X |
Ln(X) |

1 |
0 |

2 |
0.693147 |

3 |
1.098612 |

10 |
2.302585 |

Geometric mean = 2.78 |
Arithmetic mean = 1.024. EXP[1.024] = 2.78 |

The base of natural logarithms is 2.718. The expression: EXP[1.024] means that 2.718 is raised to the 1.024th power. Ln(X) is the natural log of X.

Naturally, you get the same result using logs base 10 as shown below.

X |
Log(X) |

1 |
0.0000 |

2 |
0.30103 |

3 |
0.47712 |

10 |
1.00000 |

Geometric mean = 2.78 | Arithmetic mean = 0.44454. 10 ^{0.44454} = 2.78 |

If any one of the scores is zero then the geometric mean is zero. The geometric mean does not make sense if any scores are less than zero.

The geometric mean is less affected by extreme values than is the arithmetic mean and is useful as a measure of central tendency for some positively skewed distributions.

The geometric mean is an appropriate measure to use for averaging rates. For example, consider a stock portfolio that began with a value of $1,000 and had annual returns of 13%, 22%, 12%, -5%, and -13%.The table below shows the value after each of the five years.

Year | Return | Value |

1 | 13% | 1,130 |

2 | 22% | 1,379 |

3 | 12% | 1,544 |

4 | -5% | 1,467 |

5 | -13% | 1,276 |

The question is how to compute annual rate of return? The answer is to compute the geometric mean of the returns. Instead of using the percents, each return is represented as a multiplier indicating how much higher the value is after the year. This multiplier is 1.13 for a 13% return and 0.95 for a 5% loss. The multipliers for this example are 1.13, 1.22, 1.12, 0.95, and 0.87. The geometric mean of these multipliers is 1.05. Therefore, the average annual rate of return is 5%. The following table shows how a portfolio gaining 5% a year would end up with the same value ($1,276) as the one shown above.

Year |
Return |
Value |

1 |
5% |
1,050 |

2 |
5% |
1,103 |

3 |
5% |
1,158 |

4 |
5% |
1,216 |

5 |
5% |
1,276 |