MEASURES OF DISPERSION

MEASURES OF DISPERSION Measure of Cetral Tedecy: Measures of Cetral Tedecy ad Dsperso ) Mathematcal Average: a) Arthmetc mea (A.M.) b) Geometrc mea (G.M.) c) Harmoc mea (H.M.) ) Averages of Posto: a) Meda b) Mode Arthmetc Mea: () Smple arthmetc mea dvdual seres by () Drect method: If the seres ths case be.e., Sum of the seres = Number of terms + + +... + = 3 = =...,,, 3, ; the the arthmetc mea s gve () Smple arthmetc mea cotuous seres If the terms of the gve seres be,,..., ad the correspodg frequeces be f, f,... f, the the arthmetc mea s gve by,

f + f +... + f f = = = f + f +... + f f Cotuous Seres: If the seres s cotuous the s are to be replaced by m s where m s are the md values of the class tervals. =. Mea of the Composte Seres: If,( =,..., k) are the meas of k-compoet seres of szes,( =,,..., k) respectvely, the the mea of the composte seres obtaed o combg the compoet seres s gve by the formula Geometrc Mea: If..., geometrc mea (G.M.) s gve by = + +... + k + +... + k k = =. =,, 3, are values of a varate, oe of them beg zero, the G.M. = ( /.. 3... ) I case of frequecy dstrbuto, G.M. of values f, f,..., f s gve by... G.M. (.... ),, of a varate occurrg wth frequecy f / = f f N, where N = f + f +... + f. Cotuous Seres: If the seres s cotuous the s are to be replaced by m s where m s are the md values of the class tervals. Harmoc Mea: The harmoc mea of tems...,,, s defed as H.M. =. + +... + If the frequecy dstrbuto s f, f, f 3,..., f respectvely, the f + f + f3 +... + f H.M. = f f f + +... +.

Meda: The meda s the cetral value of the set of observatos provded all the observatos are arraged the ascedg or descedg orders. It s geerally used, whe effect of etreme tems s to be kept out. () Calculato of meda () Idvdual seres: If the data s raw, arrage ascedg or descedg order. Let be the umber of observatos. If s odd, Meda = value of th + tem. If s eve, Meda = value of th tem + value of + () Dscrete seres: I ths case, we frst fd the cumulatve frequeces of the varables th tem arraged ascedg or descedg order ad the meda s gve by Meda = th + observato, where s the cumulatve frequecy. () For grouped or cotuous dstrbutos: I ths case, followg formula ca be used. (a) For seres ascedg order, Meda = N C l + f Where l = Lower lmt of the meda class f = Frequecy of the meda class N = The sum of all frequeces = The wdth of the meda class C = The cumulatve frequecy of the class precedg to meda class. (b) For seres descedg order Meda = N C u f, where u = upper lmt of the meda class, N = = f. As meda dvdes a dstrbuto to two equal parts, smlarly the quartles, qutles, decles ad percetles dvde the dstrbuto respectvely to 4, 5, 0 ad 00 equal parts. The j th quartle N j C Q j. f s gve by = l + 4 ; j =,, 3 upper quartle. Q s the lower quartle, Q s the meda ad Q 3 s called the

() Lower quartle () Dscrete seres : + Q = sze of 4 th tem () Cotuous seres : N C Q l 4 = + f (3) Upper quartle () Dscrete seres : 3( + ) Q 3 = sze of 4 th tem () Cotuous seres : 3N C Q l 4 3 = + f Mode: The mode or model value of a dstrbuto s that value of the varable for whch the frequecy s mamum. For cotuous seres, mode s calculated as, Mode f f 0 = l + f f0 f Where, l = The lower lmt of the model class f = The frequecy of the model class f 0 = The frequecy of the class precedg the model class f = The frequecy of the class succeedg the model class = The sze of the model class. Emprcal relato : Mea Mode = 3(Mea Meda) Mode = 3 Meda Mea. Measure of dsperso:the degree to whch umercal data ted to spread about a average value s called the dsperso of the data. The four measure of dsperso are () Rage () Mea devato (3) Stadard devato (4) Square devato

