Prof. YoginderVerma. Prof. Pankaj Madan Dean- FMS Gurukul Kangri Vishwavidyalaya, Haridwar

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1 Paper:5, Quattatve Techques or aagemet Decsos odule:5 easures o Cetral Tedecy: athematcal Averages (A, G, H) Prcpal Ivestgator Co-Prcpal Ivestgator Paper Coordator Cotet Wrter Pro. S P Basal Vce Chacellor aharaja Agrase Uversty, Badd Pro. YogderVerma Pro Vce Chacellor Cetral Uversty o Hmachal Pradesh. Kagra. H.P. Pro. Pakaj ada Dea- FS Gurukul Kagr Vshwavdyalaya, Hardwar Dr. Sajay shra Assocate Proessor, Departmet o Busess Admstrato, JP Rohlkhad Uversty, Barelly.

2 QUADRANT- I odule 5 easures o Cetral Tedecy: athematcal Averages (A, G, H) Objectves Ater studyg ths module you would be able to uderstad the: Itroducto Cocept o measures o cetral tedecy; Arthmetc ea; Geometrc ea; Harmoc ea; ethods o calculatg A, G & H; erts, demerts ad uses o A, G & H; ad Relato betwee A, G & H. The Iteratoal oetary Fud (IF) o Tuesday rased projectos or Ida s ecoomc growth by 0. percetage pots to 7.6 percet or 06-7 ad The projectos came at a tme whe the Fud sad global ecoomc growth wll be subdued ths year, ollowg a slowdow the US ad Brta s vote to et the Europea Uo. It, however, retaed global ecoomc growth at 3. percet or 06 ad 3.4 percet or 07. Busess Stadard, New Delh October 05, 06. Statemets lke these whch talk about the growth rates o atos/states/dustres/sectors/areas/etc. are qute commo that we read daly ewspapers/magazes/jourals/etc. or hear t o TV chaels or dscussos amog ourselves. Smlarly our daly lves we ote make statemets lke: the average come o Area A s Rs 5,000/- per moth; the commerce studets study o a average 4 hrs daly ater college; average wages o workers o Factory X are Rs 0,000/- per moth; etc. A careul aalyss o these statemets reveals that they are talkg about some value or gure, ot etreme but some cetral value, aroud whch most o the observatos cluster. Ths cetral value, aroud whch most o the data pots cluster, s used as a represetatve value or the data.

3 Hece these cetral values whch represet the data are kow as easures o Cetral Tedecy. Whe people talk about a average value or the mddle value or the most requet value, they are talkg ormally about the some measure o cetral tedecy. Geerally, t s very dcult rather mpossble or a huma md to remember the huge ad uweldy set o umerc values whch t comes across le o daly bass; ad eve t remembers them the also t s ot possble to draw some vald cocluso rom these tes/hudreds/thousads/lakhs/etc o gures. easures o Cetral Tedecy are the statstcal tool whch helps codesg, smplyg ad makg the data more uderstadable. Hece easures o Cetral Tedecy occupy a place o pre emece all statstcal aalyses. The measures o cetral tedecy whch we dscuss here ths module are: A. Arthmetc mea B. Geometrc mea C. Harmoc mea A. Arthmetc ea The arthmetc mea or average as reerred commo parlace s the most commo measure o cetral tedecy. It s obtaed by addg all the observatos ad the dvdg the sum by the umber o observatos. Depedg o the type o data.e. ugrouped (uclassed) data or grouped (classed) data, deret methods or calculatg the arthmetc mea are used. Arthmetc ea o Ugrouped Data: There are two methods or calculatg arthmetc mea or ugrouped data. ) Drect method ) Idrect or short cut method ) Drect method: Arthmetc ea (A..) = Sum o observatos Number o observatos I we have X, X, X3, X observatos, the X+ X+ X3+ +X X A.. (X) = =

4 Eample:- Fd the arthmetc mea o marks obtaed by 0 studets a test. The marks are as ollows:- 6, 8, 87, 78, 54, 56, 67, 65, 68, 69. Soluto ) A.. = ( )/0 = (690)/0 = 69 The average marks are 69. Idrect or short cut method: I ths method a arbtrary assumed mea s used. Devatos o dvdual observatos rom ths assumed mea are take or calculatg arthmetc mea. Let A be the arbtrary assumed mea ad d the ew varable deed as ollows: d= -A, the A.. (X) = A+ = d / Eample:- Fd the arthmetc mea o marks obtaed by 0 studets a test. The marks are as ollows:- 63, 6, 67, 68, 64, 66, 67, 65, 68, 70. Soluto S. No. X d=-a let A = = = = = = = = = = =0 d=60 0 A.. (X) =A+ = d / = 60+60/0 = 66

5 Arthmetc ea o Grouped Data: There are two methods or calculatg arthmetc mea o grouped data. ) Drect method ) Idrect or step-devato method ) Drect method: Suppose we have data orm o X, X..X observatos wth correspodg requeces,. The arthmetc mea wll be A..= X+X+..X = = /N Eample 3:- Calculate the average umber o chldre per amly rom the ollowg data. No. o chldre No. o amles Soluto No. o chldre () No. o amly () =0 5 5= = = = = =30 = 40.= 59 A.. = = /N = 59/40 =.65

6 ) Idrect or step-devato method: Steps we ollow ths method are as ollowsa) Frst d out the md pots o deret classes (X) b) The decde about the value o assumed mea. Let t be A c) Calculate the value o d. I class terval s deoted by h ad A s assumed mea the d = (X-A)/h. d) ultply these devatos wth correspodg requecy ad calculate the value o d. e) Apply the ormula- A.. = A+ ( d/n )h Eample:-4 The ollowg table shows the daly come dstrbuto o 500 workers. Fd the average come. Icome No. o Workers Soluto Icome Workers d Value () () d=(x-5)/50 d =500 d=-80 A.. = A+ ( d/n )h = 5 + (-80) = 7 Thus, average come s Rs. 7.

