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Geometric and Harmonic Mean Calculator

Get the right average for growth rates, ratios and speeds: the geometric and harmonic means, or the true average of percent changes.

Level: Grades 9–10

Printable worksheet

Geometric and Harmonic Mean Calculator

Inputs
Enter

All must be greater than 0. Separate with commas, spaces or new lines.

Result

Geometric mean

6

Geometric mean 6, harmonic mean 5.142857, arithmetic mean 7.

Harmonic mean
5.142857
Arithmetic mean
7
Count
3
Product
216

See the step-by-step solution ↓

Step-by-step solution

  1. Multiply the 3 values together, since they work by multiplying

    3×12×6=216

  2. Take the cube root of the product: the one number that, used 3 times, gives the same product

    GM=21613=6

  3. Harmonic mean: add up 1 divided by each value, then divide the count by that total

    HM=313+112+16=5.1428571429

  4. For comparison, the ordinary (arithmetic) mean: the sum divided by the count

    AM=3+12+63=7

  5. For positive values, the ordinary mean is never below the geometric mean, and the harmonic mean is never above it

    7 ≥ 6 ≥ 5.142857

  6. So the geometric mean, which sits between the other two, is

    GM=6

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find the right kind of average for growth rates, ratios and speeds. The ordinary average adds numbers up and divides by how many there are. That's the right choice for amounts that add. But it's wrong for things that multiply, such as growth rates and ratios. It's also wrong for rates averaged over equal distances, such as speeds. The geometric mean and the harmonic mean are the averages built for those jobs.

Enter a list of positive values to see the geometric, harmonic and ordinary (arithmetic) means side by side. Or switch to percent changes. Then you'll get the one steady rate that, repeated each period, ends at the same place as your run of gains and losses. The steps show the product and its root, and the reciprocals (1 divided by each value) behind the harmonic mean. They also show the rule AM ≥ GM ≥ HM with your numbers.

Where it's used

Why this math matters outside the classroom:

  • Investment returns. An investor with yearly returns of +10%, −5%, +20% and +2% finds a compound average of about 6.35% a year, not the simple 6.75%.
  • Average speed. A cyclist riding equal distances at 40 km/h and 60 km/h averages the harmonic mean, 48 km/h, not 50.
  • Equal monthly buys. Someone investing $100 a month at share prices of $20 and $25 pays the harmonic mean, about $22.22 a share on average.
  • Population growth. A biologist whose culture grew 1.5-fold, then 2-fold, then 3-fold reports an average factor of ∛9, about 2.08 per day.
  • Aspect ratios. Designers pick 16:9 partly because it is near the geometric mean of the 4:3 and 2.35:1 ratios it had to show.

How to use it

  1. Choose whether you're entering positive values or percent changes.
  2. Type the numbers separated by commas, spaces or new lines. Values must be above 0; percent changes must be above −100.
  3. Read the geometric mean (or the average change per period). Below it are the harmonic and arithmetic means, or the total change.
  4. Open the steps to see the product, the root and the reciprocals.

What the inputs mean

Enter
Use percent changes for things like yearly returns or monthly growth. Enter 10 for a 10% rise and -5 for a 5% fall. The calculator turns them into growth factors (what you multiply by), 1.10 and 0.95, before averaging.
Values
Zero and negative numbers have no geometric or harmonic mean in the usual sense, so they're rejected. If your data are changes that can be negative, use the percent-change option.
Percent change each period
Enter 10 for +10% and -5 for −5%. Each must be above −100.

Formula and method

geometric mean GM = (x₁ × x₂ × … × xₙ)^(1/n) = e^((ln x₁ + … + ln xₙ) ÷ n)
harmonic mean HM = n ÷ (1/x₁ + 1/x₂ + … + 1/xₙ)
arithmetic mean AM = (x₁ + x₂ + … + xₙ) ÷ n
AM ≥ GM ≥ HM, with equality only when every value is the same
average change per period = (GM of the factors 1 + rᵢ/100 − 1) × 100

In the formulas, x₁, x₂ and so on are your values, and n is how many there are. GM, HM and AM are the geometric, harmonic and arithmetic means. rᵢ is each percent change. ln is the natural logarithm, and e is the number it's based on.

