MEASURES OF CENTRAL TENDENCY AND DISPERSION

Transcription

1 CHAPTER 5 MEASURES OF CENTRAL TENDENCY AND DISPERSION UNIT I: MEASURES OF CENTRAL TENDENCY After readng ths chapter, students wll be able to understand: LEARNING OBJECTIVES To understand dfferent measures of central tendency,.e. Arthmetc Mean, Medan, Mode, Geometrc Mean and Harmonc Mean, and computatonal technques of these measures. To learn comparatve advantages and dsadvantages of these measures and therefore, whch measures to use n whch crcumstance. UNIT OVERVIEW Defnton of Central Tendency Dfferent Measures of Central Tendeny Ideal Measures of Central Tendency Arthmetc Mean Medam Mode Geometrc Mean Harmonc Mean Propertes of AM Weghted AM Propertes of Mode Propertes of GM Weghted GM Propertes of Medan Partton Values Propertes of HM Weghted HM Decles Quartles Percentles JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 47 / 808

2 5. STATISTICS 5.. DEFINITION OF CENTRAL TENDENCY In many a case, lke the dstrbutons of heght, weght, marks, proft, wage and so on, t has been noted that startng wth rather low frequency, the class frequency gradually ncreases tll t reaches ts maxmum somewhere near the central part of the dstrbuton and after whch the class frequency steadly falls to ts mnmum value towards the end. Thus, central tendency may be defned as the tendency of a gven set of observatons to cluster around a sngle central or mddle value and the sngle value that represents the gven set of observatons s descrbed as a measure of central tendency or, locaton, or average. Hence, t s possble to condense a vast mass of data by a sngle representatve value. The computaton of a measure of central tendency plays a very mportant part n many a sphere. A company s recognzed by ts hgh average proft, an educatonal nsttuton s judged on the bass of average marks obtaned by ts students and so on. Furthermore, the central tendency also facltates us n provdng a bass for comparson between dfferent dstrbuton. Followng are the dfferent measures of central tendency: () () () (v) (v) Arthmetc Mean (AM) Medan (Me) Mode (Mo) Geometrc Mean (GM) Harmonc Mean (HM) 5.. CRITERIA FOR AN IDEAL MEASURE OF CENTRAL TENDENCY Followng are the crtera for an deal measure of central tendency: () It should be properly and unambguously defned. () It should be easy to comprehend. () It should be smple to compute. (v) It should be based on all the observatons. (v) It should have certan desrable mathematcal propertes. (v) It should be least affected by the presence of extreme observatons. 5.. ARITHMETIC MEAN For a gven set of observatons, the AM may be defned as the sum of all the observatons dvded by the number of observatons. Thus, f a varable x assumes n values x, x, x,..x n, then the AM of x, to be denoted by X, s gven by, x x x... x X n n = n n x JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 47 / 808

3 MEASURES OF CENTRAL TENDENCY AND DISPERSION 5. X = x..(5..) n In case of a smple frequency dstrbuton relatng to an attrbute, we have f x f x f x... fn xn x f f f... f = X = f f x x N f n..(5..) assumng the observaton x occurs f tmes, =,,,..n and N=f. In case of grouped frequency dstrbuton also we may use formula (5..) wth x as the md value of the -th class nterval, on the assumpton that all the values belongng to the -th class nterval are equal to x. However, n most cases, f the classfcaton s unform, we consder the followng formula for the computaton of AM from grouped frequency dstrbuton: Where, fd x A C N x A d C A = Assumed Mean C = Class Length..(5..) ILLUSTRATIONS: Example 5..: Followng are the daly wages n Rupees of a sample of 9 workers: 58, 6, 48, 5, 70, 5, 60, 84, 75. Compute the mean wage. Soluton: Let x denote the daly wage n rupees. Then as gven, x =58, x =6, x = 48, x =5, x =70 x =5, x =60, x =84 and x = , Applyng (5..) the mean wage s gven by, 9 x = x= 9 ( ) = ` 9 56 = ` 9 = ` JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 47 / 808

4 5.4 STATISTICS Example 5..: Compute the mean weght of a group of BBA students of St. Xaver s College from the followng data: Weght n kgs No. of Students Soluton: Computaton of mean weght of 6 BBA students No. of Table 5.. Weght n kgs. Student (f ) Md-Value (x ) f x () () () (4) = () x () Total 6 Applyng (5..), we get the average weght as x f x N = 6 kgs. = 6.4 kgs. Example 5..: Fnd the AM for the followng dstrbuton: Class Interval Frequency Soluton: We apply formula (.) snce the amount of computaton nvolved n fndng the AM s much more compared to Example 5... Any md value can be taken as A. However, usually A s taken as the mddle most md-value for an odd number of class ntervals and any one of the two mddle most md-values for an even number of class ntervals. The class length s taken as C. JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 474 / 808

