# Geometric mean

## Inhaltsverzeichnis

The Geometric mean G of observations n X1, X2, … , Xn is defined as

G = (X1∙X2∙ …∙Xn )1/n with xi greater or equal to 0 for i = 1,2,...n.

### Merke

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The geometric mean is used to compound the growth rates of time- series data.

### Beispiel

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On December 31, 2008, you put \$1,000 in your bank account. On December 31, 2009, you have \$1,200; on December 31, 2010, you have \$1,300. One year later, you still have the same amount of money. But at the end of 2012, your money has shrunk by \$30, to \$1,270. Finally, on December 31, 2014, the account balance is \$1,350. Find the annual growth rates and calculate their mean.

We calculate

x1 = 20 percent, x2 = 1,300/1,200 – 1 = 8.33 percent, x3 = 0 percent, x4 = -30/1,300 = 2.3077 percent and x4 = 1,350/1,270 – 1 = 6,2992 percent.

Therefore, the average growth rate is calculated as

G = (X1∙X2∙ …∙Xn )1/n

= (1.2∙1.0833∙1∙0.976923*1.062992)1/5

= 1.350.2

= 1,0619

= 6.19 percent.

So we find a growth rate of 6.19 percent per year. The account will therefore show, on average, the following amounts at the end of each year:

 Year Amount of money December 31, 2008 1,000.00 December 31, 2009 1,061.90 December 31, 2010 1,127.63 December 31, 2011 1,197.43 December 31, 2012 1,271.55 December 31, 2013 1,350.00

Tab. 3: Money in your bank account: average growth rate.

### Methode

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• If an amount of money rises, the growth rate is superior to 1.
• If it stays constant, the growth rate is equal to 1.
• If it shrinks, the growth rate is inferior to 1.

Alternatively, we could have calculated the average growth rate like this:

G = (capital at the end/capital at the beginning)1/n

= 1,350/1,000)1/5

= 1,350.2

= 1,061 percent.