Geometric Mean Calculator

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What is the geometric mean?
When should I use the geometric mean?
How to calculate the geometric mean?
Geometric mean formula?
Geometric mean negative numbers
Geometric mean with zero
Example of geometric mean

What is the geometric mean?

The geometric mean is a value that represents the middle of a set of numbers, a measure of central tendency. But instead of using the addition and then division on the arithmetic average, we use the product and the nth root.
There are other ways to show this concept, such as the arithmetic average, median, and a mode.

When should I use the geometric mean?

You may use the geometric mean with data that grow up exponentially:
xi = x0 *(1+r)t
For example, when x0=50, r = 4, (1 + r) = 5:
{50 250 1,250 6,250 31,250 156,250 781,250}

Period (t)AverageMedianGM
532,5503,7502,795
6139,5076,2506,250
Ratio: t6/t54.31.72.2
Ration: t9/t84.532.2

One period changes the average dramatically, from 32,550 to 139,507.
When using median or geometric mean measured, there are big changes between the period, but much smaller.
In this example, the change of the geometrical mean between any two consecutive periods is always 2.2.

How to calculate the geometric mean?

1. Multiply all the numbers.
2. Count how many numbers there are.
3. Divide the addition by the count.

Alternative Method
1. Calculate the average natural log (ln) of the numbers.
2. Calculate the exponential of the average ln.

What is the geometric mean formula?

GM = n√(𝛑xi) = (𝛑xi)1/n = expΣln(xi)/n

Geometric mean negative numbers

You can not calculate the geometric mean with negative numbers!
If you treat the list of numbers that contains negative numbers as a list of percentages, you may transform the list into a positive dataset of ratios. In this case, all the numbers will be positive, and it will be possible to calculate the geometric mean.
1. Transform each number list to proportion. F(x) = 1 + x/100. For example 5 is transformed to 1 + 5/100 = 1.05, and -3 is transformed to 1 - 3/100 = 0.97.
2. Calculate the geometric mean (GM) for the transformed list.
3. Convert the result back to percentages. GM' = 100(GM -1).

Geometric mean with zero

The geometric mean, like average, is a measurement that is calculated based on all the data.
If one of the values is zero, the geometric mean will be zero, In this case the geometric mean looses the information of the other non-zero values.
There many possible solutions to this problem, following part of the solutions:

  1. Replace any zero with one.
  2. Replace any zero with half of the detection limit.
  3. Replace any zero with the detection limit minus one significant digit. For example if the detection limit is 0.023, use 0.0229.
  4. Add one to each value, and then reduce one from the result.

The detection limit is the smallest value which your equipment can measure.

Example of geometric mean

What is the geometric mean of 4 and 9?
√(4 * 9) = √36 = 6
Rectangle example 1. If the dimensions of a rectangle are 4 cm × 9 cm, a square with the dimensions of 6 cm x 6 cm has the same are.
2. The ratio 9 to 6 equals the ratio 6 to 4.

Rectangular prism example √(3 * 4 * 9) = √72 = 4.16
1. If the dimensions of a rectangular prism are 3 cm × 4 cm * 6 cm , a cube with the dimensions of 4.16 cm x 4.16 cm x 4.16 cm has the same volume.
2. The ratio 9 to 6 equals the ratio 6 to 4.