5 MEASURES OF CENTRAL TENDENCY AND DISPERSION 5.5 Table 5.. Computaton of AM Class Interval Frequency(f ) Md-Value(x ) x A d c = x f d () () () (4) (5) = ()X(4) (A) Total 08 4 The requred AM s gven by x A fd N C = = = Example 5..4: Gven that the mean heght of a group of students s nches. Fnd the mssng frequences for the followng ncomplete dstrbuton of heght of 00 students. Heght n nches No. of Students Soluton: Let x denote the heght and f and f 4 as the two mssng frequences. JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 475 / 808

6 5.6 STATISTICS Table 5.. Estmaton of mssng frequences CI Frequency Md - Value (x ) (f ) x A d c x 67 () () () (4) (5) = () x (4) f 67 (A) f 4 70 f Total + f + f 4 +f 4 As gven, we have f f4 00 f f4 69 and x fd A C N ( f4 ) f f4 5 f 4 7 On substtutng 7 for f 4 n (), we get f 7 69, f 4 Thus, the mssng frequences would be 4 and 7. Propertes of AM ()..() If all the observatons assumed by a varable are constants, say k, then the AM s also k. For example, f the heght of every student n a group of 0 students s 70 cm, then the mean heght s, of course, 70 cm. f d JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 476 / 808

7 MEASURES OF CENTRAL TENDENCY AND DISPERSION 5.7 () the algebrac sum of devatons of a set of observatons from ther AM s zero.e. for unclassfed data, ( x x ) 0 and for grouped frequency dstrbuton, f ( x x ) 0 For example, f a varable x assumes fve observatons, say 58, 6, 7, 45, 9, then x =46.4. Hence, the devatons of the observatons from the AM.e. ( x x ) are.60, 6.60, 9.40,.40 and 7.40, then ( x x ) ( 9.40) + (.40) + ( 7.40) = 0. }...(5..4) () (v) AM s affected due to a change of orgn and/or scale whch mples that f the orgnal varable x s changed to another varable y by effectng a change of orgn, say a, and scale say b, of x.e. y=a+bx, then the AM of y s gven by y a bx. For example, f t s known that two varables x and y are related by x+y+7=0 and x 5, then the AM of y s gven by =.. 7 x y If there are two groups contanng n and n observatons and x and x as the respectve arthmetc means, then the combned AM s gven by n x n x x n n (5..5) Ths property could be extended to k> groups and we may wrte x n x n.(5..6) =,,...n. Example 5..5: The mean salary for a group of 40 female workers s ` 5,00 per month and that for a group of 60 male workers s ` 6800 per month. What s the combned mean salary? Soluton: As gven n = 40, n = 60, x = ` 5,00 and x salary per month s = ` 6,800 hence, the combned mean n x n x x n n = 40 ` 5, ` 6, = ` 6, MEDIAN PARTITION VALUES As compared to AM, medan s a postonal average whch means that the value of the medan s dependent upon the poston of the gven set of observatons for whch the medan s wanted. Medan, for a gven set of observatons, may be defned as the mddle-most value when the observatons are arranged ether n an ascendng order or a descendng order of magntude. JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 477 / 808

8 5.8 STATISTICS As for example, f the marks of the 7 students are 7, 85, 56, 80, 65, 5 and 68, then n order to fnd the medan mark, we arrange these observatons n the followng ascendng order of magntude: 5, 56, 65, 68, 7, 80, 85. Snce the 4 th term.e. 68 n ths new arrangement s the mddle most value, the medan mark s 68.e. Medan (Me) = 68. As a second example, f the wages of 8 workers, expressed n rupees are 56, 8, 96, 0, 0, 8, 06, 00 then arrangng the wages as before, n an ascendng order of magntude, we get ` 56, ` 8, ` 8, ` 96, ` 00, ` 06, ` 0, ` 0. Snce there are two mddlemost values, namely, ` 96, and ` 00 any value between ` 96 and ` 00 may be, theoretcally, regarded as medan wage. However, to brng unqueness, we take the arthmetc mean of the two mddle-most values, whenever the number of the observatons s an even number. Thus, the medan wage n ths example, would be M ` ` = ` 98 In case of a grouped frequency dstrbuton, we fnd medan from the cumulatve frequency dstrbuton of the varable under consderaton. We may consder the followng formula, whch can be derved from the basc defnton of medan. N N l M l N N u l C (5..7) Where, l = lower class boundary of the medan class.e. the class contanng medan. N = total frequency. N l = less than cumulatve frequency correspondng to l. (Pre medan class) N u = less than cumulatve frequency correspondng to l. (Post medan class) l beng the upper class boundary of the medan class. C = l l = length of the medan class. Example 5..6: Compute the medan for the dstrbuton as gven n Example 5... Soluton: Frst, we fnd the cumulatve frequency dstrbuton whch s exhbted n Table JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 478 / 808

