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Geometric Mean Measures of Central Tendency 8) ~btain'themooe for the following distributions without using the usual formulas: i) x:O-10 10-29 20-30 30-40 40-50 50-60 60-70 f: I 6- 15 20 . 15 6 I ii) x: 48-52 52-56 56-60 60-64 64-68 68-72 72-76 f: 4 8 16 18 15 4 2 (Answers: i - 35, ii - 61.93) 9) Estimate the median when arithmetic mean is 27.9 and mode is 25.2. Give the assumptions, if any. (Answer: 27) 10) What are the modal values of the following distributions? a) Hair Colour : Black Brunette Red Head a~o~,de Frequency : 11 24 6 18 b) BloodGroup : AB 0 A B Frequency : 4 12 35 ,I6 (Answers: a - ~runettib - A) - - Note : These questions and exercises will help you to understand the unit better. Try to write answers for them. But do not ~~:u211tlf~,~ dswers to the University. These are for your practice only. UNIT 13 GEOMETRIC, HARMONIC AND MOVING AVERAGES ; . Structure 13.0 Objectives 1 3.1 Introduction 13.2 . Geometric Mean 13.2.1 Computation 13.2.2 Weighted ~eometridMean 13.2.3 Properties of Geometric Mean 13.2.4 Uses and Limitations 13.3 Harmonic Mean 13.3.1 Computation 13.3.2 Weighted Harmonic Mean \ 13.3.3 Properties of Harmonic Mean 13.3.4 Uses and Limitations 13.4 Harmonic Mean Versus Arithmetic Mean 13.5 Moving Average . 13.5.1 What is Moving Average? 13.5.2 Computation 13.6 Choice of a Suitable Average 13.7 Let Us Sum Up 13.8 Key Words 13.9 Answers to Check Your Progress 13.10 Terminal Questions/Exercises .13.0 OBJECTIVES After studying this unit, you should be able to: @ define and compute geometric mean and harmonic mean mumerate the properties of geometric mean and harmonic mean e apprerl .:r :'.le limitations and uses of geometric mean and harmonic mean .e explain the concept of moving average e use moving average in determining trend of tim.e series make a choice of suitable average in,a given situation. , As you know, the averages can be classified as mathematical averages, positional averages and special averages. You have already studied about arithmetic mean which belongs to the sategory of mathematical averages, median and mode, which belong to the category of averages. In this unit you will study abaut the tgo other mathematical averages viz., Geometric Mean and Harmonic Mean. You will also study special average viz., Moiring'Average, and how tochoose a suitable average amongst all types of averages in a given situation. 13.2 . GEOMETRIC MEAN . In the situations where we deal with that change over a period of time, we may be interested to know the average rate of change. In such cases the simple arithmetic Tean is' nut suitable and we have to resort to the geometric mean. 13.2.1 Computation Like other averages, computation procedure of geometric mean is different for grouped data &d ungrouped data. Now let us study these metll6cFs'. Measures of Central Tendency Ungrouped data If there are two items in the data series, the square root of the product of these two items is the geometric mean. If there are three items, the cube root of the product of three items is their geometric mean. If there are 'n' items in the series, its geometric mean is the nth root of the product of those items. Let us express it symbolically: Geometric Mean = Qx xz ...... x,. where X,, X,, Xnrefer to the 'n' items of the series. For example, we have three numbers 4, 8, and 16, the geometric mean of these three numbers would be: . 0-M= 3- = K2 =8 'Ih.15, geometric mean is an average based on the product of items. When the number of, items is three or more, finding their product and extracting its roots becomes difficult. Therefore, computations can be simplified by the use of logarithm. The procedure is as follows: 1) Obtain the logarithm of the different values of the variable and take their total ); log x. 2) Divide it by 'n' (the number of items) and take the antilogarithm of the value so obtained. That gives the Geometric Mean. Symbolically it can be expressed as follows: 1 log G.M. = nI log ( x I xz ...... xn) ! - log XI + log xz + .... log xn n - Zlog x n Therefore, G.M.