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Log transformations


Whereas ΣX is the sum of the observations in a list (X1 + X2 + X3, ... + Xn),
ΠX is the product of the observations in a list (X1 × X2 × X3, ... × Xn).

So, for a geometric mean, instead of dividing by the number of observations (n), you have to take the nth root.

In other words: n√ΠX instead of (ΣX)/n

So, if you only had two observations, 6 and 24, n = 2
Their arithmetic mean = (6+24)/2 = 30/2 = 15
Their geometric mean = √(6×24) = √144 = √(12×12) = 12

If you had three observations, 2, 6 and 4, n=3
Their arithmetic mean = (2+6+4)/3 = 12/3 = 4
Their geometric mean = 3√(2×6×4) = 3√48 = 3√(4×4×4) = 4

An alternative, and more popular, way of doing the same thing, is to log 'transform' the data, find their mean, then 'detransform' the result to give the geometric mean.

You may remember that

X = 0.0010.010.1110100100010,000
log X = -3-2-101234

Therefore, for 0.001 × 100 × 10,000 × 1000,
log 0.001 = -3
log 100 = 2
log 10,000 = 4
So their sum (-3 + 2 + 4) = 3

Taking the 'antilog'

log X= -3-2-10123456
antilog X= 0.0010.010.1110100100010,000100,0001,000,000

The antilog of 6 = 1,000,000

To find the geometric mean of 10,000, 0.001, 100, 1000:
We add up their logs (4 + -3 + 2 + 3) = 6
Divide by the number of observations (6 / 4) = 2
And take the antilog (antilog of 2) = 100

So the geometric mean of 10,000, 0.001, 100, 1000 = 100
Which is mathematically identical to 4√(10,000 × 0.001 × 100 × 1000)
But can be much easier, and more convenient, to handle.