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This is a simple, generalized statistics calculator that computes statistical values such as the mean, population and sample standard deviation, and geometric mean among others. Many of these values are more well described in other calculators also available on this site. Also, many of these values are more thoroughly explained in the respective calculator pages linked below. In addition, the calculator will compute the variance as the standard deviation squared.
The geometric mean is a type of average that is calculated by taking the nth root of the product of n numbers. Unlike the arithmetic mean (which is the sum of numbers divided by the count), the geometric mean is particularly useful when dealing with values that have widely varying scales or when you want to find an average that accounts for proportional growth.
The geometric mean is especially valuable in cases where you need to average ratios, percentages, or values that are exponentially related. For example, if you're comparing car ratings where one car scores 8/10 for fuel efficiency and 2/10 for safety, the arithmetic mean would be 5, but the geometric mean would be 4, which better reflects the overall rating when considering both factors proportionally.
Key Property: The geometric mean is always less than or equal to the arithmetic mean, and it's particularly useful when dealing with percentage changes or multiplicative relationships.
Formula:
(Πi=1N xi)1/N = N√(x1 × x2 × x3 × ... × xn)
Formula Components:
Example Calculation:
Calculate the geometric mean of: 1, 5, 7, 9, 12
Geometric Mean = 5√(1 × 5 × 7 × 9 × 12)
= 5√(3780)
≈ 5.194
Compare this to the arithmetic mean: (1 + 5 + 7 + 9 + 12) / 5 = 34 / 5 = 6.8
Notice how the geometric mean (5.194) is lower than the arithmetic mean (6.8), which is always the case when values vary significantly.
Applications of Geometric Mean:
When calculating average growth rates over time, the geometric mean provides the correct average rate. For example, if an investment grows 10% in year 1, 20% in year 2, and 15% in year 3, the geometric mean gives the average annual growth rate.
Used in psychology, economics, and other social sciences when averaging ratios, rates, or indices that have different scales or units.
When working with aspect ratios (like 16:9, 4:3), the geometric mean helps find an average ratio that maintains the proportional relationship.
In geometry, the geometric mean appears in various contexts, such as finding the mean proportional in similar triangles or calculating dimensions in geometric progressions.
Used in finance to calculate average returns on investments, especially when dealing with compound interest or when returns vary significantly over time periods.