# GEOMETRIC MEAN

Geometric mean :

Geometric mean is one of the measures of central tendency which can be defined as follows.

For a given set of n positive observations, the geometric mean is defined as the n-th root of the product of the observations.

Let the  variable "x" assume "n" values  as given below

All the above values are being positive, then the GM of x is given by

For a grouped frequency distribution, the GM is given by

Where, N  =  ∑f

## Properties of geometric-mean

1)  Logarithm of G for a set of observations is the AM of the logarithm of the observations; i.e. logG  =  (1/r)∑logx.

2) If all the observations assumed by a variable are constants, say K > 0, then the GM of the observations is also K.

3)  GM of the product of two variables is the product of their GM‘s i.e. if z  =  xy, then

GM of "z"  =  (GM of "x") x (GM of "y")

4)  GM of the ratio of two variables is the ratio of the GM’s of the two variables i.e. if z  =  x / y, then

GM of "z"  =  (GM of "x") / (GM of "y")

5) Like arithmetic mean, GM also possess some mathematical properties.

6)  It is rigidly defined.

7)  It is based on all the observations.

8)  It is difficult to comprehend.

9)  It is difficult to compute.

10)  It has limited applications for the computation of average rates and ratios and such like things.

## Geometric-mean - Practice problems

Problem 1 :

Find the GM of 2, 4 and 8.

Solution :

For the given data, the formula to find geometric-mean is given by

Fitting the given data in to the above formula, we get

GM  =  (2 x 4 x 8)

GM  =  (2)

GM  =  2²

GM  =  4

Problem 2 :

Find the GM of 3, 6 and 12.

Solution :

For the given data, the formula to find geometric-mean is given by

Fitting the given data in to the above formula, we get

GM  =  (3 x 6 x 12)

GM  =  (6³)

GM  =  6

Problem 3 :

Find the GM for the following distribution:

Solution :

After having gone through the stuff given above, we hope that the students would have understood "Geometric-mean".

Apart from the stuff given on this web page, if you need any other stuff in math, please use our google custom search here.

You can also visit our following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6