Geometric Mean in Mathematics

Geometric Mean

Definition: The geometric mean is a type of average that is used to calculate the central tendency of a set of numbers by using the product of their values.

Formula: The geometric mean of a set of n numbers {x1, x2, ..., xn} is calculated as:

gm = √(x1 × x2 × ... × xn)

Alternative Formula: The geometric mean can also be calculated using the following formula:

gm = e^(∑(ln(x)) / n)

where ln is the natural logarithm.

Properties:

• The geometric mean is always less than or equal to the arithmetic mean.
• The geometric mean is sensitive to the units of measurement.
• The geometric mean is used to calculate the average of rates, ratios, and indexes.

Applications:

• Finance: The geometric mean is used to calculate the average rate of return on an investment.
• Economics: The geometric mean is used to calculate the average growth rate of an economy.
• Engineering: The geometric mean is used to calculate the average rate of flow in a system.

Examples:

• The geometric mean of the numbers {2, 4, 8} is √(2 × 4 × 8) = 4.
• The geometric mean of the numbers {10, 20, 30} is e^(∑(ln(x)) / 3) = 20.

Geometric Mean

• The geometric mean is a type of average that calculates the central tendency of a set of numbers using the product of their values.

Formula

• The geometric mean of a set of n numbers {x1, x2,..., xn} is calculated as gm = √(x1 × x2 ×...× xn).
• The geometric mean can also be calculated using the alternative formula: gm = e^(∑(ln(x)) / n).

Properties

• The geometric mean is always less than or equal to the arithmetic mean.
• The geometric mean is sensitive to the units of measurement.
• The geometric mean is used to calculate the average of rates, ratios, and indexes.

Applications

• In finance, the geometric mean is used to calculate the average rate of return on an investment.
• In economics, the geometric mean is used to calculate the average growth rate of an economy.
• In engineering, the geometric mean is used to calculate the average rate of flow in a system.

Examples

• The geometric mean of the numbers {2, 4, 8} is √(2 × 4 × 8) = 4.
• The geometric mean of the numbers {10, 20, 30} is e^(∑(ln(x)) / 3) = 20.

Learn about the geometric mean, a type of average used to calculate the central tendency of a set of numbers. Understand its formula, alternative formula, and properties.