The geometric mean of a series of n positive observations is defined as the nth root of their product.
Calculation of Geometric Mean
(a) Individual series
If there are n observations, X1, X2, ...... Xn, such that Xi > 0 for each i, their geometric mean (GM) is defined as
Average Rate of Growth of Population
The average rate of growth of price, denoted by r in the above section, can also be interpreted as the average rate of growth of population. If P0 denotes the population in the beginning of the period and Pn the population after n years, using Equation (2), we can write the expression for the average rate of change of population per annum as
Similarly, Equation (4), given above, can be used to find the average rate of growth of population when its rates of growth in various years are given.
Remarks: The formulae of price and population changes, considered above, can also be extended to various other situations like growth of money, capital, output, etc.
Example : The population of a country increased from 2,00,000 to 2,40,000 within a period of 10 years. Find the average rate of growth of population per year.
Solution: Let r be the average rate of growth of population per year for the period of 10 years. Let P0 be initial and P10 be the final population for this period. We are given P0 = 2,00,000 and P10 = 2,40,000.
Thus, r = 1.018 - 1 = 0.018.
Hence, the percentage rate of growth = 0.018 ××100 = 1.8% p. a.
Example : The gross national product of a country was Rs 20,000 crores before 5 years. If it is Rs 30,000 crores now, find the annual rate of growth of G.N.P.
Solution: Here P5 = 30,000, P0 = 20,000 and n = 5.
Hence r = 1.084 - 1 = 0.084
Thus, the percentage rate of growth of G.N.P. is 8.4% p.a
Example : Find the average rate of increase of population per decade, which increased by 20% in first, 30% in second and 40% in the third decade.
Solution: Let r denote the average rate of growth of population per decade, then
Hence, the percentage rate of growth of population per decade is 29.7%.
Suitability of Geometric Mean for Averaging Ratios
It will be shown here that the geometric mean is more appropriate than arithmetic mean while averaging ratios. Let there be two values of each of the variables x and y, as given below:
We note that their product is not equal to unity. However, the product of their respective geometric means, i.e., 1/√6 and √6 , is equal to unity. Since it is desirable that a method of average should be independent of the way in which a ratio is expressed, it seems reasonable to regard geometric mean as more appropriate than arithmetic mean while averaging ratios.
Properties of Geometric Mean
Merits, Demerits and Uses of Geometric Mean
Exercise with Hints
Hint: The number of bacteria at midnight is GM of 4 ´ 106 and 9 ´ 106.
Hint: We know that the square of the difference of two numbers is always positive, i.e., (a - b)2 ³0. Make adjustments to get the inequality (a + b)2³4ab and then get the desired result, i.e., AM ³ GM.
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