log X log X

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1 LECTURE # 26 Geometric mea, Harmoic mea & relatioship betwee them Geometric mea Harmoic mea Relatio betwee the arithmetic, geometric ad harmoic meas Some other measures of cetral tedecy GEOMETRIC MEAN: The geometric mea, G, of a set of positive values, 2,, is defied as the positive th root of their product. G... (Where i > 0) Whe is large, the computatio of the geometric mea becomes laborious as we have to extract the th root of the product of all the values. The arithmetic is simplified by the use of logarithms. Takig logarithms to the base 0, we get Hece log G 2 G 2 [ log log... log ] log ati log log Example: Fid the geometric mea of umbers: 45, 32, 37, 46, 3, 36, 4, 48, 36. Solutio: We eed to compute the umerical value of But, obviously, it is a bit cumbersome to fid the ith root of a quatity. So we make use of logarithms, as show below: log log log G Hece G ati log

2 The above example pertaied to the computatio of the geometric mea i case of raw data. Next, we cosider the computatio of the geometric mea i the case of grouped data. GEOMETRICMEAN FOR GROUPED DATA: I case of a frequecy distributio havig k classes with midpoits, 2,,k ad the correspodig frequecies f, f2,, fk (such that fi ), the geometric mea is give by G f 2... k f2 fk Each value of thus has to be multiplied by itself f times, ad the whole procedure becomes quite a formidable task! I terms of logarithms, the formula becomes log G 2 2 f log Hece [ f log f log... log ] f log G atilog f k k Obviously, the above formula is much easier to hadle. Let us ow apply it to a example. Goig back to the example of the EPA mileage ratigs, we have: Mileage Ratig No. of Cars Class-mark (midpoit) log f log

3 G atilog atilog This meas that, if we use the geometric mea to measures the cetral tedecy of this data set, the the cetral value of the mileage of those 30 cars comes out to be miles per gallo. The questio is, Whe should we use the geometric mea? The aswer to this questio is that whe relative chages i some variable quatity are averaged, we prefer the geometric mea. EAMPLE: Suppose it is discovered that a firm s turover has icreased durig 4 years by the followig amouts: Year Turover Percetage Compared With Year Earlier 58 2, , , , , The yearly icrease is show i a percetage form i the right-had colum i.e. the turover of 5 is 25 percet of the turover of 58, the turover of 60 is 0 percet of the turover of 5, ad so o. The firm s ower may be iterested i kowig his average rate of turover growth. If the arithmetic mea is adopted he fids his aswer to be: Arithmetic Mea: i.e. we are cocludig that the turover for ay year is 53.75% of the turover for the previous year. I other words, the turover i each of the years cosidered appears to be per cet higher tha i the previous year. If this percetage is used to calculate the turover from 58 to 62 iclusive, we obtai: 53.75% of 2,000 3, % of 3,075 4, % of 4,728 7, % of 7,26,76 Whereas the actual turover figures were Year Turover 58 2, , , , ,500

4 It seems that both the idividual figures ad, more importat, the total at the ed of the period, are icorrect. Usig the arithmetic mea has exaggerated the average aual rate of icrease i the turover of this firm. Obviously, we would like to rectify this false impressio. The geometric mea eables us to do so: Geometric mea of the turover figures: 4 ( ) % Now, if we utilize this particular value to obtai the idividual turover figures, we fid that: 5.37% of 2,000 3, % of 3,027 4, % of 4,583 6, % of 6,37 0,500 So that the turover figure of 62 is exactly the same as what we had i the origial data. Iterpretatio: If the turover of this compay were to icrease aually at a costat rate, the the aual icrease would have bee 5.37 percet.(o the average, each year s turover is 5.37% higher tha that i the previous year.) The above example clearly idicates the sigificace of the geometric mea i a situatio whe relative chages i a variable quatity are to be averaged. But we should bear i mid that such situatios are ot ecoutered too ofte, ad that the occasio to calculate the geometric mea arises less frequetly tha the arithmetic mea.(the most frequetly used measure of cetral tedecy is the arithmetic mea.) The ext measure of cetral tedecy that we will discuss is the harmoic mea. HARMONIC MEAN; The harmoic mea is defied as the reciprocal of the arithmetic mea of the reciprocals of the values. HARMONIC MEAN I case of raw data: H. M. I case of grouped data (data grouped ito a frequecy distributio): H. M. f (where represets the midpoits of the various classes).

