BUSINESS STATISTICS (PART-9) AVERAGE OR MEASURES OF CENTRAL TENDENCY: THE GEOMETRIC AND HARMONIC MEANS. INTRODUCTION We have so far discussed three measures of cetral tedecy, viz. The Arithmetic Mea, Media & Mode. However, there are two other meas that are used occasioally i busiess & ecoomics. They are the Geometric & Harmoic meas. I algebra Geometric Mea & Harmoic Mea are calculated i case of Geometric & Harmoic Progressios but i Statistics we eed ot bother about the Progressios. The Geometric Mea is a suitable measure of cetral tedecy if variate values are ratios, percetages or proportios. For example GM is used: (i) (i) to fid the average rate of populatio growth or the rate of iterest, ad, i costructio of idex umbers Whereas, the Harmoic Mea is a suitable measure of cetral tedecy whe data pertais to speed, rates ad time. Now we shall first discuss how to compute Geometric Mea i differet situatios. 2. GEOMETRIC MEAN FOR UNGROUPED DATA The geometric mea may be explaied with the simple example. Example If we have three observatios x, x 2, x 3 whose values are respectively 4, 8 ad 6, the the GM of the observatios is qubic root of the product of three values.
That is, GM = 4 x8x6 3 =8 Let us take aother example, Example 2 The populatio of a coutry icreased by 20% i 995 over 994 ad 30% i 996 over 995. The the average growth rate may be obtaied by the GM. GM = 20x30 2 24.49% i.e. GM is the square root of product of the two percetages. Let us cosider oe more example, Example 3 Decadal percetage growth of urba populatio i Idia (excludig Assam ad J & K) from 94 to 200 is give below Years 94 95 96 97 98 99 200 Decadal percet 8.25 9.08 32.09 4.49 25.85 37.9 46.02 icrease Average percet growth rate of urba populatio with the last seve decades may be obtaied by the GM GM = 8.25x9.08x32.09x4.49x25.85x37.9x46.02 7 Takig logarithms, we get GM x, x 7 2,... x 9.9755 7 0.965.2806.5063.679.425.5787.6630. 425 Takig atilog of both sides, we get geometric mea GM = 26.62.
Defiitio & Geeral formula Suppose x, x2,..., x are observatios, all the values beig positive, the geometric mea is give as, GM x, x2,... x That is, GM is -th root of the product of observatios. For large value of, -th root is ot easy to compute. To overcome this difficulty, geometric mea may be computed through logarithm, by usig followig formula GM Ati log i log x i 3. A COMPARATIVE STUDY OF THE PERFORMANCE OF GM Let us cosider the last example i.e., the example 3. We ote that the AM =30.09, Media = 32.09 whereas, Mode = 26.62 of decadal percetage icrease i the populatio. I order to see which of the above averages is most suitable i the followig table, we give the populatios of differet years computed usig these averages ad compare it with the actual populatios obtaied usig actual growth rate. To simply the calculatio, let us assume the populatio i 93 was 0,000. Year Populatio usig Estimate Populatio usig actual growth rate GM AM M e =32.09 =26.62% =30.09% 94 0825 2662 3009 3209 95 2890 6032 6923 7447 96 7026 20299 2205 2279 97 24090 25702 28639 29296 98 3037 32544 37256 38697 99 489 4207 48466 54 200 632 5276 63049 6756
We observe from the above table that the estimated populatios of differet years obtaied through usig the GM are much closer to populatios of correspodig years, obtaied through usig the actual growth rate; tha the oe obtaied usig AM & Media. Thus we see that the AM & Media are ot the proper averages to be used to kow the average growth rate of the populatio. That is GM is the most frequetly used average to obtai average growth rate. 4. HARMONIC MEAN FOR UNGROUPED DATA Example 4 A cyclist pedals from his house to his college at a speed of 0 km/h ad back from the college to his house at 5 km/h Fid the average speed. Let the distace from the house to the college be km. I goig from house to college, the distace ( km) is covered i /0 hours, while i comig from college to house, the distace is covered i /5 hours. Thus a total distace of 2 km is covered i x x 0 5 hours. Total dis ta ce travelled Hece average speed = Total time take 2x 2 km/h x x 0 5 Which is the HM of observatios 0 ad 5. So i this case the average speed is give by the harmoic mea of 0 ad 5 ad ot by the arithmetic mea. Defiitio ad Geeral formula Harmoic mea is the iverse of the arithmetic mea of the reciprocals of the o-zero observatios of a set.
Suppose are variate values, oe of which is zero, the the harmoic mea is give as HM i xi i xi 5. GEOMETRIC AND HARMONIC MEAN FOR GROUPED DATA I case of discrete frequecy distributio, where each of i occurs i times (i=, 2,,), the geometric mea is give by GM f f2 f x, x2,..., x N Where = f i i It ca be computed directly usig the above formula or alteratively, we use logarithm fuctio. That is, N f i log x i N log = f log x f log x... log Therefore, 2 2 i f x Ad, the harmoic mea is These formulae also give GM & HM for cotiuous frequecy distributio where, are take the mid values of the differet classes.
6. RELATION BETWEEN AM, GM AND HM Example 5 The AM, GM ad HM for the umbers 4,8,6 are AM = 9.3 GM = 8 H M =6 So we ote that AM > GM > HM For ay set of positive observatios, we have the followig iequality AM GM HM The equality sig holds whe all the observatios are equal 7. CONCLUSION 7. Requisites for a Ideal Measure of Cetral Tedecy Accordig to Professor Yule, the followig are the characteristics to be satisfied by a ideal measure of cetral tedecy. It should be: (i) rigidly defied, (ii) readily comprehesible ad easy to calculate, (iii) based o all the observatios, (iv) suitable for further mathematical treatmet, (v) affected as little as possible by fluctuatios of samplig.
7.2 Merits ad Demerits of GM & HM Geometric Mea Like arithmetic mea it also depeds o all the observatios. It is affected by the extreme values but ot to the extet of AM. However, there is oe great drawback with it, that it caot be calculated if ay oe or more values are zero or egative. I case a eve umber of observatios are egative, a absurd value of geometric mea will be available from a practical poit of view. Hece, if there is a zero or egative value i the set of variate values it should ot be used. Harmoic Mea It fulfills almost all properties of a good measure of cetral tedecy except whe ay observatio is zero, it caot be calculated. Its mai advatage is that it gives more weightage to smaller observatios ad less weightage to larger observatios. I dealig with busiess problems, harmoic mea is rarely used. A Geeral review of the differet measures of cetral tedecy Measures of cetral tedecy give oe of the very importat characteristics of data. Ay oe of the various measures of cetral tedecy may be chose as the most represetative or typical measure. The arithmetic mea is widely used ad uderstood as a measure of cetral tedecy. The cocepts of weighted arithmetic mea, geometric mea, ad harmoic mea are useful for specific types of applicatios. The media is geerally a more represetative measure for ope-ed distributios ad highly skewed distributios. The mode should be used whe the most demaded or customary value is eeded.