By Timothy Banas
Updated Aug 30, 2022
In mathematics, a "mean" is an average that summarizes a data set. A well‑chosen mean provides insight without misrepresenting the underlying values. For example, a meteorologist might report that the average temperature on January 22 in Chicago is 25°F, based on historical data. While this figure cannot predict the exact temperature for next January 22, it offers a reliable guide for travelers to pack appropriately.
The arithmetic mean is the most common form of average. It is found by adding all data points together and dividing by the count of those points.
Example: Arithmetic mean of 11, 13, 17, and 1,000 = (11 + 13 + 17 + 1,000) ÷ 4 = 260.25
The geometric mean, on the other hand, multiplies all data points and then takes the nth root, where n is the number of values.
Example: Geometric mean of 11, 13, 17, and 1,000 = 4th root of (11 × 13 × 17 × 1,000) ≈ 39.5
Outliers—values that are markedly different from the rest—can distort the arithmetic mean. In the example above, 1,000 is an outlier. The arithmetic mean (260.25) is far removed from the bulk of the data, making it a poor representation of the set. The geometric mean (39.5), however, is much closer to the majority of values, mitigating the influence of the outlier.
Use the arithmetic mean when your data are normally distributed and free of extreme outliers. It works well for average temperatures, sports statistics like batting averages, and typical daily measurements.
Opt for the geometric mean when dealing with multiplicative processes or data that span several orders of magnitude. Biologists use it to estimate bacterial population sizes that can jump from 20 to 20,000. Economists prefer it for income distributions, where a few high earners can skew the arithmetic average.
Choosing the right mean ensures that your analysis reflects reality rather than a misleading arithmetic distortion.
