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Full text of "Sankhiki Ke Siddhant"

mm 



1. Practical Problems In Statistics 

2, Indian Statistics ( co-author ) 



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5KT f^fWT Practical Problems in Statistics (second edi- 
tion) % fer ^r^r 1 1 

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qmr 

(INTRODUCTION AND DEFINITION) 

Ni; ^ifts^MJiw ( statistical 

methods); grf^T$ $ <m*TM; gJWfit % *nT; Sirfwft ^ ?RT IT^f 
I S^ (? ? ?) 



(FUNCTION AND IMPORTANCE OF STATISTICS) 

q 
U) 



(PLANNING A STATISTICAL ENQUIRY) 



(statistical units) 7f^p3T rf&F$ (degree o accuracy) 
I S^S ^o 



(COLLECTION OF DATA) 

^rg^R (direct personal investigation); 
( indirect oral investigation ) ; 



(by schedule questionnaire); ***F?k srRmil 33RT (by local 
t eports ); srfflfafw mw%\ ( representative data )~^f%^l^f^^ 
(deliberate sampling); Wftflfr (random sampling); ERRR 
JFR3T (selection of sample ); stfe*R& i^RTfecfT f3pw (law of 
statistical regularity ); JRgfa ^iwr iwt ( law of inertia of 
large numbers ) fgrak TW\ *W?*3 ( collection of secondary 
data ); fstfPI SfRsft *<wta ( using secondary data); 
( necessary attributes of data ) 



( reliability of data ); ^Fpft^^n ( suitability of data); 
(adequacy of data) R^T^ 1 S3 (^^ 



*PT 

(EDITING OF COLLECTED DATA) 
( accuracy ) ; ^TOR^C ( approximation ) 
f^Rgnr ( Statistical errors )m i%WT ( errors of origin ); 
f^TSPR ( errors of manipulation ); WRten-f%WT ( errors of 
inadequacy ); fi^ ^ en^ f^WT (absolute and relative 
errors ) fofrf f%wr ; ^i^r RW?; 
( biassed and unbiassed errors ) ; 5RS(TRft 1 ?S 



(CLASSIFICATION AND TABULATION OF DATA) 

3r*ft^*!! 3<$ % W3^fR ( by attributes ); wf^ % 
( by class intervals ); ^nra^ff ^ ( exclusive method ); 
( inclus ive method ) ^rrpgt^ ^^PT; WPnPta? ; firfer 
( single tabulation ) fs>*j<q* 



(double-tabu lation);^^^ 5 ^!^ (treble tabulation ); 
^iSTR^FT ( manifold tabulation ); WT OTC^^R ( simple tabula- 
tion ); ^sreFWffcFT ( complex tabulation) 



(STATISTICAL AVERAGES) 

M % JpJT; frfasr TO^C % RM; 
(Mode) ^R^TfT; ^jf^S^ f^TOJFTT; ^jfw^ % 
I 

(median) T^^TO; OTsnFWT^fl 1 ^T WFR f^^FT^RT; 

(discrete) ^ w rW; 'Scfcr ( continuous ) 



( quartiles, deciles and per- 
cen tiles ;) faftm TOR $ 
( arithmetic average ) 

^cicf 



( direct method ) $fg-i% ( short-cut method ) 
% 



; *TfRcT 

( geometric mean ) qf^rra 

j^ij^; j^^cfj % ^fp^^ ^f^f cf*JT 

( harmonic mean ) 



( limitations of averages ) 
( Standardized death and birth rates 
i ss ( 






(DISPERSION AND SKEWNESS) 

( dispersion )fw?^ ( range ); ^r 
( quaitile deviation ); ^g*k fN^Ff % ^n*r ^r qrfJfaf; 
( mean deviation ); RM*c rf<iprf ^F ^M-f^crR" cT4f *TFT 

fir$i*RT; JIM fH^t % $im ^T^I ^fw ; WIT HT 
( standard deviation and its coefficient ) i%W^ ^F%4l EI 



( skewness ) 

^^1; T%fRTT ^JT inq 1 (measurement of skewness ) fwrai % 
( coefficient of skewness ); Rlc*FR cf^T ^^IcW i 

2 ( m 



(INDEX NUMBERS) 

