Average Calculator — arithmetic, geometric & harmonic mean

How it works

"Average" usually means the arithmetic mean, but there are three classical (Pythagorean) means, each suited to different data, per Wolfram MathWorld:

• Arithmetic mean — add the values and divide by the count: (Σx) ÷ n. The familiar average.
• Geometric mean — the nth root of the product of the n values. It is the right average for things that multiply, such as growth rates and investment returns.
• Harmonic mean — n divided by the sum of the reciprocals. It is the right average for rates, such as average speed over equal distances.

For any set of positive numbers, the three always order as harmonic ≤ geometric ≤ arithmetic (the AM–GM–HM inequality), with equality only when all the values are the same. The geometric and harmonic means need every value to be positive; the calculator shows "—" for them otherwise. It also reports the median for comparison.

Worked example

For 2, 4, 4, 4, 5, 5, 7, 9 (8 values), the sum is 40, so the arithmetic mean is 40 ÷ 8 = 5. The geometric mean (≈ 4.60) and harmonic mean (≈ 4.20) are slightly smaller, as the inequality predicts. As an example of when the harmonic mean matters: driving one leg at 40 mph and an equal-distance leg at 60 mph gives an average speed of the harmonic mean, 48 mph — not 50.

Frequently asked questions

What is the average of a set of numbers?

Most often it is the arithmetic mean: add all the numbers and divide by how many there are. For 2, 4, 4, 4, 5, 5, 7, 9 that is 40 ÷ 8 = 5.

What is the difference between the mean, geometric mean and harmonic mean?

The arithmetic mean adds and divides; the geometric mean multiplies and takes a root; the harmonic mean averages reciprocals. Use the geometric mean for growth rates and the harmonic mean for rates like speed.

When should I use the geometric mean?

Use it for quantities that compound or multiply — average investment return, population growth, or ratios — where averaging the factors, not the amounts, is what matters.

When should I use the harmonic mean?

Use it for averaging rates over equal weights, most famously average speed across equal distances. Travelling equal distances at 40 and 60 mph averages 48 mph, the harmonic mean.

Why is the arithmetic mean always the largest?

For positive numbers the three means satisfy harmonic ≤ geometric ≤ arithmetic. They are equal only when every value is identical; the more spread out the values, the larger the gaps.

What is the difference between the average and the median?

The average (arithmetic mean) uses every value and is pulled by outliers; the median is the middle value and resists them. The calculator shows both so you can compare.