9 MEASURES OF CENTRAL TENDENCY AND DISPERSION 5.9 Table 5..4 Computaton of Medan Class boundary Less than cumulatve frequency (l ) 9 (N l ) (l ) 0(N u ) N 08 We fnd, from the Table 5..4, = 54 les between the two cumulatve frequences 9 and 0.e. 9 < 54 < 0. Thus, we have N l = 9, N u = 0 l = and l = Hence C = =0. Substtutng these values n (5..7), we get, 54 9 M = = = Example 5..7: Fnd the mssng frequency from the followng data, gven that the medan mark s. Mark : No. of students : 5 8? 6 Soluton: Let us denote the mssng frequency by f. Table 5..5 shows the relevant computaton. JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 479 / 808

10 5.0 STATISTICS Table 5..5 (Estmaton of mssng frequency) Mark Less than cumulatve frequency 0(l ) (N l ) 0(l ) +f (N u ) 40 9+f 50 +f Gong through the mark column, we fnd that 0<<0. Hence l =0, l =0 and accordngly N l =, N u =+f. Also the total frequency.e. N s +f. Thus, N N l M l N N u l f 0 0 ( f ) f 6 5 f f 5f 0 f 0 f 0 So, the mssng frequency s 0. C Propertes of medan We cannot treat medan mathematcally, the way we can do wth arthmetc mean. We consder below two mportant features of medan. () If x and y are two varables, to be related by y=a+bx for any two constants a and b, then the medan of y s gven by y me = a + bx me For example, f the relatonshp between x and y s gven by x 5y = 0 and f x me.e. the medan of x s known to be 6. Then x 5y = 0 JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 480 / 808

11 MEASURES OF CENTRAL TENDENCY AND DISPERSION 5. y = x y me = x me y me = y me = () For a set of observatons, the sum of absolute devatons s mnmum when the devatons are taken from the medan. Ths property states that x A s mnmum f we choose A as the medan. PARTITION VALUES OR QUARTILES OR FRACTILES These may be defned as values dvdng a gven set of observatons nto a number of equal parts. When we want to dvde the gven set of observatons nto two equal parts, we consder medan. Smlarly, quartles are values dvdng a gven set of observatons nto four equal parts. So there are three quartles frst quartle or lower quartle denoted by Q, second quartle or medan to be denoted by Q or Me and thrd quartle or upper quartle denoted by Q. Frst quartle s the value for whch one fourth of the observatons are less than or equal to Q and the remanng three fourths observatons are more than or equal to Q. In a smlar manner, we may defne Q and Q. Decles are the values dvdng a gven set of observaton nto ten equal parts. Thus, there are nne decles to be denoted by D, D, D,..D 9. D s the value for whch one-tenth of the gven observatons are less than or equal to D and the remanng nne-tenth observatons are greater than or equal to D when the observatons are arranged n an ascendng order of magntude. Lastly, we talk about the percentles or centles that dvde a gven set of observatons nto 00 equal parts. The ponts of sub-dvsons beng P, P,..P 99. P s the value for whch one hundredth of the observatons are less than or equal to P and the remanng nnety-nne hundredths observatons are greater than or equal to P once the observatons are arranged n an ascendng order of magntude. For unclassfed data, the p th quartle s gven by the (n+)p th value, where n denotes the total number of observatons. p = /4, /4, /4 for Q, Q and Q respectvely. p=/0, /0,.9/0. For D, D,,D 9 respectvely and lastly p=/00, /00,.,99/00 for P, P, P.P 99 respectvely. In case of a grouped frequency dstrbuton, we consder the followng formula for the computaton of quartles. Np N l Q l C (5..8) N N u l The symbols, except p, have ther usual nterpretaton whch we have already dscussed whle computng medan and just lke the unclassfed data, we assgn dfferent values to p dependng on the quartle. JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 48 / 808

