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Example: Th one of Phavmaceutical Company, 120
phaumacists are working,out of uwhich 50 gis &
70 boys with mean heights 0f14SL 148cm resp
ind out mean fov boys & qirls height togethe
CAns: 14h6 75cm)
aking Log of any number
Log value
Con sists
Charaetemstic MantisSa
Step 4 Finding of Chavra ctei stic
Charactemstic =K-T
where
K Total No.ofF ddigit in
q Iven Value/ number
Note: Tf No.contain /value contain
decimal point fhen Cons]der
oligits which are on Left Side
of decimal point Ci.e. befoe
decimal point only
Characteristic = Z +I
where,
Z Total No.0f 2eros Come
oufter clecimal Pointli.e on
right sde of decimal poin t)
iofe I NOte Only when value / numbr conta) ns
rte en0. Values.i.e.zen before
decimoal point.
- CCD](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/biostatisticsresearchmethodologyuniti-mean-220116171642/85/Biostatistics-and-Research-Methodology-Unit-I-Measures-of-Central-Tendency-1-Mean-20-320.jpg)
Biostatistics and Research Methodology: Unit I - Measures of Central Tendency 1.Mean
- 1. @ Chaitali C. Dongaonkar 1 BIOSTATISTICS AND RESEARCH METHODOLOGY UNIT – I
- 2. @ Chaitali C. Dongaonkar 2 Contents 1. Learning Objectives 2. Introduction to Measures of Central Tendencies 3. Types of Statistical Averages 4. Mean – Arithmetic Mean 5. Mean – Combined Mean 6. Taking Logarithm of any number 7. Taking Antilogarithm of any number 8. Mean – Geometric Mean 9. Mean – Harmonic Mean
- 3. Onit 1L MMeasures of Centrau Tendencies: Leaning ObJectives: To Know meaning of Mean, Meclidn & Mode Tounderstand Calculations of vamous tupes of mean like Amthmetic. Geometic an od Hoamonic Todescibe meits & demerits of mean,median &mode Tnmoluction AverageSingle wod/Single fiqure which gives Complete plcture of pheno menon eg If we collect dlata of heights ot 2000 Students in College we will unable to emember al 2000 numbers SO, we Calculate one single number nown as Average" 2. Avemage Ke presentatrve closer to most member3 ofdata Hence This mepresentative known osAvexage" 3.Central Tendency: This Average Lies more at the Cente of the data also known as A Centsa lendloney -@CcD
- 4. 2 To Obtain Single number which wi Tepres ent whole cdata. 4.0blective 5. CompaTislon: Compamslon of lbulkydata is cntical So taking average and compamsion of data made simpler &guicker 6 Taking averrage: NOT last stage of dlatoa analysis but Stating point of Calculations heeded fo further analysis. Statistical Avemrage A Posttional Average Mathematical Average Median CMe) Mode Hamoni c Anthmatic Mean CX) Geometic Mean CG) CM.) Mean H) MEAN Three types of Means O Amthmatic Mean C) Geome.tricMean C) Hamonic Mean CH) -ccp
- 5. lbO Amthmatic Me.an A.M.of Set of observations is defined as the Sum ofallo bseavacfions divided by the total number 0f obsevations Denotedl by o u Athmahc meah tor an ungvouped data: Tndividual semes Tf ,2,A3,, . . -,4n be the values of n' obsevations then orithmartic ean X = = 2i where ; = Aj2,3,, - n O gn=numberof Items/0bservaTOns Thing to be Remember Sum of given data No.0f given data AM XK ExamplesS OFind Mean of given data 12,19,15,22,17 Ans n X =219+1S+22 +I17 5 Mean of given dato 1S - cCD