() Rage : It s the dfferece betwee the values of etreme tems a seres. Rage = X ma X m The coeffcet of rage (scatter) = ma ma + Rage s ot the measure of cetral tedecy. Rage s wdely used statstcal seres relatg to qualty cotrol producto. Rage s commoly used measures of dsperso case of chages terest rates, echage rate, share prces ad lke statstcal formato. It helps us to determe chages the qualtes of the goods produced factores. m m. Quartle devato or sem ter-quartle rage: It s oe-half of the dfferece betwee the thrd quartle ad frst quartle.e., Q3 Q Q.D. = ad coeffcet of quartle devato the thrd or upper quartle ad Q s the lowest or frst quartle. Q Q 3 =, where Q 3 s Q3 + Q () Mea Devato: The arthmetc average of the devatos (all takg postve) from the mea, meda or mode s kow as mea devato. Mea devato s used for calculatg dsperso of the seres relatg to ecoomc ad socal equaltes. Dsperso the dstrbuto of come ad wealth s measured term of mea devato. () Mea devato from ugrouped data (or dvdual seres) Mea devato M =, where M meas the modulus of the devato of the varate from the mea (mea, meda or mode) ad s the umber of terms. () Mea devato from cotuous seres: Here frst of all we fd the mea from whch devato s to be take. The we fd the devato dm = M of each varate from the mea M so obtaed. Net we multply these devatos by the correspodg frequecy ad fd the product f.dm ad the the sum f dm of these products. Lastly we use the formula, mea devato f M = f dm =, where = Σf.

(3) Stadard Devato: Stadard devato (or S.D.) s the square root of the arthmetc mea of the square of devatos of varous values from ther arthmetc mea ad s geerally deoted by σ read as sgma. It s used statstcal aalyss. () Coeffcet of stadard devato: To compare the dsperso of two frequecy dstrbutos the relatve measure of stadard devato s computed whch s kow as coeffcet of stadard devato ad s gve by Coeffcet of S.D. σ =, where s the A.M. () Stadard devato from dvdual seres σ = ( ) N where, = The arthmetc mea of seres N = The total frequecy. () Stadard devato from cotuous seres σ = f( ) N where, = Arthmetc mea of seres = Md value of the class f = Frequecy of the correspodg N = Σf = The total frequecy Short cut Method: fd N () σ = () fd N σ = d N d N where, d = A = Devato from the assumed mea A f = Frequecy of the tem N = Σf = Sum of frequeces

(4) Square Devato: () Root mea square devato S = N = f ( A), where A s ay arbtrary umber ad S s called mea square devato. () Relato betwee S.D. ad root mea square devato : If σ be the stadard devato ad S be the root mea square devato. The, S + d = σ. Obvously, S wll be least whe d = 0.e., = A Hece, mea square devato ad cosequetly root mea square devato s least, f the devatos are take from the mea. Varace: The square of stadard devato s called the varace. Coeffcet of stadard devato ad varace : The coeffcet of stadard devato s the rato of the S.D. to A.M..e., σ. Coeffcet of varace = coeffcet of S.D. 00 = 00. Varace of the combed seres : If, are the szes,, the meas ad σ,σ the stadard devato of two seres, the = [ ( σ + d ) + ( σ + d )] σ + σ, Where d =, d = ad = + +

Very Short Aswer Questos. Fd the mea devato about the mea for the followg data: ) 38, 70, 48, 40, 4, 55, 63, 46, 54, 44 ) 3, 6, 0, 4, 9, 0 Sol. ) Mea 38 + 70 + 48 + 40 + 4 + 55 + 63+ 46 + 54 + 44 = 0 500 = = 50 0 The absolute values of mea devatos are =, 0,, 0, 8, 5, 3, 4, 4, 6. Mea devato about the Mea = 0 = 0 = + 0 + + 0 + 8 + 5 + 3 + 4 + 4 + 6 0 84 = = 8.4 0 ) Mea () = 6 = 3 + 6 + 0 + 4 + 9 + 0 4 = = = 7 6 6 The absolute values of the devatos are = 4,, 3, 3,, 3 Mea devato about the Mea = 6 = 6 4 + + 3+ 3+ + 3 6 = = =.6666.67 6 6