; q^f ^ S^T; ^cf! % 
( price 



quotations ); WNK^T 1^13" ( selection of base ); 
*RFTT ( calculation of price relatives ); ^^I-WM^ ^ Jj- ^qj. 
3TO $ *HFTT; ^M *fiT 5PFT; ^IRcT 3R^ ^t f%M ( methodsof 
weighting); Jj^igTffit ^ w^TigTOT ^T H^w ( relation between 
price relatives and link relatives ); wpsRfT ^Ct^T ( rever- 
sibility test ); ^R:^ ^cJp^cfi ( time reversibility ) w& ^^P'RIT 
(factor reversibility); ftm ^T WK^I ^ ( Fisher's ideal 
formula ); f^^-o^^-^RT? ^^T ( construction of cost of 
living index numbers ); ^rs^T?;^!/^ 1 ^^!; **W ^ ( aggregate 
expenditure method ); qfeTR '%*& ^Ifcf (family budget method); 



re (indices of industrial productioa ); 
N ( indices of business conditions ); ^j 1 - 



^T fWf 
(DIAGRAMATIC REPRESENTATION OF DATA) 

WPft % R^f % W ^ SRSS 5R% % ^FT; 

% R^f; f%yfT f% : f ( dimensional diagrams ) ^ r%?T f^f; 

q^ f%?i ( simple-bar-diagrams ) ^cFft^xF^^-MW ( sub- 
divided bar-diagrams ); ^-ft^T-RW; Wlcf ( rectangles ); ^f 
(squares); ffr (circles); mfl^T i%^; ^ (cubes); 
( pictograms ); HRf^^f ^ ( cartograrns ); 



(GRAPHIC REPRESENTATION OF DATA) 



) (plotting of historigrams on natural scale) 
H3 ! ^T % M^; ft^ 5Flf%^-f^ ( absolute historigram one 
variable ) fz WRR; ^TT ( false base line ); ft *n ?rfir 
( historigrams twoor more Variables ); 

(frequency 



diagrams ) ^g--f^Rf (bar*diagrams); ^^Rf-cf^ (discontinuous 
curves ); ^m ^ ( continuous curves ); f%fir?r SfWC 
^^ (theoritical frequency curves ) 
( normal frequency curve ); f 
( skew frequency curve ) ; fWT rr| ^"I^Rcn ?T^ ( V shaped 
or extremely asymetrical frequency curve ); 



^5 ( U shaped frequency curve ); ^epft ^TKcFr *ra ( cumula* 
tive frequency curve ); sigqm *l$f % ftp<?t^ (graphs on ratio 
sca l es ) %cfT,^%^f ^ %^t spfi (logarithmic scale and logari- 
thmic curves ); ^FJ *%<rf * f^PTC; SFWRraft I S 



(ANALYSIS OF TIME SERIES) 

(secular trend); WW f^"^^ (seasonal 
variations); ! ^r gp^T^W (cyclical fluctuations ); % ^r 
( random or irregular .fluctuations ); ^t^- 
( measurement of secular trend ) 
^ ( trend fitting by inspection ); 
( method of moving averages ); 
fiT f^^cf ( theory of moving avarege method ); 
t(% ( method of least .squares ) ^r^T^T^R ^ 
( measurement of short period fluctuations ); 

measurement of seasonal fluctuations ) 
^ ^t ^Tfe^ Ffl^ <3ftr ( method of monthly 
averages to compute a seasonal index) w$=r ^Rfcfr qft ^^qr 
^^ c$" ^f-qy^qf ^% ( method of moving avearges to compute 
a seasonal index ); ^^ngrof ^<li% ( method of link rela- 
tives ); ^r^ ^< ^fcwft ^r^^^ff ft fl:N ( measurement of 
cyclical and irregular fluctuations ); sr^isroft | 33 (^?v ^v) 



(THEORY OF CORRELATION) 

qfR^TO; ^RIcH^ cfTT ^^micq^ ^^t^^ ( positive 

and negative correlation); f%%<* fm ( scatter diagram ); 



3 ( correlation graph ); 
efficient of correlation ) 555*5 
*T 33 ( Karl Pearson's formula) 
and short-cut method ); ^rar-s?^ 5J 
of correlation in a time series ); 
(correlation of long time changes) ; 
( correlation of short time oscillations ); 
5^r^> R^^T=TT ( calculation of co-efficient of correlation in a 
grouped series) ; rg cR-JT ^jM ^ 
( probable error of coefficient of correlation ); 




( calculation of coefficient of correla- 
tion by ranks method ); ^Ifft f^^ffl jppfr '(coefficient of 
concurrent deviations ); ^^T^oft l ?S (V*,? ^U) 



(INTERPOLATION) 