12 5. STATISTICS Another way to fnd quartles for a grouped frequency dstrbuton s to draw the ogve (less than type) for the gven dstrbuton. In order to fnd a partcular quartle, we draw a lne parallel to the horzontal axs through the pont Np. We draw perpendcular from the pont of ntersecton of ths parallel lne and the ogve. The x-value of ths perpendcular lne gves us the value of the quartle under dscusson. Example 5..8: Followng are the wages of the labourers: ` 8, ` 56, ` 90, ` 50, ` 0, ` 75, ` 75, ` 80, ` 0, ` 65. Fnd Q, D 6 and P 8. Soluton: Arrangng the wages n an ascendng order, we get ` 50, ` 56, ` 65, ` 75, ` 75, ` 80, ` 8, ` 90, ` 0, ` 0. Hence, we have Q (n ) th 4 value = 0 th value 4 =.75 th value = nd value dfference between the thrd and the nd values. = ` [ (65 56)] = ` 6.75 D 6 = (5 + ) 0 6 th value P 8 = 6.60 th value = 6 th value dfference between the 7 th and the 6 th values. = ` ( ) = ` 8.0 (0 ) 8 00 = 9.0 th value th value = 9 th value dfference between the 0 th and the 9 th values = ` ( ) = ` 0.0 Next, let us consder one problem relatng to the grouped frequency dstrbuton. JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 48 / 808

13 MEASURES OF CENTRAL TENDENCY AND DISPERSION 5. Example 5..9: Followng dstrbuton relates to the dstrbuton of monthly wages of 00 workers. Wages n (`) : less than more than No. of workers : Compute Q, D 7 and P. Soluton: Ths s a typcal example of an open end unequal classfcaton as we fnd the lower class lmt of the frst class nterval and the upper class lmt of the last class nterval are not stated, and theoretcally, they can assume any value between 0 and 500 and 500 to any number respectvely. The deal measure of the central tendency n such a stuaton s medan as the medan or second quartle s based on the ffty percent central values. Denotng the frst LCB and the last UCB by the L and U respectvely, we construct the followng cumulatve frequency dstrbuton: N 00 For Q, Wages n rupees (CB) Table 5..7 Computaton of quartles No. of workers (less than cumulatve frequency) L U 00 snce, 57<75 <84, we take N l = 57, N u =84, l =899.50, l =099.50, c = l l = 00 n the formula (5..8) for computng Q. Therefore, Q = ` = ` Smlarly, for D 7, Thus, 7N 7 00 = 70 whch also les between 57 and D 7 = ` = ` Lastly for P, P = ` [ = ` N = and as 5 < < 8, we have ] JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 48 / 808

14 5.4 STATISTICS 5..5 MODE For a gven set of observatons, mode may be defned as the value that occurs the maxmum number of tmes. Thus, mode s that value whch has the maxmum concentraton of the observatons around t. Ths can also be descrbed as the most common value wth whch, even, a layman may be famlar wth. Thus, f the observatons are 5,, 8, 9, 5 and 6, then Mode (Mo) = 5 as t occurs twce and all the other observatons occur just once. The defnton for mode also leaves scope for more than one mode. Thus sometmes we may come across a dstrbuton havng more than one mode. Such a dstrbuton s known as a mult-modal dstrbuton. B-modal dstrbuton s one havng two modes. Furthermore, t also appears from the defnton that mode s not always defned. As an example, f the marks of 5 students are 50, 60, 5, 40, 56, there s no modal mark as all the observatons occur once.e. the same number of tmes. We may consder the followng formula for computng mode from a grouped frequency dstrbuton: Mode =l + f f f 0 f f 0 c.(5..9) where, l = LCB of the modal class..e. the class contanng mode. f 0 = frequency of the modal class f = frequency of the pre-modal class f = frequency of the post modal class C = class length of the modal class Example 5..0: Compute mode for the dstrbuton as descrbed n Example. 5.. Soluton: The frequency dstrbuton s shown below: Table 5..8 Computaton of mode Class Interval Frequency (f ) (f 0 ) (f ) Gong through the frequency column, we note that the hghest frequency.e. f 0 s 8. Hence, f = 58 and f = 65. Also the modal class.e. the class aganst the hghest frequency s JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 484 / 808

15 MEASURES OF CENTRAL TENDENCY AND DISPERSION 5.5 Thus l = LCB= and c= = 0 Hence, applyng formulas (.9), we get Mo = 4. whch belongs to the modal class. (40 49) When t s dffcult to compute mode from a grouped frequency dstrbuton, we may consder the followng emprcal relatonshp between mean, medan and mode: Mean Mode = (Mean Medan).(5..9A) or Mode = Medan Mean (.9A) holds for a moderately skewed dstrbuton. We also note that f y = a+bx, then y mo =a+bx mo.(5..0) Example 5.: For a moderately skewed dstrbuton of marks n statstcs for a group of 00 students, the mean mark and medan mark were found to be and What s the modal mark? Soluton: Snce n ths case, mean = and medan = 5.40, applyng (5..9A), we get the modal mark as Mode = Medan Mean = = 46. Example 5..: If y = +.50x and mode of x s 5, what s the mode of y? Soluton: By vrtue of (.0), we have y mo = = GEOMETRIC MEAN AND HARMONIC MEAN For a gven set of n postve observatons, the geometrc mean s defned as the n-th root of the product of the observatons. Thus f a varable x assumes n values x, x, x,.., x n, all the values beng postve, then the GM of x s gven by G= (x x x.. x n ) /n... (5..) For a grouped frequency dstrbuton, the GM s gven by f f f f G= (x x x.. x n n ) /N... (5..) Where N = f In connecton wth GM, we may note the followng propertes : JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 485 / 808

