer at hand. Arithmetic average is much easier to calculate in your head or on paper compared to geometric, harmonic, or weighted average.

When Not to Use Arithmetic Average (Alone)

Extreme Values

When the data set contains extreme values which are unevenly distributed between the two tails of the data, aritmetic average becomes biased to one or the other side and may no longer represent the real central tendency in a data set (without the extremes).

This problem can be addressed by using median alongside or in place of arithmetic average.

Errors in Data

Working with clean data sets is often impossible in practice – real world data sets often contain errors or missing values.

For example, when you download stock market data from a website or trading software, some pieces of data can be missing. Some programs automatically set missing rows to zero (or some other arbitrary value, like 999,999.99).

Although this is not a problem with arithmetic average itself, it can bias the result. As with extreme values, median is also affected by this, but usually less significantly.

In any case, before using arithmetic average (or any statistical tool), make sure your inputs are correct (or at least that you are aware of the errors and the biases they may cause in your results).

High Dispersion

When data set is very volatile or dispersed (it has high variance or standard deviation), knowing arithmetic average only can provide a very incomplete picture about the data (although that can be said of any data set).

There is nothing wrong with using arithmetic average for dispersed data sets. In fact, replacing arithmetic average with some other measure like geometric average will not solve all your problems here.

However, it is always a good idea to add other statistical measures to your analysis to check the volatility or skewness in the data set (arithmetic average and other measures of central tendency can't identify and describe such characteristics).

Percentage Changes

Arithmetic average is unsuitable when working with percentage changes over multiple time periods, especially when the changes are volatile. A time series of monthly or yearly investment returns is a good example.

In this case, the basis for the percentages is very likely to differ significantly from period to period and arithmetic average is quite useless. Geometric average is better here.

For more explanation why, and an example, see why arithmetic average fails to measure average percentage returns over time.

Different Weights

Arithmetic average is also inappropriate when individual items have different weights or different importance in the data set.

Arithmetic average assigns equal weight to all items. When you need to reflect different weights (for example in a portfolio of stocks or a stock index), weighted average is more useful. See reasons and example here: Why you need weighted average for calculating total portfolio return.

Using Arithmetic Average with Other Statistics

In many cases it helps to use arithmetic average together with other measures, such as median or standard deviation. You can easily calculate these and other statistics in Excel using the Descriptive Statistics Calculator.

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