Geometric mean





Geometric mean in math is a type of mean or average, which denotes the central tendency or typical value of a set of numbers.

In other words, it is a measure of central tendency, just like median. It is different from Arithmetic mean, because the numbers are multiplied, and the nth root of resulting product is taken.

If x1, x2, x3,.....xn are the numbers for which we have to find the G.M G, then

G = nx1x2x...xn 

In other words G.M is the nth root of the product of n values in the data.

G.M is used often in comparing different items, when each item has multiple properties that have different numeric ranges. G.M is often useful for highly skewed data. It is also natural for summarizing ratios.

Note:

We should not use G.M if, the data has any negative values.

Examples:

1. Find the G.M of 5,7,3,1.

Solution:

By definition, the G.M is the 4th root of product of 5,7,3 and 1.

G = 45x7x3x1 

   =4105 

   =3.2011


2. Find the geometric mean of 1/2 and 40.

Solution:

By definition, G.M of 1/2 and 40 is the square root of product of 1/2 and 40.

G = √(1/2)x40 

   =√20

   =2√5

   =4.4721



1. Find 5 geometric means between 576 and 9

Solution: Let G1,G2,G3,G4,G5
Let "r" be the common ratio
Here the first term is 576 that is a = 576 and b = 9
G1 = ar, G1 = 576r
G2= ar2,G2= 576r2
G3= ar3,G3= 576r3
G4= ar4,G4= 576r4
G5= ar5,G5= 576r5
G6= 9

Here the last term is G6= 576 r6
9 = 576 r6
9/576 = r6
1/64 = r6
1 6/2 6 = r 6
(1/2)6 = r 6
r = 1/2

G1 = ar = 576(1/2) = 288

G2= ar2 = 576 (1/2)2 = 576 x (1/4)= 144
G3= ar3 = 576 (1/2)3 = 576 (1/8) = 72

G4= ar4 = 576 (1/2)4 = 576 (1/16) = 36

G5= ar5 = 576 (1/2)5 = 576 (1/32) = 18








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