T *FT OT^T I f%^^^T^% (graphic 



method); sfto JT^^T ^(%??t ( algebraic methods ) 

( method of curve fitting ); 



(method of finite defferences or Newton's 
method); 11^ ^t ^fcf ( Lagrange's method ); 



(INTERPOLATION OF DATA) 

( preliminaries 



to interpretation ) f^^T-WlF^^^i ( false generalisations )* 



cr*rr 






(INDIAN STATISTICS) 

( population census )-- 



pf'TOpfT % cl^fo; ^TRclk Sf^^T 4t 
f* ( vital statistics ); ^ffgY^l^ ^^^ (industrial statis- 
tics ) $&$$ ^3ff ^t ^5f|<5^T ( census of manufacturing 
industries ); ^^tPt^ ^cTi% W ( statistics of industiial 
output ) I fgfa ^^ ( agricultural statistics ) %^f swqj ( area 
statistics ); ^'^it ^>^^cf qiT ^ ^'^it eRffsrer ^1T ^; ^!WR 
( yeild statistics ); ^Ffl 1 Clicf ( traditional method ); |f~ 
( random sampling method ) I ij^ Q*h; ( price 
statistics ) ^l| % WPI 5^ ^ ( harvest prices ; ; 



J ( Economic adviser's wholesale price index number ); 

^Rt* (retail price index numbers ) ; 
( wage statistics ) Wf^fJr^ Hf^ ^R^;ffi[ R^r 
^15 ^g^sn^T ( agricultural labour enquiry); 
(national income ) ^i^ 
qR^RIt?; *IRST ^ ^F^ ^FH; ?rpnWFT 
( national sample survey ) 



( statistical terms ) 
( mathematical tables ) 



( Introduction and Definition ) 

Jf *f 

% 



% % wnwf ^i JRFT ^ ^ i ^ % 

'^TRcf *f TlH^ft ^RTWt t, ^T % ^ t SffcT ^t ^fiqjfl" ^f % 

^T % 



^|cr % 
IRPR ^^Twf % sqpfrr % ^cf % 



% ^ ^ 1 1 

WIWi W H?^ Rffcji^ | | 
f ft^TT 1%^^" ^^T^l! % 
Weft ft | c^j%?ff |5t ^riqf ?jfc ^I^PT WFT 3 ^R3^f % 



% f&E ^" 



( precision ) I 
I 

f% 



if 

% ft^Mfa =rfi gjp^ i qf 3=5 1 T% mf % 



sfo) ( Statistics ) 

sratn Jf mi % 

ftf^a 1 ' ^C^ ^ fetf ^^ ft, 

(Statistics) ^^ 1 1 wge^JT ^r JRta ^T ^|^ ^qrr?tf ( events ) 
^ ^R^r c[5Tf SRFT ( cause and effect ) 
I crrf% f^f-xrf[-f^T ft^ ^TRft ^rrat % 
^% | ^% WT%^m (statements) ^t 4t 

% ^5pr ^t ^ra" I, iwr ( law ) ^p* 1 1 
i sf^r 1 1 



f: W^TT 

% ^FT 3^ OT^t q-RHiqr % $ ^e f i ^ 1 

i ^r^ ^ % 



( wavelength ) 
ffi \ ^ %^ 
% ^ ^ ^w ^ ^iwt, at 



( Statistics ) I ^r ^K ( Statistics ) 



5TT I ^^-OTSRST crft *rRT ^ *TOT I SR|% % s^fcr ( homo- 
geneous ) 1 1 | Oft c^f% eft 



ft, ^P| 

rff % f qr If 5r?g^ gvajf ^r e^f ffer 1 1 %*rar 
T 5ft ^r^crr 1 1$% ^T f 4-^T^r ^4^ ^% ^ ^ri|q; ^ 
( accurate ) ff | ^r% OT^QI cT^ ^TT^?r ( collec- 
tion and estimation ) *r ^%%r TR^clT ^T ffcn 
% f^r-cRg ( subject matter ) 1 

% 



I iw ^i^^ qr OT wra 1 Tgwr 



fMt ^ fSlft^r ^C^ % f^ ft^F 

Rr f f i 

( Statistical Methods ) 



5ft f R -S^ ^TR^T =37 I 

( colbction, estimation, and enumeration } 



I ^ ^TRrft ( data ) tfsi^tef ^t ^rdt | 3 gt ft 

1 33 *&f 4t i^r ^^JT ^ .^IPWT ^rww ^r|f itcir 



^ ( 

% 1%^ ^tfepsN 1 ^ ( statistical method ) ^r 
I I eff^ste df^^T ( statistical methods } 



( quantitative data ) ^T ^f^ ( collection ), 
( classification ) 3 CfR^W (tabulation) 



V 

fiwrsrr % gift sNfr 35*1 srwpre Tfigsar (accuracy) 



T Sqfqta ^f OT32 

I i 



r (averages) 

(dispersions) ^t g^FTT $ ^l ^r% i 
(^) f%r%r ^rTHiMf % 
(correlation) % 

(v) ir^3cr WPfi' epi "R%Fr (interpretation) 

% 



(forecasting) I 

qR^^f ? 