. Fd the mea devato about the meda for the followg data. ) 3, 7, 6,, 3, 0, 6,, 8,, 7 ) 4, 6, 9, 3, 0, 3, Sol. Gve data the ascedg order :0,,,, 3, 3, 6, 6, 7, 7, 8 Mea (M) of these observatos s 3. The absolute values of devatos are M = 3,,,,0,0,3,3,4, 4,5 Mea devato about Meda = = M 3+ + + + 0 + 0 + 3 + 3+ 4 + 4 + 5 = 7 = =.45 ) 4, 6, 9, 3, 0, 3, Epressg the gve data the ascedg order, we get, 3, 4, 6, 9, 0, 3. Meda (M) of gve data = 6 The absolute values of the devatos are = 4, 3,, 0, 3, 4, 7 Mea Devato about Meda = 7 = M 4 + 3+ + 0 + 3 + 4 + 7 3 = = = 3.9. 7 7 3. Fd the mea devato about the mea for the followg dstrbuto. ) 0 3 f 3 8 ) 0 30 50 70 90 f 4 4 8 6 8 Sol. )

f f f 0 3 30.87 5.6 3 0.87 0.44 8 6 0.3.4 3 56.3 3.56 Σf 534 Mea () = = =.87 N 45 N = 45 Σ f = 534 Σf = 3.95 Mea Devato about the Mea = 4 = f 3.95 = = 0.7. N 45 ) f f f 0 4 40 40 60 30 4 70 0 480 50 8 400 0 0 70 6 0 0 30 90 8 70 40 30 Σf 4000 Mea () = = = 50 N 80 N = 80 Σf = 4000 Σf = 80 Mea Devato about the Mea = 5 = f 80 = = 6. N 80

4. Fd the mea devato about the meda for followg frequecy dstrbuto. 5 7 9 0 5 f 8 6 6 Sol. Wrtg the observatos ascedg order. Cumulatve f frequecy M f M CF) 5 8 8 6 7 M 6 4 > N/ 0 0 9 6 4 0 8 3 6 0 5 0 5 6 6 8 48 N = 6 Σf M = 84 Hece N = 6 ad N 3 = Meda (M) = 7 Mea Devato about Meda = 6 = M 87 = = 3.3. 6

Short Aswer Questos. Fd the mea devato about the meda for the followg cotuous dstrbuto. ) Marks obtaed 0-0.0-0 0-30 30-40 40-50 50-60 No.of boys 6 8 4 6 4 ) Class terval 0-0.0-0 0-30 30-40 40-50 50-60 60-70 70-80 Frequecy 5 8 7 8 0 0 0 Sol. ) Class terval frequecy f C.F. Mdpot M f M 0-0 6 6 5 0.86 37.6.0-0 8 4 5.86 0.88 0-30 4 8 5.86 40.04 30-40 6 44 35 7.4 4.4 40-50 4 48 45 7.4 68.56 50-60 50 55 7.4 54.8 N = 50 57.6 Hece L = 0, N 5 =, f = 4, f = 4, h = 0 Meda (M) = N f 5 4 0 L + h = 0 + 0 = 0 + = 0 + 7.86 = 7.86 f 4 4 Mea Devato about Meda = 6 = f M 57.6 = = 0.34. N 50

) Class terval frequec y f C.F. Mdpot M f M 0-0 5 5 5 4.43 07.5.0-0 8 3 5 3.43 5.44 0-30 7 0 5.43 50.0 30-40 3 35.43 37.6 40-50 8 60 45.43 40.04 50-60 0 80 55 8.57 7.40 60-70 0 90 65 8.57 85.70 70-80 0 00 75 8.57 85.70 N=00 48.6 Here N = 00, N 50 =, L = 40, f = 3, f = 8, h = 0 Meda (M) = N f 50 3 80 L + h = 40 + 0 = 40 + = 40 + 6.43 = 46.43 f 8 8 Mea Devato about Meda = 8 = f M 48.6 = = 4.9. N 00

