GEOMETRIC MEAN Lecture 06

 

GEOMETRIC MEAN

Lecture 06

Introduction

The Greek philosopher Pythagoras first conceptualized the geometric mean, which applied in the following situations. 

i. To determine the mean data expressed in percentage.

ii. To determine the average of the series that are not independent of each other.

iii. To determine the average of data expressed in ratio.

iv. To find the average of highly skewed data.

Definition

The nth root of the product of n values.

Let X1, X2,..., Xn be the values of a data set. Then the geometric mean is defined as:

This formula is useful when the number of observations in given data is small. When a set of data contains a large number of observations, then we need an alternative way for computing the geometric mean. The alternative of computing the geometric mean is given as under:

Taking the advantages of two properties of the logarithm:

When frequencies are given, then the geometric mean can be obtained as:

Note: The geometric mean of two positive numbers is always less or equal to their arithmetic mean.

Example 3. 13: Compare the geometric and arithmetic means for the following data:

i.   4 and 6

ii.  5 and 5

iii.  - 6 and 8

iv.   0 and 8

Solution:

i. The arithmetic mean of 4 and 6.

The geometric mean of 4 and 6.

The geometric mean is less than the arithmatic mean.

ii. The arithmetic mean of 5 and 5.

The geometric mean of 5 and 5.

In this case, the arithmatic mean & geometric mean are identical. 

iii. The arithmetic mean of -6 and 8.

The geometric mean of -6 and 8.

iv. The arithmetic mean of 0 and 8.

The geometric mean of 0 and 8.

Note: Don’t use a geometric mean if you have any negative or zero values in your data.

Example 3.14: Find the geometric mean of the marks obtained by 5 students in a term exam.

                                    10, 30, 45, 76, 80

Solution:

X	Log X
10	1.000
30	1.477
45	1.653
76	1.881
80	1.903
Total	7.914

Example 3.15: Calculate the geometric mean for the following data.

Values	15	25	35	40	50
Frequency	5	20	40	10	4

Solution:

Example 3.16: Find the geometric mean of the following frequency distribution.

Age	10 - 15	15 - 20	20 – 25	25- 30	30 - 35	35 - 40
No. of person	2	10	17	18	12	9

Solution:

Example 3.17: The profit rate of a commercial firm increases 10% in the first quarter, 12.5% in the second, 11% in the third, and 13% in the fourth quarters, respectively. Find the average increase.

Solution:

Average increase = 1.1155 – 1 = 0.1155

Average increase = 11.55%

MERITS AND DEMERITS OF GEOMETRIC MEAN

Merits of Geometric Mean

i. It is rigorously defined by proper mathematical formula.

ii. It is based on all the observations of the data.

iii. It is suitable for average percentages and ratios.

iv. It is less affected by extreme values.

v. It gives equal weight to all the observations.

Demerits of Geometric mean

i. It is not simple to understand.

ii. It cannot be computed if any value is zero or negative.

iii. It has limited application in the statistical analyses.

 

• Read More: Harmonic Mean