SRI 






(analysis) IW ^R ^t ^f^' % %=T^T 133? 

t I 



%*n 



i% ? o o cafraqt it % ^t f%^ft 133? 

, .8. ^TTO ^T^^ ^fcT cn% ft^ 1 1 I ^K^q" ^ t H^clT 

1 % 



cRT 



I 
f^lPF'WI 



i 
1 1 m wii ^rs^^H ^OT ^rR^tmcF t t 03* 

f cf^* ?rf^ ft ^T^t f% ^THSTt eit N)-<&^ ^TR^RT ^^^T q|f ft 

% ^ ^^r ^w ^T ^TFT ^?n WCTT f 



mr%r 

ClM^t ^T 3<T%T ^TH^ IT ^RIT I 

% ^S^R ( 



% ^RF cj^ W& I ) 



( Definition of Statistics ) 
( statistics ) ^ ^T Jrq^q^ ^T^-W^TTRJcT (state- 
arithmetic) ^ f^Tf ^T I %&Bt SFTfcT % 



(Dr. Bowley) % 

r 

(Statistics is the science of the measurement of social 
organism, regarded as a whole in all its manifestations) 



1 1 5*r TR^T^T % ^TS^R ^tfeq^ % %^f ir 
r % ^^f^ra" if i m wrgf%^ 

?Tff 



ft 
ff i W 



*TT**jf (Averages) ^[ f^irr^ ^fT 3TF e^^F 1 1 (Statistics 
may rightly be called the science of averages) I 

?rff I R^T ^"wr ^rtf^fl 1 % 



ft ^t 



(probability) =qrr 

(correlation) % fw?cf | ^ ^|f ^T ^T ^cTI f% 
% W I I "% TR^NT % 

ferr?ff 

% 



(science of counting) | | 
(science of averages) 3$ 

FRcTTC ^ftTO ft ^cTT | I 5f|cf 

^f^^t | ) 

% i%^ 

(estimation) 

(Bodding- 

ton) ^ ^tfe^r sft ^m^^f ^fk ^^Tf^T^ff (estimates and proba- 
bilities) ^T f%irr^' ^ R TOTITO f^T 1 1 3?r ^ qR^qm ^r 3^ 
% i%^l q^-f^r ^ i%^R ^% ^t it 1 1 

5Er[r Wt ^ m ^tfeqpsft ^ 133? 



( all-inclusive ) 



it 



I ? ? 



(King) 



T t ? 3 % 

f% '^tfwft 3RT 

^TTFR: (Lovit) $ y$$mw. Weft I 
(King) % ^r^H ^rfe^^ 5ri7^7{T (enumeration) 
{ collection of estimates ) 

it^^c srerirsff (phenomenon) qfi; 
(\fcRf ^T f%ffR I I (The science of statistics is the 
method of judging collective, natural or social pheno- 
menon from the results obtained by the analysis of enu- 
meration or collection of estimates) | srffts, (Lovit) $ WTTO 

^ffelMirajf % m%*$ (collection), 
( classification ) ^ ^I^^^T ( tabulation ) 
T (phenomena) ^g 



% 



(science of statistics) 
(exposition) | I 



laws) 33 






% f^SFcT ( statistical 

% 



% 



fer 



% 



WT (Divisions of Statistics) 



(?) eifWEte $(%*?! (Statistical Methods) ?% 
% wrfw ^ c^gR ffr sn% Eferr % f%w (rules of procedure) 
WflFT fespflt TC fRR I%^T ^MI I I ^ 

% HW I 
(Applied Statistics) 



I \ 



I 

escriptive Applied Statistics) 
cf ^TfqJit q 1 ^ T%=3K f%^T ^IM t I 
(Scientific Applied Statistics) t 