. Fd the mea devato about the mea for the followg cotuous dstrbuto. Heght ( cms) Number of boys 95-05 05-5 5-5 5-35 35-45 45-55 9 3 6 30 0 Sol. Heght (C.I) No.of boys (f ) Mdpot d A h = f d f 95-05 9 00 3 7 5.3 7.7 05-5 3 0 6 5.3 98.9 5-5 6 0 6 5.3 37.8 5-35 30 30 (A) 0 0 4.7 4.0 35-45 40 4.7 76.4 45-55 0 50 0 4.7 47.0 N=00 Σf d = 47 8.8 Σfd 47 Mea () = A + h = 30 + 0 = 30 4.7 = 5.3 N 00 Mea Devato about Mea = 6 = f 8.8 = =.9. N 00

3. Fd the varace for the dscrete data gve below. ) 6, 7, 0,, 3, 4, 8, ) 350, 36, 370, 373, 376, 379, 38, 387, 394, 395 Sol. ) Mea 6 + 7 + 0 + + 3 + 4 + 8 + 7 = = = 9 8 8 ( ) 6 3 9 7 4 0 3 9 3 4 6 4 5 5 8 3 9 Σ = 74 Varace 8 ( ) = 74 ( σ ) = = = 9.5. 8

) 350, 36, 370, 373, 376, 379, 38, 387, 394, 395 ( ) 350 7 79 36 6 56 370 7 49 373 4 6 376 379 4 385 8 64 387 0 00 394 7 89 395 8 34 83 Mea 350 + 36+ 370 + 373 + 376 + 379 + 385 + 387 + 394 + 395 3770 () = = = 377 0 0 Varace 0 ( ) = 83 ( σ ) = = = 83.. 0

4. Fd the varace ad stadard devato of the followg frequecy dstrbuto. 6 0 4 8 4 8 30 f 4 7 8 4 3 Sol. f f ( ) ( ) f ( ) 6 3 69 338 0 4 40 9 8 34 4 7 98 5 5 75 8 6 4 8 9 5 5 00 8 4 9 8 34 30 3 90 363 N=40 760 736 Mea 760 () = = 9 40 Varace Σf ( ) 736 ( σ ) = = = 43.4 N 40 Stadard devato (σ) = 43.4 = 6.59.

5. Fd the mea devato from the mea of the followg data, usg the step devato method. Marks 0-0 0-0 0-30 30-40 40-50 50-60 60-70 No.of studets 6 5 8 5 7 6 3 Sol. To fd the requred statstc, we shall costruct the followg table. Class terval Mdpot f d =( - 35)/0 f d -meda f - meda 0-0 5 6 3 8 8.4 70.4 0-0 5 5 0 8.4 9 0-30 5 8 8 8.4 67. 30-40 35 5 0 0.6 4 40-50 45 7 7.6 8. 50-60 55 6.6 9.6 60-70 65 3 3 9 3.6 94.8 N=50 Σf d = 8 659. h( Σfd ) Here N = 5, Mea() = A + N 0( 8) = 35 + = 33.4 marks 50 Mea Devato from mea = Σf = (659.) = 3.8 (early) N 50

Log Aswer Questos. Fd the mea ad varace usg the step devato method of the followg tabular data, gvg the age dstrbuto of 54 members. Age years ( ) 0-30 30-40 40-50 50-60 60-70 70-80 80-90 No.of members (f ) 3 6 3 53 40 5 Sol. Age years C.I. Md pot ( ) (f ) d = A C A = 55, C = f d d f d 0 0-30 5 3 3 9 9 7 30-40 35 6 4 44 40-50 45 3 3 3 50-60 55 A 53 0 0 0 0 60-70 65 40 40 40 70-80 75 5 0 4 04 80-90 85 3 6 9 8 N=54 5 8 765 Σfd 5 Mea () = A + C = 55 + 0 = 55 0.77 = 54.73 N 54 Varace (µ) = Σ f d Σ f d 765 5 765 5 = = N N 54 54 54 (54) 54 765 5 44630 5 44405 = = = =.406 (54) 93764 93764

