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Geometric Mean Calculator

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Geometric Mean

Definition

The geometric mean is the nth root of the product of n numbers. It's particularly useful for analyzing proportional changes and rates of growth.

Formula

Geometric Mean=x1×x2×...×xnn=(∏i=1nxi)1/n\text{Geometric Mean} = \sqrt[n]{x_1 \times x_2 \times ... \times x_n} = \left(\prod_{i=1}^n x_i\right)^{1/n}

Example

For the values: 100, 120, 108, 135, 150

Geometric Mean=100×120×108×135×1505=121.89\text{Geometric Mean} = \sqrt[5]{100 \times 120 \times 108 \times 135 \times 150} = 121.89which is the average growth rate over the 5 periods.

Key Points

  • Best for analyzing proportional changes or growth rates
  • All values must be positive for real-valued results
  • Less sensitive to extreme values than arithmetic mean

Comparison of Different Means

Try Arithmetic MeanTry Harmonic Mean
TypeFormulaExample
Arithmetic Mean1n∑i=1nxi=x1+x2+...+xnn\frac{1}{n}\sum_{i=1}^n x_i = \frac{x_1 + x_2 + ... + x_n}{n}2+4+6+84=5\frac{2 + 4 + 6 + 8}{4} = 5
Geometric Mean∏i=1nxin=x1×x2×...×xnn\sqrt[n]{\prod_{i=1}^n x_i} = \sqrt[n]{x_1 \times x_2 \times ... \times x_n}2×4×6×84≈4.4\sqrt[4]{2 \times 4 \times 6 \times 8} \approx 4.4
Harmonic Meann∑i=1n1xi=n1x1+1x2+...+1xn\frac{n}{\sum_{i=1}^n \frac{1}{x_i}} = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n}}412+14+16+18≈3.8\frac{4}{\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8}} \approx 3.8

Arithmetic Mean

Best For:
  • Simple averages of quantities
  • Calculating central tendency
  • Equal weight to all values
  • Linear relationships
Common Uses:
  • Test scores
  • Heights/weights
  • Daily temperatures
  • Income levels
Limitations:
  • Sensitive to outliers
  • Not ideal for ratios or rates

Geometric Mean(Current)

Best For:
  • Growth rates
  • Percentage changes
  • Ratios and proportions
  • Exponential relationships
Common Uses:
  • Investment returns
  • Population growth
  • Interest rates
  • Price indices
Limitations:
  • Only works with positive numbers
  • More complex calculation

Harmonic Mean

Best For:
  • Rates and speeds
  • Per-unit quantities
  • Reciprocal relationships
  • Density measurements
Common Uses:
  • Average speed over different distances
  • Price per unit calculations
  • Density measurements
  • Circuit calculations (parallel resistors)
Limitations:
  • Only works with positive numbers
  • Gives more weight to smaller values

When to Use Each Mean

Use Arithmetic Mean when you need a simple average and all values should have equal weight

Use Geometric Mean when dealing with growth rates, returns, or multiplicative changes

Use Harmonic Mean when working with rates, speeds, or other measures where using reciprocals makes sense

Related Links

Mean, Median, Mode Calculator

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Harmonic Mean Calculator

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Five Number Summary Calculator

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Line Chart

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