16 5.6 STATISTICS () () Logarthm of G for a set of observatons s the AM of the logarthm of the observatons;.e. logg logx (5..) r f all the observatons assumed by a varable are constants, say K > 0, then the GM of the observatons s also K. () GM of the product of two varables s the product of ther GM s.e. f z = xy, then GM of z = (GM of x) (GM of y) (5..4) (v) GM of the rato of two varables s the rato of the GM s of the two varables.e. f z = x/y then GM of z GM of x GM of y (5..5) Example 5..: Fnd the GM of, 6 and. Soluton: As gven x =, x =6, x = and n=. Applyng (5..), we have G= ( 6 ) / = (6 ) / =6. Example 5..4: Fnd the GM for the followng dstrbuton: x : f : Soluton: Accordng to (5..), the GM s gven by G = (x x x x 4 ) f f f f /N 4 = ( ) /0 = ().50 = 4 = 5.66 Harmonc Mean For a gven set of non-zero observatons, harmonc mean s defned as the recprocal of the AM of the recprocals of the observaton. So, f a varable x assumes n non-zero values x, x, x,,x n, then the HM of x s gven by n H= (/x ) (5..6) JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 486 / 808

17 MEASURES OF CENTRAL TENDENCY AND DISPERSION 5.7 For a grouped frequency dstrbuton, we have H= N f x (5..7) Propertes of HM () () If all the observatons taken by a varable are constants, say k, then the HM of the observatons s also k. If there are two groups wth n and n observatons and H and H as respectve HM s than the combned HM s gven by n n H n n H (5..8) Example 5.5: Fnd the HM for 4, 6 and 0. Soluton: Applyng (5..6), we have H Example 5..6: Fnd the HM for the followng data: x: f: Soluton: Usng (5..7), we get H = Relaton between AM, GM, and HM For any set of postve observatons, we have the followng nequalty: JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 487 / 808

18 5.8 STATISTICS AM GM HM.. (5..9) The equalty sgn occurs, as we have already seen, when all the observatons are equal. Example 5..7: compute AM, GM, and HM for the numbers 6, 8,, 6. Soluton: In accordance wth the defnton, we have AM GM = (6 8 6) /4 = ( 8 4 ) /4 = HM The computed values of AM, GM, and HM establsh (5..9). Weghted average When the observatons under consderaton have a herarchcal order of mportance, we take recourse to computng weghted average, whch could be ether weghted AM or weghted GM or weghted HM. Weghted AM = wx w Weghted GM = Ante log w logx w.. (5..0).. (5..) Weghted HM = w w x.. (5..) Example 5..8: Fnd the weghted AM and weghted HM of frst n natural numbers, the weghts beng equal to the squares of the correspondng numbers. Soluton: As gven, x. n w. n Weghted AM = wx w JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 488 / 808

19 MEASURES OF CENTRAL TENDENCY AND DISPERSION n =... n... n =... n n = = n(n +) n(n +)(n +) 6 n(n ) (n ) w Weghted HM = w x = = = =...n n... n... n... n nn n n(n 6 ) n A General revew of the dfferent measures of central tendency After dscussng the dfferent measures of central tendency, now we are n a poston to have a revew of these measures of central tendency so far as the relatve merts and demerts are concerned on the bass of the requstes of an deal measure of central tendency whch we have already mentoned n secton 5... The best measure of central tendency, usually, s the AM. It s rgdly defned, based on all the observatons, easy to comprehend, smple to calculate and amenable to mathematcal propertes. However, AM has one drawback n the sense that t s very much affected by samplng fluctuatons. In case of frequency dstrbuton, mean cannot be advocated for open-end classfcaton. Lke AM, medan s also rgdly defned and easy to comprehend and compute. But medan s not based on all the observaton and does not allow tself to mathematcal treatment. However, medan s not much affected by samplng fluctuaton and t s the most approprate measure of central tendency for an open-end classfcaton. JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 489 / 808