% cr^TTcf^ c^T^IRsp gtfef^ % f^TC ^f tcf ^1^ 5RT 

i%qi sFTcrr t ^t 1 jjrfgRR (forecasting) 

i 



^rp^sr ^ 5^ ir^T^r ^T ^pffr% ^ $ i%9Fff % 



(Statistics & Mathematics) 
( applied mathematics ) 

% 



( essentially ) ^Tf^ffcr % ( mathematical concepts ) 



STOW 

(Theory of Probability) < 
% fl^rr WRT ^f^r 1 1 ?*r TOR-STO^ % 
*ft |tr 1 1 3snfW?f sRfeft (Bernoulli), tffe (Gauss) 



( Mathematical Statistics ) titi&Fft wh; ^ftjcr ^frft ^T 133*. 



t IcF" ^RScfT ^T^^ 17T 

( empirical science ) I I 

I f% 

T ^R 1 ^ I I 

% ds 

^T^fT Tf =T ft, ^ 5RK 



f I 
sjfc ^r^W^l (Statistics & Economics) 



<TC ft^T | I ^f^T T%^t T%^T?cr ept Bq^ K t ^fRT ^TfcTT 

I 

% 



% ^P: 



(economic policy) f^iT^T ^^t Weft t, st 
ft ^fTcIT | I i^TT ^SR^cT ^T g^ 1 ^^^ W^F 

i%tr 1%^ ^ IT^T^ ^t 5nf%-^r Prft^cf ^TT ^f^r ^rff 
(economic planning) t ^t R^TT 



$ o 



^rfferal: ^ sr^r itcrr 
( Econometrics ) ^t f 1 ?W w ? %M % 

if ^^rr ^cR t ^Tf^ ^" HT'Ht^ (measurable) 



1% 



f%cRT 



qr 



f%^TT 



% 



RIT 1 ?! if ^^T t I 



5r?g?r 

W^ q 

(fro ^ ? 



tc ^frr, 



^fff, 



(B) wfepft frr^f w f^iTR 1 

(*?r) erfeJT^t *HJI^. 

(IT) sffei^ T^JJI ftrmH 1 1 



5aref % $5 ^ fes Rwr 5fi SRRTT 1 1 
J 

5RT 



( Functions and Importance of Statistics ) 

% 35R ( Functions of Statistics ) w 

fftft t f% 



SR3T ft 

ft ^%^7T I cf^ ^t ^TPT ^fk g^lT^R 1 ^ ??T t Wcf ^ % 
3? ept ^f^ft ^T ^T^T TW ^TTcfT | ^% ^TF^ ^ 
% M t c[^fjTT I f =T ^q! % 

?IT ^^ gi%^T^T^ ft 
( complex ) sfh S?(%SB ^^rr $f 

q- Jf ^q-R^TcT (%^f ^TcfT | I 

ejrcr 



*IT 

% ^T^- g-iro^T WN^ WT^R ft ^^T | I 
ftt I, f*R^t ^ S^RI^r^ M if ^T ^^ ^T^ t OT% ^tf 
|% %$Ftifi ( Index numbers ) | 

<rft*RT ^m ^W ^t ^^[^ ( concrete ) ^ 
I ^t^T^t ^ ^CFTfrT ^T ^ T^ f TO ^ ^WRRT ^fcft | f% 



II 




% 

f^cTT 3ffh ^RR^cRTT |& W ft 'Sfffit f .srf^E % ^P(P 
| ^ q^ sq-ftRf e$ WRRT ( bias ) ^ q^TO ( prejudice ) 



SIT 3R& f I 1% 215^ 5rr|; ( Tycho Brahe ) 
mft % %^r^ ( Kepler ) ^ $$ $ ^^r wf^ 

% H[^ it Rq-q R^^r ^ i ;% RW ^T ^t RWT M^ (deductive 
methods) % ^f R^^T ^TT ^W ? ^Tffe^ U^T ^TfCcT i%^T ^IT ^T^IT f I 






, 

( forecasting ) 



% 

cfT t f% % qRcR^ ^ff cf^ w^M^ ( acci- 
dental ) 3T ^^ ( significant ) | 



1% tf wr^r wsfqRj ( significant ) | m 
^lf | 7 f^f% 5RT RR^R ( eliminate ) 
t fra% faws ( error ) 



(Functions of a Statistician) 

(statistician) % 
123? 