7 erts o Arthmetc ea: ) It s easy to uderstad ad calculate ) It s based o all observatos ) It s rgdly deed v) It s capable o urther mathematcal treatmet v) It s least aected by samplg luctuato. Demerts o Arthmetc ea: ) It s uduly aected by etreme values. ) I case o ope eded classes t caot be calculated. B. Geometrc ea: Whe we are terested measurg average rate o chage over tme the we use geometrc mea. Geometrc mea s deed as the th root o the product o tems (or) values. Calculato o Geometrc ea (G..) - Idvdual seres: I,,...,, 3 be observatos studed o a varable X, the the G. o the observatos s deed as Applyg log both sdes G..= 3... log G. log.... = [log log... log ]

8 = log G. at log log Calculato o G.. - Dscrete seres: I varable X wth requeces G..=,,...,, 3,,...,, 3 be observatos o a respectvely the the G. s deed as N Where N =.e. total requecy Applyg log both sdes () we get G.= atlog log N Calculato o G.. -Cotuous Seres: I cotuous seres the G.. s calculated by replacg the value o Where by the md pots o the class s.e. G.= atlog log m N m s the md value o the th class terval. erts o Geometrc ea: ) It s rgdly deed. m.

9 ) It s based o all the observatos. 3) I G ad G are geometrc meas o two groups havg ad observatos, respectvely, the the geometrc mea G o the combed group o (+) values s gve by log G = (log G + log G) / ( + ) Uses o Geometrc ea: Geometrcal ea s especally useul the ollowg cases. ) The G. s used to d the average percetage crease sales, producto, or other ecoomc or busess seres. For eample, rom 99 to 994 prces creased by 5%,0%,ad 8% respectvely, the the average aual come s ot % whch s calculated by A. but t s 0.9 whch s calculated by G.. ) G. s theoretcally cosdered to be best average the costructo o Ide umbers. C. Harmoc ea: The Harmoc ea (H..) s deed as the recprocal o the arthmetc mea o the recprocals o the dvdual observatos. Calculato o H. -Idvdual seres: I,,...,, 3 be observatos o a varable X the harmoc mea s deed as H....

10 H. Calculato o H.. -Dscrete seres: I occurg wth requeces,,...,, 3,,...,, 3 be observatos respectvely, the H.. s deed as H.... H. Calculato o H. Cotuous seres: I case o cotuous seres H. ca be calculated by takg md values ( m ) place o ' s. Hece H. s gve by H., where m m s the md value o the th class terval Eample:- 5 A cyclst pedals rom hs house to hs college at a speed o km.p.h ad back rom the college to hs house at 5 km.p.h Fd the average speed. Soluto Let the dstace rom the house to the college be kms. So the total dstace travelled by cyclst gog to college ad the comg back to house s kms. Sce the speed o cyclst gog rom house to college s km.p.h. thereore the tme take to cover ths dstace s / hours. Smlarly the tme take to reach house rom college s / 5 hours. Thus a total dstace o kms s covered ( + 5 )hours.

11 Speed = Dstace/Tme Hece, Average Speed = Total Dstace Total Tme = ( X + X 5 ) = ( + 5 ) =3.33 km.p.h erts o Harmoc ea: ) Its value s based o all the observatos o the data. ) It s less aected by the etreme values. 3) It s strctly deed. Demerts o Harmoc ea: ) It s ot smple to calculate ad easy to uderstad. ) It caot be calculated oe o the observatos s zero. 3) The H. s always less tha A. ad G.. Uses o Harmoc ea: The H. s used to calculate the averages where two uts are volved lke rates, speed, etc. Relato betwee A.., G.. ad H..

12 The relato betwee A., G., ad H. s gve by A. G. H. Note: The equalty codto holds true oly all the tems are equal the dstrbuto. Prove that a ad b are two postve umbers the A. G. H. Soluto: Let a ad b are two postve umbers the The Arthmetc mea o a ad b = a b The Geometrc mea o a ad b = ab The harmoc me o a ad b = ab a b Let us assume A. G. a b ab a b ab a b 4 a b 0 whch s always true. A. G. () let us assume G H ab

13 ab ab a b a b ab a b 4 a b 0 ab Whch s always true. G. H () rom () ad () we get A. G. H. Summary The measures o cetral tedecy gve us a dea about the cetral value aroud whch the data values cluster. That s why these values are cosdered to be represetatve values.e. the values whch represet the data. Arthmetc mea s the most commo measure o cetral tedecy whch s obtaed by addg all the observatos ad the dvdg the sum by the umber o observatos. Geometrc mea s used or measurg the average rate o chage over tme. It s deed as the th root o the product o tems (or) values. Harmoc ea (H..) s deed as the recprocal o the arthmetc mea o the recprocals o the dvdual observatos