20 5.0 STATISTICS Although mode s the most popular measure of central tendency, there are cases when mode remans undefned. Unlke mean, t has no mathematcal property. Mode s also affected by samplng fluctuatons. GM and HM, lke AM, possess some mathematcal propertes. They are rgdly defned and based on all the observatons. But they are dffcult to comprehend and compute and, as such, have lmted applcatons for the computaton of average rates and ratos and such lke thngs. Example 5..9: Gven two postve numbers a and b, prove that AH=G. Does the result hold for any set of observatons? Soluton: For two postve numbers a and b, we have, a b A And Thus G H a ab b ab a b a b ab AH a b = ab = G Ths result holds for only two postve observatons and not for any set of observatons. Example 5..0: The AM and GM for two observatons are 5 and 4 respectvely. Fnd the two observatons. Soluton: If a and b are two postve observatons then as gven a b 5 a+b = 0..() and ab 4 ab = 6..() (a b) (a b) 4ab = = 6 JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 490 / 808

21 MEASURES OF CENTRAL TENDENCY AND DISPERSION 5. a b = 6 (gnorng the negatve sgn).() Addng () and () We get, a = 6 a = 8 From (), we get b = 0 a = Thus, the two observatons are 8 and. Example 5..: Fnd the mean and medan from the followng data: Marks : less than 0 less than 0 less than 0 No. of Students : 5 Marks : less than 40 less than 50 No. of Students : 7 0 Also compute the mode usng the approxmate relatonshp between mean, medan and mode. Soluton: What we are gven n ths problem s less than cumulatve frequency dstrbuton. We need to convert ths cumulatve frequency dstrbuton to the correspondng frequency dstrbuton and thereby compute the mean and medan. Table 5..9 Computaton of Mean Marks for 0 students Marks No. of Students Md - Value f x Class Interval (f ) (x ) () () () (4)= () () = = = = 45 5 Total JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 49 / 808

22 5. STATISTICS Hence the mean mark s gven by f x x= N 670 = 0 =. Table 5..0 Computaton of Medan Marks Marks (Class Boundary) No.of Students (Less than cumulatve Frequency) N 0 Snce 5 les between and, we have l = 0, N l =, N u = and C = l l = 0 0 = 0 Thus, Medan = Snce Mode = Medan Mean (approxmately), we fnd that Mode = x x..4 Example 5..: Followng are the salares of 0 workers of a frm expressed n thousand rupees: 5, 7,,, 7, 5, 4, 8, 0, 6, 5, 9, 8,,,,,, 5, 4. The frm gave bonus amountng to `,000, `,000, ` 4,000, ` 5,000 and ` 6,000 to the workers belongng to the salary groups,000 5,000, 6,000 0,000 and so on and lastly,000 5,000. Fnd the average bonus pad per employee. JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 49 / 808

23 MEASURES OF CENTRAL TENDENCY AND DISPERSION 5. Soluton: We frst construct frequency dstrbuton of salares pad to the 0 employees. The average bonus pad per employee s gven by fx N Where x represents the amount of bonus pad to the th salary group and f, the number of employees belongng to that group whch would be obtaned on the bass of frequency dstrbuton of salares. Table 5.. Computaton of Average bonus No of workers Bonus n Rupees Salary n thousand ` Tally Mark (f ) x f x (Class Interval) () () () (4) (5) = () (4) TOTAL Hence, the average bonus pad per employee 7000 ( `) 0 (`) = 550 SUMMARY The best measure of central tendency, usually, s the AM. It s rgdly defned, based on all the observatons, easy to comprehend, smple to calculate and amenable to mathematcal propertes. However, AM has one drawback n the sense that t s very much affected by samplng fluctuatons. In case of frequency dstrbuton, mean cannot be advocated for open-end classfcaton. Medan s also rgdly defned and easy to comprehend and compute. But medan s not based on all the observaton and does not allow tself to mathematcal treatment. However, medan s not much affected by samplng fluctuaton and t s the most approprate measure of central tendency for an open-end classfcaton. Mode s the most popular measure of central tendency, there are cases when mode remans undefned. Unlke mean, t has no mathematcal property. Mode s also affected by samplng fluctuatons. Relatonshp between Mean, Medan and Mode Mean Mode = (Mean Medan) Mode = Medan Mean JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 49 / 808

