幾何平均

Use geometric mean for data that changes exponentially, such as economic growth, biological populations, or compound interest. The arithmetic mean would be skewed toward larger values in multiplicative datasets. Geometric mean accurately represents the central tendency when values grow or shrink by consistent percentages over time.

Multiply all data values together, then take the nth root of the product, where n is the number of values. Alternatively, convert each value to its logarithm, add all logarithms, divide by n, then take the antilog of the result. Both methods yield the same geometric mean value.

The geometric mean of a dataset is always quantitatively less than or at most equal to the arithmetic mean. This relationship holds because geometric mean accounts for multiplicative relationships in data, while arithmetic mean treats all values equally. The difference is most pronounced when data values vary widely.

No, geometric mean cannot be used if the dataset contains zero or negative values. Zero would make the product zero, resulting in a geometric mean of zero. Negative values create mathematical complications with roots and logarithms, making geometric mean undefined or meaningless for such datasets.

Geometric mean calculates the average growth factor for compound interest over multiple periods. For example, with annual rates of 10%, 5%, and 2%, the geometric mean of 1.10, 1.05, and 1.02 equals 1.056, indicating an average growth rate of 5.6% per annum. This accurately represents the true average return across time intervals.

Population and financial investments grow multiplicatively over time, not additively. Geometric mean captures this exponential growth pattern accurately, whereas arithmetic mean would overestimate or misrepresent the true central tendency. This makes geometric mean essential for realistic statistical analysis of biological and economic datasets.

Convert each data value to its natural logarithm, then calculate the arithmetic mean of all logarithmic values. Finally, take the antilog (exponential function) of this mean to obtain the geometric mean. This method is computationally useful for large datasets and provides the same result as the nth root approach.