- 6. (2) he a.bsorbance values ot aspisin Solution were obtained in 9 folols using UV Specto photometer as 0-273,0:275, 0 271, 0.27S, 0.274,0. 27s,0-279, O.278 0and 0:28I. find mean value of absorbance Ans n X =U=0:273+0.27S +O.271 +0 27S+ O.274 t0 27S +0.279 +0.278 + 0. 281 2 481 0 27G Mean value of absorbance was foundto be 0 2+6 Tryyourself Example1: The Telativehumidity in the tablef prduction department as given below. Calculate The mean percent melative humidity. CAns: 66) Days Monday Tuesday Wednesda4 Thurs day fiday Saturoay Percentreahve Humiclity G 2 61 75 G5 -@cCD
- 7. DisCTete Series oY UhgaDupec requency distribatio CWeighted AMD 0Direct Method: A,2,3,M, ----Rn fi,fa,fa,fhr fS-Voluesoffrequencies (-Values of OT weights X f i CEf =n) fi Note if weights ov frequencies ave same Ci.e. f, f2 f,= fn) then Ose simple mean. CAM) Examples OCalculate AM of maves obtained in Phama.ceutics by Students of T: Y. B.Phom class Marks 40 8 52 58 64 61 A7 No.of Studentss 5 2 85 3 2 1 Ans Marks ObtaIned C) No.0f Stuclents Cf) f 5 200 =877 33 X = 56 37 52 364 58 46 64 320 69 207 148 +8 7-8 n f =33 PA =1977 Average marks secured by stuclents are 56.87 - CCD
- 8. Shovtcut Method: X a +2fd (d--a where a- assumed mea /ATbitary Mecan Middle Value of"x d= deviation n Number ofitems/obsemvations f =tTeguency EXample: C Previous Question) No of d =-a MaTks Obtained Students) C) fd Cf) Ca =58) -18 90 48 -10 20 52 -42 Jo-58 64 5 30 61 33 7h 16 32 78 2 0 20 n=33 f d f 33 = - 5I X = a+ 2Fo n = 59 t (3 = 58 + C-1:54) X 56:46y -CCD
- 9. it) Contihuous Senes oT ouped treguency distibution: Step deviation/Assumecd/Tndlirect Method: when class imtervals aTe in continuous Seies. X = A + (2T)h N where A = asSumed mean /Arbitary mean Class mark in which (Nl2)th tem falls orSimply micldle Value of X. h class width /5i2e Ch- UL-LL) N Total frequency Both are f Class treguency Step deviation egual X = Mid Value d =XA h X = UL FLL (6Direct Method: i m; fi where f = treguency m mid point of class m = L + L L 2 UL Upper Limit ofclass) LL Lower Limit of class ) - CCD
- 10. Example Calculate Avrithmatic mean of following distibution of patient of Covicd-19 as per their we ights We ight CX) 30-40 40-So 50-60 60-7o70-20 80 -70 No.of patients s 12 1 4 7 Ans DStepdeviation /Tndimect Method No.of X d Patients Cf) Weight f d Cwiiddpt (d -XA) 30-40 5 3S -IS h0 -SO -16 SO 6 S O 2 65A O 607O0 7-0 80 7S 80-90 I7 S5 2 34 N - f=63 Zfd 10 - At( h h = UL LL = 0 3 0 = 10 :. Y A +( h N A Middle tem of xCmicd point) A = G5 6 5 + x 1o = 65 + 100 6 3 X = 66.59 -cCD X=65 + | 58
- 11. 2) Direct Methocl No.of Pattents CP) Mid poimt weight CA) f m Cm) 30-40 35 175 40-So 360 SO 60 5S 38S 60-70 | 2 6S 730 70-8 0 7s Oso 80-9o 17 85 144s N-2f63 Zf.m-41s fm .m -OLtLL 2 epuiteo 63 GG 59 Averagewe.ight of patientsof Covid -19 S 66.S8. TryitYourSelf EXample1: Calculate the mean Overal fartality Rate InSmall pox trom age -wise fatality Rate qiven below. Agegopup: O- 10 10-2.0 20-30 30-40 40-SO 50-60 FatalityRate 8ls 12 30 0 6 - @CCD
