Geometric

Geometric Mean

In business and economic problems, very often we are faced with questions pertaining to percentage of change over time. Neither the mean, the median nor mode is the appropriate average to use in these instances.

Simple G.M. Given n observations x1,x2,…………,xn G.M = (x1x2,…………xn) 1/n When the number of observations is three or more the task of multiplying the number and extracting the root becomes quite difficult. To simplify the calculations logarithms are used. log G.M. = (log x1 + log x2 +…… + log xn)/n log G.M. = ∑ log xi / n

Weighted G.M. If however x1,x2,…………,xn have frequencies f1,f2,…….fn respectively. G.M. = (x1f1 x2f2 ,…………xn fn ) 1/N Log G.M. = (f1log x1 + f2log x2 +…… +fn log xn)/N = ∑ fi (logxi )/N

For example 1) G.M of 2,4,8 is = ( 2 x 4 x 8)1/3 = (64) 1/3 = 43/3 = 4 (Ans)

Illustration 1) The annual rates of growth of output of a factory in 5 years are 5.0,7.5,2.5,5.0, and 10 respectively. What is the average rate of growth per annum for the period. Ans) G.M. = ( 5 x 7.5 x 2.5 x 5 x 10) 1/5 = (4687.5)1/5 = 5.4% average growth

Illustration - 1 The annual rates of growth of output of a factory in 5 years are 5.0,7.5,2.5,5.0, and 10 respectively. What is the average rate of growth per annum for the period. Annual rate of growth

Output relatives at the end of the year

Log X

5.0 7.5 2.5 5.0 10.0

105.0 107.5 102.5 105.0 110.0

2.0212 2.0314 2.0107 2.0212 2.0414 ∑ log xi = 10.1259

G.M. = antilog (∑ log xi / n) = antilog ( 10.1259/5) = 105.9 The average rate of growth per annum for the period is 105.9 – 100 = 5.9 %.

Application of G.M. 1) The G.M. is used to find the average per cent increase in sales,production,population or other economic or business data. For example, from 2002 to 2004 prices increased by 5%,10% and 18% respectively. The average annual increase is not 11% as given by the arithmetic average but 10.9% as obtained by the G.M.This average is also useful in measuring the growth of population, because population increases in geometric progression. If the series of numbers are in geometric progression, G.M is appropriate average to use.

2) G.M. is theoretically considered to be the best average in the construction of index number. It makes index numbers satisfy the time reversal test and gives equal weights to equal ratio of change.

Advantages 1) Given the product and the number of observations, G.M. can be calculated, even if the magnitude of each observation is not known. 2) Unlike A.M., G.M. is not affected very much by the presence of extremely large and small observations. 3) It can be treated algebrically.G.M of the composite group can be determined, if the G.M and the number of observations in each of the group are known

Disadvantages 1) It can not be calculated, if any of the observations is zero. 2) It is difficult to calculate. The calculation of G.M needs knowledge of logarithms. 3) Like A.M. , G.M. may not be an actual value of variable.

Harmonic Mean Harmonic mean of a set of observations is the reciprocal of the arithmetic mean of their reciprocals. Like G.M, H.M. is defined only when no observation is zero.

Formula n n Simple H.M. = ---------------------------------- = --------1/x1 + 1/x2 + ……. + 1/xn ∑ 1/xi

N

N

Weighted H.M = ----------------------------- = ---------f1/x1 + f2/x2 + ……. + fn/xn ∑ fn/xi

Example (i) Calculate H.M. of numbers 10,20,25,40 and 50. Solution:-

X

1/X

10 20 25 40 50

0.100 0.050 0.040 0.025 0.020 ∑ 1/X = 0.235

n H.M. = ------------ = 5 / 0.235 = 21.28 (Ans) ∑ 1/xi

(i) Calculate H.M. from the following frequency distribution. X

f

0 – 10 10 – 20 20 – 30 30 – 40 40 - 50

8 15 20 4 3

Total

50 =N

Variable

X

f

f x 1/x

0 – 10 10 – 20 20 – 30 30 – 40 40 - 50

5 15 25 35 45

8 15 20 4 3

1.600 1.000 0.800 0.114 0.067

50

∑ fi/xi = 3.581

N H.M. = ------------ = 50/ 3.581 = 13.96 (Ans) ∑ fi/xi

Uses The H.M. is a measure of central tendency for the data expressed in rates such as kms. per hour, rupees per kg, interest rate per annum etc. If the given rates are stated as “x units per y”,then for finding the average rates, use i) H.M. when x’s are given. ii) A.M. when y’s are given.

For example i)

A person goes from A to B on cycle at 20 m.p.h and returns at 24 m.p.h.Find his average speed. Ans) Say distance from A to B is n miles. Here miles are given. Total distance cover Average speed = -------------------------------Total time required

Total distance cover Average speed = -------------------------------Total time required n+n 2n = ----------------------- = -----------------------n/20 + n/24 n( 1/20 + 1/24) 2 = ----------------------= 21.82 m.p.h ( 1/20 + 1/24)

Important Note In order to find the average speed shown in miles per hour(m.p.h) i) Use H.M. when miles( distance covered) are given. ii) Use A.M. hours( time of journey) are given. The given miles or hours, as the case should be used as weights when finding average.

For example a)

A man travelled 12 hours at 4 m.p.h. and again 10 miles at 5 m.p.h.What was the average speed?

Ans) Total distance covered Average speed = -------------------------------Total time required 12 x 4 + 10 x 5 98 = --------------------------= ----------= 4.45 m.p.h 12 + 10 22 Σ fixi Weighted A.M.

= ---------------------Σ fi = N

b) A man travelled 12 miles at 4 m.p.h and again 10 miles at 5 m.p.h . What was the average speed? Ans) Total distance covered Average speed = ------------------------------------Total time required 12 + 10 22 = ----------------------------------------= ----------= 4.40 m.p.h 12/4 + 10/5 5 Σ fi N Weighted H.M. = ----------------- = ---------------∑ fi/xi

∑ fi/xi

Advantages 1) H.M. gives the largest weight to the smallest item and the smallest weight to the largest item. Hence, when there are a few extremely large or small values, H.M. is preferable to A.M., as an average.

Disadvantages 1) It can not be calculated, if any of the observations is zero. 2) It is difficult to calculate. 3) Like A.M. , H.M. may not be an actual value of variable.