Geometric mean

The geometric mean of positive numbers is their product raised to the reciprocal of their count. It averages multiplicative scale and is invariant under changes of measurement units.

Logarithm: A logarithm gives the exponent to which a base must be raised to produce a number. Taking logarithms turns the geometric mean into an arithmetic mean of log-values.
Compound annual growth rate: Compound annual growth rate is the constant yearly growth rate that connects an initial value to a final value over a period. It is a geometric mean of period-by-period growth factors.
Positive real number: A positive real number is a real number greater than zero. The stated product-based definition applies to positive inputs.
Median: The median is the middle value in an ordered data set, or the midpoint of its two central values. Unlike the geometric mean, it depends on rank rather than multiplication.
Arithmetic mean: The arithmetic mean is the sum of values divided by their count. It averages additive differences, while the geometric mean averages multiplicative ratios.
Investment return: Investment return is the gain or loss on an investment over a specified period. Geometric averaging captures how sequential returns compound into an ending wealth value.
Nth root: An nth root of a number is a value that, raised to the nth power, gives that number. Raising the product to the reciprocal of the count is taking an nth root.
Root mean square: The root mean square is the square root of the arithmetic mean of squared values. It weights large magnitudes more strongly instead of averaging multiplicative scale.
Weighted geometric mean: A weighted geometric mean raises each positive value to a specified weight before multiplying and normalizing. It extends the product formula when observations contribute in unequal proportions.
Geometric mean return: Geometric mean return is the constant per-period return equivalent to a sequence of compounded investment returns. It applies the mean directly to returns after converting them into growth factors.

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