- 12. ) Discrete Semes in Groupeodl freguencydistmbution when class intevals are in discrete semes i.e. NoT Continuous. Ose Same fovmula as continous class intervals but Cohvet discTe te class intervals into Continuouc class intervals. Thing to be remember 9. How to Convet discrete class intervals into Continuousclass intemvals 2 Ans: StepT Tdentify class intervals in Continuous Series oT in discrete Semes. eq Class Tntevals 30-3S 3-60 o)-4 6S-S0 0 - S S Here, Upper limit of fietclass is Lows limit ot second class, So,1t is considereol LS ContinuouS Class Class 1.30,31,32,39,3 4 2. 35,36,37,38,39 Co tinuous Upper limit NOT included 6O interval i.e.35Uclass -LLS 35 This is Cons id ere.d as Continuous qroupecl Semes ClassLntervals Here, Upper limt ot 31-3 e Class first class is olifferent Than Lower limt of Second class. so, it js Considered as "oliscvete class infteval i.e. 35 UL 36 LL 1.31,32,33,34,35 (36)-% o 2. 36,37,38,5140 6 -4S 6 5Oo disconthnuous 5D- 55 L.LUL both included -cCD
- 13. Step Canvemting Discontinuous class intemials O into continüous class intervals. Take o Considew any two class interrvals find out gab between 0L Of one class & LL ofnext class UL Of fiet class = 35 LL Of Next class = 36 difference 3G - 35 Divide difference obtainedl by 2. acdol clifference/2 value in UL Subtract difference! 2 Value fom LL Class intemvals -05 31 3 S +05 New class intervals 3 0 5 35 5 355-40 S -0.5 36 -40 +0.5 h hSy +o.S 0 S 45 5 50 5 -0.546 0+0.5 hoAS 5 0 . 5 S 55 5+05 Toy tt youTself 50.5 55 5 Example 1: Convert diScontinuous class intenials Into Continuous class ihtervals. Class intemvals class intervals IS - 2O 25 3 0 35-40 c 4S 5 0 20 3 0 -ho 5o 235 5 -60 - CCD
- 14. Lmpotat while Considemng class intervals treguency UL value of any class should be considered tn next Class interval e Class inteTVal Values to be included O-9 CO 1,2,3,,S,6,7, 10-19 ULOf Fi33t 20-21 class to be 2t020 320 30 50 3 9 h04 1innext class 30-0 COnsidered 50 as LL Example 1: Find the mean of frollowing clata 13S 139 140- 144 14S -14 1 Class: I30 - 134 freg 1S 2S S Ans: 1. Observe class intervals in Continuo us or in discontinuous semes. Here 1is disConhinuous Semes. s0,Convertit thto Cohtimuous semes oifference -L next Etass- lass L of fiat class = 34 LL Of Second class 135 , difference LL of second Class DLOf fia clasS =13 S 134 difference 1 2. Divide diffevence by 2. o 5 -cD
- 15. 3. Now, Subtract o. 5 from LL and odd o. 5 in UL.SO, It will give continous class interrval. then find out meah by using formulah Mid Pointn) .f class New class fm 130-1341215-134 5 13 2 5 6G0 13S-139 154S139,S37 S 140-154 139-S 14hS 14 2 20 55 2S 3550 14S-1491455 149 5 47 1S 220S Zf=60 Efm-9 470 Mid point = LL of Neuw UL Of New C tLss Class f : m; ( f =N) f 6O 141 16 Mean 0f giveh d ata is A116 EXample 2: Following an aray of 5 paTacetamol Tablets pmduced by ponduction manager in dif Ferent 6 5 OuTches. Prepavme freguency distibution table anol tindout mean. 5 27 50 4S 32 36 413 4 4 8 27 46 3 t 34 42- 4S 31 8 27 9 18 17 32 93 3S 37 47 29 46 2G G 3 35 33 42 31 0 5 42 44 41 36 27 39 5 56 53 88 $5 31 2 38 54 36 3 3 8 56 596I 65 64 2 64. -@ CCD