X A V( µ ) = V = V(X) V(a + ) = a V() C C V(X) = C V( µ ) = 00.406 = 4.06.. The coeffcet of varato of two dstrbutos are 60 ad 70 ad ther stadard devatos are ad 6 respectvely. Fd ther arthmetc meas. σ Sol. Coeffcet of varato (C.V) = 00 ) ) 60 = 00 = 35 6 70 = 00 y =.85 y 3. From the prces of shares X ad Y gve below, for 0 days of tradg, fd out whch share s more stable? X 35 54 5 53 56 58 5 50 5 49 Y 08 07 05 05 06 07 04 03 04 0 Sol. Varace s depedet of charge of org. X Y X Y 5 8 5 64 4 7 6 49 5 4 5 3 5 9 5 6 6 36 36 8 7 64 49 4 4 6

0 3 0 9 4 6 ΣX = 0 ΣY = 50 ΣX = 360 ΣY = 90 ΣX 360 0 V(X) = (X) = = 36 = 35 0 0 ΣY 90 50 V(Y) = (Y) = = 9 5 = 4 0 0 Y s stable. 4. The mea of 5 observatos s 4.4. Ther varace s 8.4. If three of the observatos are, ad 6. Fd the other two observatos. Sol. 4 6 36 y y Σm S.D. = () = 4.4 + + 6 + + y 4.4 = 5 9 + + y = + y = 3...()

+ 4 + 3 + + y 4+ + y S.D. = (4.4) = 9.36 5 5 S.D. = Varace Varace = 4+ + y 5 9.36 4+ + y 8.4 + 9.36 = 5 4+ + y = 5 7.6 + y = 38 4 + y = 97...() From () ad (), + (3 ) = 97 + 69 + 6 = 97 6 + 7 = 0 3 + 36 = 0 9 4 + 36 = 0 ( 9) 4( 9) = 0 ( 9)( 4) = 0 = 4,9 Put = 4 () y = 3 4 = 9 Put = 9 () y = 3 9 = 4 If = 4, the y = 9. If = 9 the y = 4.

5. The arthmetc mea ad stadard devato of a set of 9 tems are 43 ad 5 respectvely. If a tem of value 63 s added to that set, fd the ew mea ad stadard devato of 0 tem set gve. Sol. X = 43 = 9 = 9 = 9 = 9 = 43 9 = 387 New Mea = 0 9 + 387 + 63 = = = 45 0 0 0 = = 9 9 = = S.D = () 5 = (43) 9 9 9 9 = = 9 = = 5 + 849 = 874 9 9 = 874 9 = 6866 0 9 0 = = = + = 6866 + 3969 = 0835 New S.D. = 0 = 0835 () = (45) 0 0 = 083.5 05 = 58.5 = 7.6485.

6. The followg table gves the daly wages of workers a factor. Compute the stadard devato ad the coeffcet of varato of the wages of the workers. Wages (Rs.) 75 5 75 35 375 45 475 55 5-75- 5-75- 35-375- 45-475- 55-575 Number of workers 9 4 3 4 6 Sol. We shall solve ths problem usg the step devato method, sce the md pots of the class tervals are umercally large. Here h = 50. Take a = 300. The y 300 = 50 Mdpot Frequecy f y f y f y 50 3 6 8 00 44 88 50 9 9 9 300 4 0 0 0 350 3 3 3 400 4 8 6 450 6 3 8 54 500 4 4 6 550 5 5 5 N = 7 Σf y = 3 Σf y = 39 Σfy 3 550 Mea = A + h = 300 + 50 = 300 = 78.47 M 7 7 Varace h ( σ ) = N f y ( f y ) N Σ Σ 500 = 7(39) (3 3) 7 7 [ ]

