GEOMETRIC MEAN

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

As in case of arithmetic mean, the sum of deviations of logarithms of values from the log GM is equal to zero. This property implies that the product of the ratios of GM to each observation, that is less than it, is equal to the product the ratios of each observation to GM that is greater than it. For example, if the observations are 5, 25, 125 and 625, their GM = 55.9. The above property implies that 55.9/5 x 55.9/25 = 125 /55.9 x 625/55.9Similar to the arithmetic mean, where the sum of observations remains unaltered if each observation is replaced by their AM, the product of observations remains unaltered if each observation is replaced by their GM.

Merits, Demerits and Uses of Geometric Mean

Merits

It is based on all the items of the data..It is rigidly defined. It means different investigators will find the same result from the given set of data.It is a relative measure and given less importance to large items and more to small ones unlike the arithmetic mean.Geometric mean is useful in ratios and percentages and in determining rates of increase or decrease.It is capable of algebraic treatment. It mean we can find out the combined geometric mean of two or more series.

Demerits

It is not easily understood and therefore is not widely used.It is difficult to compute as it involves the knowledge of ratios, roots, logs and antilog.It becomes indeterminate in case any value in the given series happens to be zero or negative.With open-end class intervals of the data, geometric mean cannot be calculated.Geometric mean may not correspond to any value of the given data.

Uses

It is most suitable for averaging ratios and exponential rates of changes.It is used in the construction of index numbers.It is often used to study certain social or economic phenomena.

Exercise with Hints

A sum of money was invested for 4 years. The respective rates of interest per annum were 4%, 5%, 6% and 8%. Determine the average rate of interest p.a.

The number of bacteria in a certain culture was found to be 4 ´ 106 at noon of one day. At noon of the next day, the number was 9 ´ 106. If the number increased at a constant rate per hour, how many bacteria were there at the intervening midnight?

Hint: The number of bacteria at midnight is GM of 4 ´ 106 and 9 ´ 106.

If the price of a commodity doubles in a period of 4 years, what is the average percentage increase per year?

Hint

A machine is assumed to depreciate by 40% in value in the first year, by 25% in second year and by 10% p.a. for the next three years, each percentage being calculated on the diminishing value. Find the percentage depreciation p.a. for the entire period.

Hint

A certain store made profits of Rs 5,000, Rs 10,000 and Rs 80,000 in 1965, 1966 and 1967 respectively. Determine the average rate of growth of its profits.

Hint

An economy grows at the rate of 2% in the first year, 2.5% in the second, 3% in the third, 4% in the fourth ...... and 10% in the tenth year. What is the average rate of growth of the economy?

Hint

The export of a commodity increased by 30% in 1988, decreased by 22% in 1989 and then increased by 45% in the following year. The increase/decrease, in each year, being measured in comparison to its previous year. Calculate the average rate of change of the exports per annum.

Hint

Show that the arithmetic mean of two positive numbers a and b is at least as large as their geometric mean.

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.

If population has doubled itself in 20 years, is it correct to say that the rate of growth has been 5% per annum?

Hint

The weighted geometric mean of 5 numbers 10, 15, 25, 12 and 20 is 17.15. If the weights of the first four numbers are 2, 3, 5, and 2 respectively, find weight of the fifth number. Hint: Let x be the weight of the 5th number, then