- 16. Ans .Step1 Deciding class width for given data. Any suitable class width can be taken. Here Cosider class width as 5 Step 2 Observe qiven clata and identify lowest Value and highest value. Here, Lowest value is 25L Highest Value is 74. Step3 Now, set classintervals Statingtoom Lowest value to Highest value, fietly Class intervals will be in dis crete dlishibuton ahd then convevt it into Continuous one Here, class width considered as 5.so,iet Class intemval wil| become Cstasting tom Lowest value 25) 25 29. 1.e. Class width = 05 1Sclass interval = 25 -29 i.e. 25,26,27,29,29 = 05 Step Consicler class intervalsHI Highestvalue iven in distobution Here.it is 74.So,we wil prepaTe Class intevals starting tom Lowest Value. i.e 25 to Highest value i.e. 74. Important Always PTe pare Cliscrete/ discontinuous class Intervals when distibution data is given& then Convet it into Continuous class intervals.s it should include all values qiven CCD
- 17. Now, we wil| wnte class intervals. & we wil| fino freguency of same by Considemng tally marks. chep 6 We will convet cliscrete class intevale into Coninuous classintervals then we can easily tind out mean. Step5 New class Mid Lntervals Pointm) m OG 2h 5 295 27 162 Freq class Tally Intervals Marks 25-29 30-34 HH1 34-31 H I 14 60-44 29 S 34:5 32 2 88 34 5-39.5 37 513 08 395 44: 5 hh:Sh1S 4 2 3 36 659 50-5hH 06 41.S - 54:5 52 55-59 I1 Go-6411 GS69 0665-9:567G7 t0 th O 03 5h:559 5 595 64 S 3 12 57 71 03 62 1 96 69-5 74:5 72 2 f-65 Efm 2- Zfi mi e g 2f 2770 fog 4 2 61 Sd,Average tablets pmcduced by produdion manager in different G5 batthes are 4261 l.e. 43 ehe -CCD
- 18. Trick 1. while Converting discrete classes into ContinuoUS classes Just add class width into previous class's VL & LL 2. Calculating midpoint Just find out mid point Of fiet class & aded class width into previous Class's midpoint Merts of AM: I. Simple to understano 2. Easy fo calculate as based on obsenvations 3. Determined by rigid tomula 2Hence, itis exacttigure 4 Considers all scOTes in disti buthions & means of different distmbuctions are useful tor Compazative PUTposes 5. Least affectecl by tluctuations as compared with Other measures Dememts OF AM: 1. Only guantitative measure, NOT forgqualitate analysis 2. NOT used in case ot open end classes 3. Preguencyof any unit missing,nmean cannot be Calculated . Sometimes mean notbe an actual item in the semies and may be meaningless tigure eg Aveoage number 0f Chlldren per family 1S 2 22 - CCD
- 19. Exdmples on Combined Mean Example 1 Calculate Combined AM of medical & paramedical StatF wosleing in govemment hospitals of Pune distict in Last flve year3. Phamacist NuTSeSMidwives ard boys Drs HI3 3122 26S 28 Yea 817 21S 2.015 24S 354 217 917 2S 912 2 016 296 27 2.017 208 298 1190 229 1o12 313 2018 2h0 1342 324 3 02 1420 2019 Ans: Step1 Calculate To tal & Mean of each Category To tal Il09 314 IS 2S 1388 5139 Cn Cn) ns) 3 0S 2 776 1038 Cn.) Cn2) Mean 221 8 962.8 C,) C2) n, +n, M tngs tns, +nsM .CombIned Mean C Combined) 0,tna t ha+htns CIo9 K221:) + C48)4 x 962-8)+ CIS2S X305) t (1388 X 2t7 6)+ C5139 X 1038 6) 1o9+ 4814 + 1S2S+1388 + S139 2,45,946 2+40,34,119 2 +4,6S,125 + 3,85,3088+ 5393,449 8 2545 13975r 11,124,+79 13,97S 76716 0 4 X Combined AM of medical & pavameolical staff woreing in gov. h0spitals of pune odismct in ast tive years1S 79G 04. - CcD