The geometric mean is the one number that, multiplied by itself n times, gives the same product as your data. That makes it the natural average for things that multiply. A value that grows by factors of 1.10, 0.95, 1.20 and 1.02 ends up exactly where it would with four equal factors of their geometric mean. For long lists, the calculator averages logarithms instead of multiplying everything out. That gives the same answer without the numbers getting too big for the computer.

The harmonic mean is 1 divided by the average of 1-over-each-value. It's the right average for rates over equal amounts of the "per" part, such as speeds over equal distances or prices over equal spending. Drive one mile at 40 mph and another at 60 mph, and you average 48 mph, the harmonic mean. That's because the slower mile takes longer.

For any positive data, the three means always line up as AM ≥ GM ≥ HM. The gap between them grows the more spread out the data is. That's why a bumpy investment's simple average return overstates its real growth.

Worked example

An investment changes by +10%, −5%, +20% and +2% over four years. What steady yearly rate would have produced the same result?

  1. Growth factors: 1.10, 0.95, 1.20 and 1.02.
  2. Multiply them: 1.10 × 0.95 × 1.20 × 1.02 = 1.27908, a total gain of 27.908%.
  3. Take the fourth root: 1.27908^(1/4) = 1.063468.
  4. Subtract 1 and turn it into a percent: 6.3468% per year.
  5. Compare with the simple average of the four percents, (10 − 5 + 20 + 2) ÷ 4 = 6.75%. That overstates the growth.

Result: The investment grew at an average of about 6.3468% a year. Compounding 6.3468% for four years gives the same 27.908% total that the actual changes did.

Understanding the result

Use the geometric mean whenever the numbers really multiply together. Examples are growth rates, inflation, interest, population change, and ratios such as one price compared with another. Use the harmonic mean for averaging rates over equal amounts of the "per" part. The ordinary average is still right for amounts that add, like heights or bills.

A big gap between the arithmetic and geometric means is a sign of spread-out or bumpy data. A 50% gain followed by a 50% loss averages 0% the ordinary way. But it's −13.4% per period with the geometric mean, because the money really did fall by 25%.

Assumptions and limitations

  • Values are positive, or percent changes are greater than −100%.
  • Every value carries equal weight.
  • Weighted geometric or harmonic means (where some values count more) aren't calculated here.
  • A data set with a zero has a geometric mean of zero and no harmonic mean. So zeros are rejected, not quietly included.

Frequently asked questions

When should I use the geometric mean instead of the average?

When the values multiply rather than add: yearly returns, growth rates, ratios and index numbers. The geometric mean gives the steady rate that would reach the same end result. The ordinary average doesn't.

What is the harmonic mean used for?

Averaging rates over equal amounts of the "per" part. Examples are average speed over equal distances, or average price per share when you invest equal sums each time. It's also used to combine ratios like price-to-earnings. It gives smaller values more pull.

Why is the geometric mean always smaller than the arithmetic mean?

For positive numbers, this is a known rule called the AM–GM inequality. Spreading values apart raises their sum faster than their product. So the geometric mean is never larger, and the two are equal only when all the values are the same. For 3, 12 and 6, the arithmetic mean is 7 and the geometric mean is 6.

Can the geometric mean handle negative returns?

Yes, as long as you enter them as percent changes. A −5% return becomes a growth factor of 0.95, which is positive. Only a loss of 100% or more (a factor of zero or less) has no geometric average.

How this calculator is checked

Method: Standard definitions of the geometric mean (nth root of the product, computed through logarithms), harmonic mean (reciprocal of the mean reciprocal) and arithmetic mean, and the AM–GM–HM inequality; compound average growth from the geometric mean of growth factors. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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