24 5.4 STATISTICS Relaton between AM, GM, and HM AM GM HM GM and HM, lke AM, possess some mathematcal propertes. They are rgdly defned and based on all the observatons. But they are dffcult to comprehend and compute and, as such, have lmted applcatons for the computaton of average rates and ratos and such lke thngs. EXERCISE UNIT-I Set A Wrte down the correct answers. Each queston carres mark.. Measures of central tendency for a gven set of observatons measures (a) The scatterness of the observatons (b) The central locaton of the observatons (c) Both (a) and (b) (d) None of these.. Whle computng the AM from a grouped frequency dstrbuton, we assume that (a) The classes are of equal length (b) The classes have equal frequency (c) All the values of a class are equal to the md-value of that class (d) None of these.. Whch of the followng statements s wrong? (a) Mean s rgdly defned (b) Mean s not affected due to samplng fluctuatons (c) Mean has some mathematcal propertes (d) All these 4. Whch of the followng statements s true? (a) Usually mean s the best measure of central tendency (b) Usually medan s the best measure of central tendency (c) Usually mode s the best measure of central tendency (d) Normally, GM s the best measure of central tendency 5. For open-end classfcaton, whch of the followng s the best measure of central tendency? (a) AM (b) GM (c) Medan (d) Mode 6. The presence of extreme observatons does not affect (a) AM (b) Medan (c) Mode (d) Any of these. 7. In case of an even number of observatons whch of the followng s medan? (a) Any of the two mddle-most value JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 494 / 808

25 MEASURES OF CENTRAL TENDENCY AND DISPERSION 5.5 (b) The smple average of these two mddle values (c) The weghted average of these two mddle values (d) Any of these 8. The most commonly used measure of central tendency s (a) AM (b) Medan (c) Mode (d) Both GM and HM. 9. Whch one of the followng s not unquely defned? (a) Mean (b) Medan (c) Mode (d) All of these measures 0. Whch of the followng measure of the central tendency s dffcult to compute? (a) Mean (b) Medan (c) Mode (d) GM. Whch measure(s) of central tendency s(are) consdered for fndng the average rates? (a) AM (b) GM (c) HM (d) Both (b) and (c). For a moderately skewed dstrbuton, whch of he followng relatonshp holds? (a) Mean Mode = (Mean Medan) (b) Medan Mode = (Mean Medan) (c) Mean Medan = (Mean Mode) (d) Mean Medan = (Medan Mode). Weghted averages are consdered when (a) The data are not classfed (b) The data are put n the form of grouped frequency dstrbuton (c) All the observatons are not of equal mportance (d) Both (a) and (c). 4. Whch of the followng results hold for a set of dstnct postve observatons? (a) AM GM HM (b) HM GM AM (c) AM > GM > HM (d) GM > AM > HM 5. When a frm regsters both profts and losses, whch of the followng measure of central tendency cannot be consdered? (a) AM (b) GM (c) Medan (d) Mode 6. Quartles are the values dvdng a gven set of observatons nto (a) Two equal parts (b) Four equal parts (c) Fve equal parts (d) None of these 7. Quartles can be determned graphcally usng (a) Hstogram (b) Frequency Polygon (c) Ogve (d) Pe chart. 8. Whch of the followng measure(s) possesses (possess) mathematcal propertes? (a) AM (b) GM (c) HM (d) All of these JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 495 / 808

26 5.6 STATISTICS 9. Whch of the followng measure(s) satsfes (satsfy) a lnear relatonshp between two varables? (a) Mean (b) Medan (c) Mode (d) All of these 0. Whch of he followng measures of central tendency s based on only ffty percent of the central values? (a) Mean (b) Medan (c) Mode (d) Both (a) and (b) Set B Wrte down the correct answers. Each queston carres marks.. If there are observatons 5, 0, 5 then the sum of devaton of the observatons from ther AM s (a) 0 (b) 5 (c) 5 (d) None of these.. What s the medan for the followng observatons? 5, 8, 6, 9,, 4. (a) 6 (b) 7 (c) 8 (d) None of these. What s the modal value for the numbers 5, 8, 6, 4, 0, 5, 8, 0? (a) 8 (b) 0 (c) 4 (d) None of these 4. What s the GM for the numbers 8, 4 and 40? (a) 4 (b) (c) 8 5 (d) 0 5. The harmonc mean for the numbers,, 5 s (a).00 (b). (c).90 (d) If the AM and GM for two numbers are 6.50 and 6 respectvely then the two numbers are (a) 6 and 7 (b) 9 and 4 (c) 0 and (d) 8 and If the AM and HM for two numbers are 5 and. respectvely then the GM wll be (a) 6.00 (b) 4.0 (c) 4.05 (d) What s the value of the frst quartle for observatons 5, 8, 0, 0,, 8,, 6? (a) 7 (b) 6 (c).75 (d) 9. The thrd decle for the numbers 5, 0, 0, 5, 8,, 9, s (a) (b) 0.70 (c) (d) If there are two groups contanng 0 and 0 observatons and havng 50 and 60 as arthmetc means, then the combned arthmetc mean s (a) 55 (b) 56 (c) 54 (d) 5. JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 496 / 808