5 EAMPLE: Suppose a car travels 00 miles with 0 stops, each stop after a iterval of 0 miles. Suppose that the speeds at which the car travels these 0 itervals are 30, 35, 40, 40, 45, 40, 50, 55, 55 ad 30 miles per hours respectively. What is the average speed with which the car traveled the total distace of 00 miles? If we fid the arithmetic mea of the 0 speeds, we obtai: Arithmetic mea of the 0 speeds: miles per hour But, if we study the problem carefully, we fid that the above aswer is icorrect. By defiitio, the average speed is the speed with which the car would have traveled the 00 mile distace if it had maitaied a costat speed throughout the 0 itervals of 0 miles each. Average speed Total distace travelled Total time take Now, total distace traveled 00 miles. Total time take will be computed as show below: Iterval Distace Speed Distace Distace Time Time Speed 0 miles 30 mph 0/ hrs 2 0 miles 35 mph 0/ hrs 3 0 miles 40 mph 0/ hrs 4 0 miles 40 mph 0/ hrs 5 0 miles 45 mph 0/ hrs 6 0 miles 40 mph 0/ hrs 7 0 miles 50 mph 0/ hrs 8 0 miles 55 mph 0/ hrs 0 miles 55 mph 0/ hrs 0 0 miles 30 mph 0/ hrs Total 00 miles Total Time hrs Hece Average speed mph 2.488

6 which is ot the same as 42 miles per hour. Let us ow try the harmoic mea to fid the average speed of the car. H. M. where is the o. of term. We have: / 30 / / / / / / / / / / H.M mph Hece it is clear that the harmoic mea gives the totally correct result. The key questio is, Whe should we compute the harmoic mea of a data set? The aswer to this questio will be easy to uderstad if we cosider the followig rules: RULES. Whe values are give as x per y where x is costat ad y is variable, the Harmoic Mea is the appropriate average to use. 2. Whe values are give as x per y where y is costat ad x is variable, the Arithmetic Mea is the appropriate average to use. 3. Whe relative chages i some variable quatity are to be averaged, the geometric mea is the appropriate average to use. We have already discussed the geometric ad the harmoic meas. Let us ow try to uderstad Rule No. with the help of a example: EAMPLE: If 0 studets have obtaied the followig marks (i a test) out of : 3,,,, 6, 5,, 7, 2, The the average marks (by the formula of the arithmetic mea) are:

7 This is equivalet to (i.e. the average marks of this group of studets are out of ). I the above example, the poit to be oted was that all the marks were expressible as x per y where the deomiator y was costat i.e. equal to, ad hece, it was appropriate to compute the arithmetic mea. Let us ow cosider a mathematical relatioship exists betwee these three measures of cetral tedecy. RELATION BETWEEN ARITHMETIC, GEOMETRIC AND HARMONIC MEANS: Arithmetic Mea > Geometric Mea >Harmoic Mea We have cosidered the five most well-kow measures of cetral tedecy i.e. arithmetic mea, media, mode, geometric mea ad harmoic mea. It is iterestig to ote that there are some other measures of cetral tedecy as well. Two of these are the mid rage, ad the mid quartile rage. Let us cosider these oe by oe: MID-RANGE: If there are observatios with x0 ad xm as their smallest ad largest observatios respectively, the their mid-rage is defied as x mid rage 0 x 2 m It is obvious that if we add the smallest value with the largest, ad divide by 2, we will get a value which is more or less i the middle of the data-set. MID-QUARTILE RANGE: If x, x2 x are observatios with Q ad Q3 as their first ad third quartiles respectively, the their mid-quartile rage is defied as Q Q mid quartile rage 3 2

8 Similar to the case of the mid-rage, if we take the arithmetic mea of the upper ad lower quartiles, we will obtai a value that is somewhere i the middle of the data-set. The mid-quartile rage is also kow as the mid-hige. Let us ow revise briefly the core cocept of cetral tedecy: Masses of data are usually expressed i the form of frequecy tables so that it becomes easy to comprehed the data. Usually, a statisticia would like to go a step ahead ad to compute a umber that will represet the data i some defiite way. Ay such sigle umber that represets a whole set of data is called Average. Techically speakig, there are may kids of averages (i.e. there are several ways to compute them). These quatities that represet the data-set are called measures of cetral tedecy.