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# What is the Geometric Mean?

July 28, 2022 written by Azhar Ejaz

The geometric mean of a variable X is the nth positive root of the product of the x1,x2,x3,……….,xn observations. in symbols, we write

G.M=(x1.x2.x3……,xn)1/n

The geometric mean of series containing n observations is the nth root of the product of values.

G.M=(x1.x2.x3……,xn)1/n

The above formula can also be written by using a logarithm

1. For Ungroup Data

G.M=Antilog (sigma logX/n)

1. For Group Data

G.M=Antilog (sigma f logX/sigma f)

For example:

Find the geometric mean of the observation 2, 4, 8, by using the basic formula

Solution: given observations 2, 4, 8

n=3

Using formula

G.M=(x1,x2,x3,……,xn)1/n

G.M=(2x4x8)1/3

G.M=(64)1/3

G.M=(43)1/3

G.M=4

For example: Find the geometric mean of observations 2, 4, 8 using the logarithmic formula for ungrouping data.

Solution: Given observations 2, 4, 8

Using formula:

G.M=Antilog (sigma logX/n)

G.M=Antilog (1.8062/3)

G.M= Antilog (0.6021)

G.M= 4.00003

G.M=4

For example: – Find the geometric mean for the following group data.

Solution: we proceed as fellows

Using formula:

G.M=Antilog (sigma f logX/sigma f)

G.M=Antilog (141.7369/85)

G.M=Antilog (1.66749)

G.M=46.50% marks

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