- 20. GIy itYousself Example: Th one of Phavmaceutical Company, 120 phaumacists are working,out of uwhich 50 gis & 70 boys with mean heights 0f14SL 148cm resp ind out mean fov boys & qirls height togethe CAns: 14h6 75cm) aking Log of any number Log value Con sists Charaetemstic MantisSa Step 4 Finding of Chavra ctei stic Charactemstic =K-T where K Total No.ofF ddigit in q Iven Value/ number Note: Tf No.contain /value contain decimal point fhen Cons]der oligits which are on Left Side of decimal point Ci.e. befoe decimal point only Characteristic = Z +I where, Z Total No.0f 2eros Come oufter clecimal Pointli.e on right sde of decimal poin t) iofe I NOte Only when value / numbr conta) ns rte en0. Values.i.e.zen before decimoal point. - CCD
- 21. Examplesind out chaacteri stic O 5h67 K - I 4 3 2 I= k- 2.8 73 k- = 1 - 1 o 034 51 = Z+ = O+T = 33 57 )0-05 791 = 2 +T = E G) 0003257-z+ = 2 +1T 3 step2 Finding of Matissa 1.Make pair of Staring 2 digits eq 54 6 7 54 Find in T column having values 0 - 99 GFind valuein I column 8601/7 aving values o -G. th0emrbc 7Find mean difference Ci3i.e. in IL column having Values c4to 1) .. 5h67 Search 54Thvalue in G Cin columnTT) G Mean dff. in 7 Cin column LE) Mantissa =73 72 F378 LOg C5467). = Chara &Mantissa 373 78 o 05467 ZtI -T+T-2 Log (o.o5467) = 27378 3 054G7 =k-I1- o Log CO64¢7)= 37 O0233 O233 -ccD
- 22. laking Antilog otAny value Step1 Tdentify Characteidic &Mantissa tom given number or Value. eg. Antilog C264 5 3 Mantiss a Characteristic Step 2 Consider Mantissoa &Take firrt tuo digits & Search for 3 digit along wIth mean difference for 4th oligit.in Antilogarithm Tabe eg G4 TST column value Value fom I Column 3 Mean difference fovm 3. Cmentioned in mean olH4 Column). 5 64 41G 3 Mean diff.in 3 hh19 Step 3: Now add 1 into Charactenstie eg 2 G 4S3 2+1 = 3 So,here wegot 3 digit. Step 4 Put decimal point in Anti log value as per obtalned digit eg. Here value is 44 19. 'Antilog (2-6453) = 441 9 - CCo
- 23. 2 Example 0 2 521 Anti loq (2 921) AntfilogC0. 402 F 2 1SIX Io-3 +100O OS29 471 215I give decimalpointafter fir3tdigit & wite 10 ol which number yo have 3 added. C-2 2= Antilog (-2. 2797) = 5 252x 1o +IO O0o O 21 O.7203 3 Antilog Co 7525) =5.G56 5649 Antilog Co-07s2 = 1 190 1189 T190 - ccD
- 24. 2 Geomehic Mean CG Geometmc mean of a set'n"observations is defined as the n th root of their product i.e. Geometncivean nn,1, Actual Calculation Antilog log logi z) TF ,, n ff 9iven . . n G Antilog : logi N Ef f freguency =0bservati on where 3 G29 - Two geometmc means ot two groups ot Sizesni&n then geometmc mean ot Combination is Antilog loaatn,:log$2 +h2 A) ,292 wo geomeme mean of fwo gooups Of sizes neach then Combination GM is -@CCD
- 25. M g M M
- 26. Where Tf,20,be non-2ero postive values of two obsevations CM AM) XCHM) Example : The weights of 10 tablets Cin gm) are given belouw 20,1 18,1 23,117,1-19, 1. 24, L25,1 2), 1:22,2C. Calculate Havomonicmean. 1/ AnS 2 0 O.9333 0 947 0 8130 O 8547 0 9403 o O 9064 I18 2 3 117 19 124 1 2S O 8 26 4 2 22 126 O793 Total = 2347 to y, H.M= 8 23 4 7 H.M. =I:2143 gm Hammonic Mean ofweght for 10 tablets - CCD