39 96 σ = 500 = 88.5 7 7 7 Coeffcet of varato = 88.5 00 3.79 78.47 =. 7. A aalyss of mothly wages pad to the workers of two frms A ad B belogg to the same dustry gves the followg data. Frm A Frm B Number of workers 500 600 Average daly wage (Rs.) 86 75 Varace of dstrbuto of waves 8 00 () Whch frm A or B, has greater varablty dvdual wages? () Whch frm has larger wage bll? Sol. () Sce varace of dstrbuto of wages frm A s 8, Sce varace of dstrbuto of wages frm B s 00, σ = 8 ad hece σ = 9. σ 9 C.V. of dstrbuto of wages of frm A = 00 = 00 = 4.84 86 σ 0 C.V. of dstrbuto of wages of frm B = 00 = 00 = 5.7 75 σ = 00 ad hece σ = 0. Sce C.V. of frm B s greater tha C.V. of frm A, we ca say that frm B has greater varablty dvdual wages. () Frm A has umber of workers.e., wage earers ( ) = 500. Its average daly wage, say = Rs.86 Sce Average daly wage = Total wages pad No.of workers, f follows that total wages pad to the workers = = 500 86 = Rs.93, 000 Frm B has umber of wage earers ( ) = 600

Average daly wage, say = Rs.75 Total daly wages pad to the workers = = 600 75 = Rs., 05, 000. Hece we see that frm B has larger wage bll. 8. The varace of 0 observatos s 5. If each of the observatos s multpled by, fd the varace of the resultg observatos. Sol. Let the gve observatos be,,..., 0 ad be ther mea. Gve that = 0 ad varace = 5.e., 0 0 0 = = ( ) = 5 or ( ) = 00... () If each observato s multpled by, the the ew observatos are y =, =,,..., 0 or = y / Therefore 0 0 0 = = = y = y = = = or = y 0 0 0 Substtutg the values of ad () we get 0 0 y y = 00.e., (y y) = 400 = = The the varace of the resultg observatos = 400 0 5 0 = =. Note: From ths eample we ote that, f each observato a data s multpled by a costat k, the the varace of the resultg observatos s k tme that of the varace of orgal observatos.

9. If each of the observatos,,..., s creased by k, where k s a postve or egatve umber, the show that the varace remas uchaged. Sol. Let be the mea of,,..., The ther varace s gve by σ =. ( ) = If to each observato we add a costat k, the the ew (chaged) observatos wll be y = + k () The the mea of the ew observatos y = y = = ( + k) = k = + = = = = + (k) = + k () The varace of the ew observatos = σ = (y y) = = + ( k k), usg () ad () = = ( ) = σ. Thus the varace of the ew observatos s the same as that of the orgal observatos. Note: We ote that addg (or subtractg) a postve umber to (or form) each of the gve set of observatos does ot affect the varace.

0. The scores of two crcketers A ad B 0 gs are gve below. Fd who s a better ru getter ad who s a more cosstet player. Scores of A : 40 5 9 80 38 8 67 66 76 Scores of B : y 8 70 3 0 4 66 3 5 4 Sol. For crcketer A: 540 = = 54 0 For crcketer B: 380 y = = 38 0 ( -meda) ( -meda) y (y - y meda) (y - y meda) 40 4 96 8 0 00 5 9 84 70 3 04 9 35 5 3 7 49 80 6 676 0 38 444 38 6 56 4 4 576 8 46 6 73 539 67 3 69 66 8 784 67 4489 3 7 49 66 44 5 3 69 76 484 4 34 56 Σ = 540 0596 Σy = 380 0680 Stadard devato of scores of A = 0596 σ = Σ( ) = = 059.6 = 3.55 0 0680 Stadard devato of scores of B = σ y = Σ(y y) = = 068 = 3.68 0

σ 3.55 C.V. of A = 00 = 00 = 60.8 54 C.V. of B = σ y 3.68 00 = 00 = 86 y 38 Sce > y, crcketer A s a better ru getter (scorer). Sce C.V. of A < C.V. of B, crcketer A s also a more cosstet player.