= Antilog =log x n For example, geometric mean of four numbers 20,65,83 and 135 will be: G.M. = Amtilog log 20 + log 65 t log 83 + log-- 135 4 = Antilog 1.7908 =61.77 Illustration 1 Compared to the previous year, the overhead expenses went up by 32% in 1987, by 40% in I 1988 and by 50% in 1989. Calculate the average rate of increase in overhead expenses over the three years. Solution The increase in overhead expenses is 32%, 40% and 50% in 1987,1988 and 1989 respectively. This means successively expenses become 132%, 142% and 150% of the previous level. Therefore, at the end of three' years the final level will be 132 140 150 per cent of the original level. 100 x 100 As these figures are multiplicative in nature, their average will be given by geometric mean, 1 x,=132x,=140x3=150andn=3 ! Now G.M. = Antilog ?log n = Antilog log 132 + log 140 t log 150 I = Antilog 6.4428 3 = Anti log 2.1476 Geometric, Harmonk = 140.5 and moving averages On an average overhead expenses become 140.5% of previous year's level. Therefore, average rate of increase in overhead expenses is 40.5% (i.e., 140.5 - 100). Grouped Data You know how to compute geometric mean for ungrouped data. Now we should discuss the . procedure fo~grouped data. As you know, the grouped data can be in the form of either discrete series or continuous series, we have to follow different procedures for these two - types of series. Discrete Series: When the data is grouped data i.e., in the form of a frequency distribution, the geometric mean is computed as follows: G.M. =I- where x,, x,, ...., xrare the different values of the variate x with their respective frequencies f,,f,, ... :..... frandn=f,+f,+ ...... fr=Xf log G.M. ='L (f ilog xi + fi log xi t fr log xr) n + .... = 1 @flogx) n G.M. =Antilog( T;f logx ) Continuous Serie's: The only change in the earlier formula of geometric mean is that you replace 'x' by 'm' which is the mid-value of classes. Zf log m Here G.M. =Antilog ( .) The procedure followed in bqth the formulas is as under: 1) Conyert the given values of variate x or the mid-value (m) in the case of continuous series into logdthms 2) Multiply them with the respective frequencies and get the product Cf log x or Zf log m, rs the case may be. 3) Divide the product by the total frequency T; f = n and take the antilogarithm of the value so obtained. Illustration 2 Find out the geometric mean for the following data : Size Frequency 7.5 - 10.5 5 10.5 - 13.5 9 13.5 - 16.5 19 16.5 - 19.5 23 19.5 - 22.5 7 22.5 - 25.5 4 25.5 - 28.5 , 1 Solution Class interval Mid-Point(m) lo^ m f f.log rn 7.5 - 10.5 9 Q.9542 5 4.7710 10.5 - 13.5 12 1,0797 9 9.71 28 13.5 - 16.5 I5 1.1761 19 22.3459 16.5 - 19.5 18 1.2553 23 id.8719 19.5 - 22.5 21 1.3222 7 9.2554 22.5 - 25.5 24 1.3802 4 5.5208 25.5 - 28.5 27 1.4314 1 1.4314 " Total 68 81.9092 Measures of Central Tendency log G.M. = (F) =1~81.9092 68 = 1.2045 G.M., = Antilog 1.2045 = 16.02 Geometric Mean for Computing Average Rate of Change . More often we are interested in the average rate of change in a variable between any two time periods such as annual rate of increase in population, annhal rate of increase in GNP, , average rate of increas~ Frofit, etc. The methods of computing such rates is similar to that of finding the geometric mean. For a given series assume Pois the value at the beginning of the period and P,, is the value at theend of the period. Now, the average growth rate (r) can be obtained by using the following compound interest formula: , P = Po ( 1 t r)", where 'n' is the time-span - Illustration 3 The population of a country was 300 millions in.1951. It became 520 millions in 1969. Calculate the percentage compund rate of growth per annum. Solution Here Po is 300: Pnis 520 and n is 18. Let 'r' be the growth rate per annum. log (1 t r) = log 520 -log 300 I 18 = Antilog 0.0133 =l.U31 . r =1.031-1 = 0.03 1 1: .: Percentage compound growth rate is 100 x r = 3.1% 13.2.2 Weighted ~eornetricMean bike weighted arithmetic mean, we can also calculate the weighted geometric mean. The , computational procedure is as follows: I I Zw Weighted G.M. = dxy,', x;4 ..... xy I perex,, x,, ..... x, are the values of the variate and W,, W,, ..... W, are the corresponding I I weights. 1 Taking logarithms, Geometric, Harmonic and moving'averages I Log Weighted G.M. = Wllog XI + W2 log x2 + ...
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