Mean Calculator

Calculate arithmetic, geometric, and harmonic means with step-by-step solutions.

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Arithmetic Mean
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Arithmetic Mean --
Geometric Mean --
Harmonic Mean --
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Arithmetic Mean Steps

Geometric Mean Steps

Harmonic Mean Steps

AM ≥ GM ≥ HM (for positive numbers)

Understanding the Different Types of Mean

A mean (or average) is a single number that represents the central tendency of a set of numbers. While the arithmetic mean is the most commonly used, the geometric and harmonic means serve important roles in specific applications where they provide more meaningful results.

Types of Means

Arithmetic Mean (AM)

The sum of all values divided by the count. Most commonly used average.

AM = (x1 + x2 + ... + xn) / n

Geometric Mean (GM)

The nth root of the product of n values. Best for rates and ratios.

GM = (x1 * x2 * ... * xn)^(1/n)

Harmonic Mean (HM)

The reciprocal of the arithmetic mean of reciprocals. Best for rates.

HM = n / (1/x1 + 1/x2 + ... + 1/xn)

The AM-GM-HM Inequality

For any set of positive numbers, the arithmetic mean is always greater than or equal to the geometric mean, which is always greater than or equal to the harmonic mean: AM ≥ GM ≥ HM. Equality holds if and only if all the numbers are equal. This is one of the most important inequalities in mathematics.

When to Use Each Mean

Arithmetic Mean

Use the arithmetic mean for most everyday calculations: test scores, temperatures, heights, or any additive quantities. It works best when values are on the same scale and not heavily skewed.

Geometric Mean

Use the geometric mean when dealing with multiplicative processes: investment returns, population growth rates, or comparing quantities on different scales. It is less affected by extreme values than the arithmetic mean.

Harmonic Mean

Use the harmonic mean when averaging rates or ratios: average speed for a round trip, price-to-earnings ratios, or any situation involving rates. It gives more weight to smaller values, making it appropriate when the data represents rates.

Practical Examples

  • Investment Returns: If an investment grows 10% one year and 20% the next, the geometric mean gives the true average annual growth rate.
  • Average Speed: If you drive 60 km/h for one leg and 40 km/h for the return, the harmonic mean (48 km/h) gives the correct average speed.
  • Grade Point Average: The arithmetic mean is appropriate for averaging test scores or grades.