Contents
- ๐ What is this?
- ๐จ๐ปโ๐ป How to use it?
- ๐ฐ Examples
- ๐ Nuances
- ๐ค Frequently asked questions
- ๐ Related materials
A fair centre for multiplication
The geometric mean gives one representative factor for a list of positive numbers. It is useful when values combine by multiplication, such as successive growth factors, ratios, indexes, or proportional measurements.
Formula and calculation method
For positive values, multiply them and take the root whose degree equals the number of values:
The calculator uses the equivalent logarithmic form. It adds logarithms and then applies the exponential, so very large and very small factors can be combined without first forming an overflowing product.
Worked examples
Values 2 and 8. Their product is 16, so the square root gives a geometric mean of 4.
Values 1, 4 and 16. The product is 64 and its cube root is 4. The arithmetic mean is 7, which shows how the two averages answer different questions.
Growth multipliers 1.10 and 0.90. Their geometric mean is about 0.994987. Repeating that factor twice has the same total effect as multiplying by 1.10 and then 0.90.
Input and interpretation
Enter only strictly positive values. A zero makes the whole product zero, while negative values require a separate convention that depends on how many values there are. This calculator deliberately uses the standard positive-data definition.
Separate values with spaces, new lines, semicolons, or vertical bars. Decimal points, decimal commas, scientific notation, and a comma followed by whitespace are accepted. The precision setting affects display only.
Questions about the geometric mean
When should I use the geometric mean?
Use it when equal multiplicative factors, ratios, or proportional changes are more meaningful than equal additions.
Why must every value be positive?
The real-valued logarithmic definition used here is valid only for numbers greater than zero. Zero and negative inputs are therefore rejected.
Is it always smaller than the arithmetic mean?
For positive values it cannot exceed the arithmetic mean, and the two are equal only when all values are the same.
How do I enter percentage changes?
Convert each change to a multiplier first: plus 10% becomes 1.10 and minus 10% becomes 0.90. Convert the resulting multiplier back to a percentage if needed.
Does rounding change the calculation?
No. It changes only the displayed digits; the calculation and copied raw values keep the available precision.
Similar calculators
You may find the following calculators on the same topic useful:
- Harmonic mean calculator. Enter positive rates or ratios to see their harmonic mean.
- Arrangements and permutations calculator. Choose how many distinct elements to arrange and get every digit of the answer.
- Improper fraction to mixed number calculator. Enter an integer numerator and positive denominator to see the whole units and simplified remainder.
- Fraction comparison calculator. Enter two integer numerators and positive denominators to compare their values.
- Prime factorization calculator. Enter a nonzero whole number to see its prime factors and their exponents.
- Simple Calculator. Simply calculate whatever you need.
- Pi Calculator. Find out the value of Pi with the desired number of decimal places.
- Calculator for the Number 'e'. Calculate the desired number of decimal places in the number 'e' (Euler's number or Napier's constant) using the 'e' number calculator.
- Quadratic Equations Calculator. Apply methods to solve quadratic equations using formulas, discriminant, and Vieta's theorem.
- Percentage Less Calculator. Calculate online how much one number is less than another in percentage.
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