Geometric mean
The product of all values under a root of degree equal to their count. Used everywhere values multiply rather than add.
How to use this page
- Fill in the fields in the form above — the result recomputes at once, with no page reload.
- Under the result you will find the interpretation scale, if the calculator has one: it says which range your value fell into.
- The formula we use is shown lower on this page — you can check the arithmetic yourself.
- To compute the same for another game or variant, change it in the select field — prices and pools update automatically.
What this page does not do: It does not point to numbers to play and does not predict a draw — every draw is independent of the previous ones.
Further reading
The geometric mean is the n-th root of the product of n numbers. It sounds scarier than it is. For two numbers, 4 and 9, you multiply them to get 36, and the square root of 36 is 6. The arithmetic mean of the same numbers is 6.5. The geometric mean is never greater than the arithmetic mean — equality holds only when all the numbers are identical.
This mean makes sense where quantities multiply. The classic example is rates of return. Assume four years: growth of 5%, growth of 12%, decline of 3%, growth of 8%. You write this as multipliers 1.05, 1.12, 0.97 and 1.08. The product is about 1.2321, and the fourth root of that number is about 1.0537, which is an average annual return of 5.37%. A plain arithmetic mean would show 5.5%, which is more than you actually earned.
Important limitation: all the numbers must be positive. A zero in the set zeroes the entire product, and a negative number means a root of even degree has no solution in real numbers. Write a 3% decline as 0.97, not as -3. An entry that does not meet these conditions will be flagged with a message.
Frequently asked questions
What is the geometric mean of the numbers 4 and 9?
Exactly 6. The product is 36, and the square root of 36 is 6. The arithmetic mean of the same numbers is 6.5.
When should I use the geometric mean instead of the arithmetic mean?
Whenever values multiply: rates of return, growth rates, index indicators. For added quantities, like ratings or temperature readings, the arithmetic mean is the right one.
Why can't I enter zero or a negative number?
Zero zeroes out the entire product, and a negative number means a root of even degree has no real solution. Write a 3 percent decline as the multiplier 0.97.
How do I calculate the average annual return over four years?
Convert each year into a multiplier, for example a 5% gain becomes 1.05. The calculator gives a result from which you subtract 1 and multiply by 100 — for the multipliers 1.05, 1.12, 0.97, 1.08 that comes out to about 5.37% per year.
See also
Arithmetic Mean
Enter numbers separated by commas and calculate the arithmetic mean online. The calculator will give you the s
Combinations, variations and permutations
Calculate the number of combinations, variations and permutations of n elements taken k at a time. Combinatori
Hypergeometric Distribution Calculator
Probability of k successes when drawing without replacement: P(X = k), P(X ≤ k), P(X ≥ k), the full table, mean and variance — with urn, card and lottery examples.