How to Compute the Geometric Mean

E. Klein | Aug 3, 2010 |

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The geometric mean is a type of mathematical average. The kind of average that most people are familiar with the regular arithmetic mean in which you add up all of the values of the data points and divide by the number of data points. For example, if there are five homes for sale in the neighborhood and the selling prices are

$100,000,
$200,000,
$85,000,
$400,000, and
$250,000,

then the arithmetic mean or average selling price is

(100,000 + 200,000 + 85,000 + 400,000 + 250,000)/5 = $207,000.

The arithmetic mean is good for determining the center of certain types of data, such as prices, ages, test scores, and other attributes. But for concepts such as average percent change or average growth rate, the geometric mean is a better tool.

Uses of the Geometric Mean, How to Compute It, and Some Examples

Suppose a culture of bacteria increases every day, and there is enough agar and room in the petri dish so that the growth is exponential (at least in the short term). A scientist observes that during the first 24-hour period, the bacteria increased by 23%. The second day they increased by 34% over the previous day, the third day by 29%, and the fourth day by 35%. If the scientist wants to find the average daily percent change over the 4-day period, she should compute the geometric mean of 1.23, 1.34, 1.29, and 1.35. (This is because a 23% increase is equivalent to growth by a factor of 1.23, etc.) The geometric average of the four numbers is

(1.23*1.34*1.29*1.35)^(1/4)
= (2.8703403)^(1/4)
= 1.3016174.

Since the average daily growth factor is 1.3016174, this means that the average daily percent increase is about 30%. The general formula for the geometric mean is

(X₁*X₂*...*X_n)^(1/n).

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Here is another example. Suppose a business's profits increased by 30% during one year. The next year, the profits decreased by 14% over the previous year. The subsequent year, the profits increased by 8% over the previous year.

Year	Profits	Actual % Change
0	$50,000	*
1	$65,000	+30%
2	$55,900	-14%
3	$60,372	+8%

To find the average yearly percent increase/decrease over the 3-year period, we need the take the geometric average of 1.30, 0.86, and 1.08. (When something decreases by 14%, it retains 86% of its value.)

(1.3*0.86*1.08)^(1/3)
= 1.20744^(1/3)
= 1.06485

Since the average yearly growth factor is 1.06485, this means the company's profits grew by an average annual rate of 6.49%.

If you instead take the arithmetic mean of 30%, -14%, and 8%, you get

(30 + (-14) + 8)/3
= 24/3
= 8.

This would indicate that the company's profits grew by an average of 8% each year, which is incorrect. For processes that are multiplicative, the geometric mean will give you the true average growth rate, while the arithmetic mean will give you an answer that is too high.

Year	Actual Profits	6.49% Growth Over Each Year	8% Growth Over Each Year
0	$50,000	$50,000	$50,000
1	$65,000	$53,245	$54,000
2	$55,900	$56,700	$58,320
3	$60,372	$60,379 (just right)	$62,985 (too high)

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Besides the geometric mean and arithmetic mean, there are other types of averages, such as the harmonic mean and root mean square. Different average formulas are appropriate for different types of data analysis.