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Harmonic Mean Calculator

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Harmonic Mean

Definition

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of a given set of numbers. It's particularly useful for rates and speeds, giving appropriate weights to each value.

Formula

H=n∑i=1n1xi=n1x1+1x2+...+1xnH = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}} = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n}}

Example: Average Speed

If a vehicle travels at:
- 60 km/h for 100 km
- 40 km/h for 100 km
- 30 km/h for 100 km

The harmonic mean speed is:

3160+140+130=40 km/h\frac{3}{\frac{1}{60} + \frac{1}{40} + \frac{1}{30}} = 40 \text{ km/h}

This is the correct average speed for the entire journey, accounting for the time spent at each speed.

Comparison of Different Means

Try Arithmetic MeanTry Geometric 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

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(Current)

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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Geometric Mean Calculator

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Line Chart

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