27 MEASURES OF CENTRAL TENDENCY AND DISPERSION 5.7. The average salary of a group of unsklled workers s ` 0,000 and that of a group of sklled workers s ` 5,000. If the combned salary s `,000, then what s the percentage of sklled workers? (a) 40% (b) 50% (c) 60% (d) none of these. If there are two groups wth 75 and 65 as harmonc means and contanng 5 and observaton then the combned HM s gven by (a) 65 (b) 70.6 (c) 70 (d) 7.. What s the HM of,/, /,./n? (a) n (b) n (c) (n +) (d) n(n +) 4. An aeroplane fles from A to B at the rate of 500 km/hour and comes back from B to A at the rate of 700 km/hour. The average speed of the aeroplane s (a) 600 km. per hour (b) 58. km. per hour (c) 00 5 km. per hour (d) 60 km. per hour. 5. If a varable assumes the values,, 5 wth frequences as,, 5, then what s the AM? (a) (b) 5 (c) 4 (d) Two varables x and y are gven by y= x. If the medan of x s 0, what s the medan of y? (a) 0 (b) 40 (c) 7 (d) 5 7. If the relatonshp between two varables u and v are gven by u + v + 7 = 0 and f the AM of u s 0, then the AM of v s (a) 7 (b) 7 (c) 7 (d) If x and y are related by x y 0 = 0 and mode of x s known to be, then the mode of y s (a) 0 (b) (c) (d). 9. If GM of x s 0 and GM of y s 5, then the GM of xy s (a) 50 (b) log 0 log 5 (c) log 50 (d) None of these. 0. If the AM and GM for 0 observatons are both 5, then the value of HM s (a) Less than 5 (b) More than 5 (c) 5 (d) Can not be determned. JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 497 / 808

28 5.8 STATISTICS Set C Wrte down the correct answers. Each queston carres 5 marks.. What s the value of mean and medan for the followng data: Marks: No. of Students: (a) 0 and 8 (b) 9 and 0 (c).68 and 7.94 (d) 4. and.8. The mean and mode for the followng frequency dstrbuton Class nterval: Frequency: are (a) 400 and 90 (b) and 90 (c) and (d) 400 and 94.. The medan and modal profts for the followng data Proft n 000 `: below 5 below 0 below 5 below 0 below 5 below 0 No. of frms: are (a).60 and.50 (b) ` 556 and ` 67 (c) ` 875 and ` 667 (d).50 and Followng s an ncomplete dstrbuton havng modal mark as 44 Marks: No. of Students: 5 8? 5 What would be the mean marks? () 45 () 46 () 47 (v) The data relatng to the daly wage of 0 workers are shown below: ` 50, ` 55, ` 60, ` 58, ` 59, ` 7, ` 65, ` 68, ` 5, ` 50, ` 67, ` 58, ` 6, ` 69, ` 74, ` 6, ` 6, ` 57, ` 6, ` 64. The employer pays bonus amountng to ` 00, ` 00, ` 00, ` 400 and ` 500 to the wage earners n the wage groups ` 50 and not more than ` 55 ` 55 and not more than ` 60 and so on and lastly ` 70 and not more than ` 75, durng the festve month of October. What s the average bonus pad per wage earner? (a) ` 00 (b) ` 50 (c) ` 85 (d) `00 JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 498 / 808

29 MEASURES OF CENTRAL TENDENCY AND DISPERSION The thrd quartle and 65th percentle for the followng data are Profts n 000 `: les than No. of frms: (a) `,500 and ` 9,84 (b) `,000 and ` 8,680 (c) `,600 and ` 9,000 (d) `,50 and ` 9, For the followng ncomplete dstrbuton of marks of 00 pupls, medan mark s known to be. Marks: No. of Students: What s the mean mark? (a) (b) (c).0 (d) The mode of the followng dstrbuton s ` 66. What would be the medan wage? Daly wages (`): No of workers: (a) ` (b) ` (c) ` 6. (d) ` 64.5 ANSWERS Set A. (b). (c). (b) 4. (a) 5. (c) 6. (b) 7. (b) 8. (a) 9. (c) 0. (d). (d). (a). (c) 4. (c) 5. (b) 6. (b) 7. (c) 8. (d) 9. (d) 0. (b) Set B. (a). (b). (b) 4. (c) 5. (c) 6. (b) 7. (d) 8. (c) 9. (b) 0. (c). (a). (c). (c) 4. (b) 5. (a) 6. (c) 7. (c) 8. (b) 9. (a) 0. (c) Set C. (c). (c). (c) 4. (d) 5. (d) 6. (a) 7. (c) 8. (c) JSNR_57089_ICAI_Busness Mathematcs_Logcal Reasonng & Statstce_Text.pdf 499 / 808