? Y 



^T *IT H 

5RT =3Ti|% $ w%r^ $ ^rfSRfcr (bias) qi TqjqRT (prejudice) 



I r%^ ^t ZT^FR % ^rft^cr ( biassed ) ?u 

K t ^|t =Tf 

it d ^tfew^r '<l"rM 
^w^ %^ ^ ^ ^i 
i ^?Tf isr ei^jft ^r fa^^rcj ^^^r ^tfew ^ T ^T ^ |,. 



% ^^ 
^sr ^4 % ssrepfa 3 ^tfeRJ, ^?tcf w$ ^ ^ ^ % ^qq 

^rr ^% I ^tfew ^r gkr 
I, $*T qrR^^ff W f^^^^f ( interpretation ) 



IJTO^T 

I % 



limitations ) % 



WcfT | % R^c^I , 

I 



cR 

^FRg: 4 ^ ^ mm 



f^TT f%^t WRtf qj ^fqi^T % S 



( welfare ) *$t ffe 



ife ^ 



R^IclT, 



(competition), 



( extent ), 



Wife ^TT 



% M^ ^^ ( survey > 



% OTFTf 



cFT | I 



, 



f%e\ 



ftrar g^twr 



ft 



BTR mi ^ff . 



economy ) 






( planned 

Jf Cmr |, 

tft HIT W 



% ^FSRT % TOFT, 



( business ) 



ft : *m 

% 



( Business Statistics 



'? ^ 



piecision 



M^ gTRp w*W 1 wf ^t ^rrt, Tif^w % ^= 
rMl % ^^r -t *lt ^tfwst % {%=TT =T| f ^?TT ^r ^^r \ 

q-qr ^fjf^ I I 

:RT ^^ f^^Rf % frraref ^ ^idf ^ 

! 3c^ RIR ^T T%TH qFrarat ( assumptions ) 



T% 



fr ^t I f% 



% ^f % 

3 ( meaningless ) f \ %a sfe6i?q- ^ JT^T^FCIT ^ ?rf?rrer t 

5 ( Econometrics ) 



( Mathematical Economics ) 
( Mathematical Statistics ) 

f% 



% 



1 1 ^ ^ * jife 

l* | | 

( Limitations of Statistics ) 



% 



M % 

(quantitative) 



f^TT SIT SSRfT |, q* 3^ ^ JraR ^t Tr| 

( arbitrary ) gtft ^ l^r%^ IgrfiRF =r|f ^ WT ^ft 1 $ 

wrf^ 



(groups) q"! *nwf ( whole ) 
T ^^% ^ qiw*? f^r% ^ | 
t qfc[-j%%^ % f%q; 1 1 % ^^f ^T 
(central tendency) fcflt f I i% 



f% 



t !" ( head ) *tt ^ ( tail ) % 

^IT ^T^cFT f% f%^t ^5Rf ^ 1^ ^r^TT 3T 
1% W*K T%^T ^| 5fR ^MT 5TPT ^T ^T^fT ^R |^ (head) 
R ^ (tail) WFl" ^ ^wn^TT 
qf 5ft sqxp f%^T ^ ^cTT I f% ^ff^^t % 

& 1 1 wv RSTRf^f t fq^T OT^ ^ f%qr 



HRT 



53^ ^i *g& ? % ^W-TF^ I ^r i^^r ?r?f 

| | 
3ft fq-=3R 



q % 



err 

(Distrust of Statistics) 
SB! Ri%cr STFRI *t f^n rar t 1 

cfR XPBR ^T 



T t 

^^ I ^ f^fT W I I JTPT: 



( manipulation ) ?^r JRSR T%^ 

q? 



ft ^TTcft I I V(k ^ IPBR ^ 

^ f 



% MIT ^RcT ^fr%^ W^^cfMf ^t ^T ^T TfcfT I 

t I ^ ' 



Nf % |%ir 



T 1% 

^IcT ^R'^f 5KT #TH TR^IRf % ^Mcf 



*There are three kinds of lies : lies, damn lies and statistics, 
B. Disraeli. 

[Statistics are like abenists they will testify for either side. 
Ltf Gnardia. 



ft STMT I T% 



(tools) 



% 



(opinion), 






'sff&ret 



(i() arfew 



fsrq f 



Oicfa? 



% 

% 



wi^r 



$T 

( Planning a Statistical Enquiry ) 



.l ft sfoa ^ ^ 5f fc t SRT st 

tf ft*TT ^T SPRIT I SRSJ: ^ Stfwft % 
(fundamentals) 
(compilation) <R f%^K ft^ ^rrai I ^ WP-^PSFT 

J ftcll I I R^ ^^ % 

; ^ ^ f^R 5 ^ 1 1 

ft 



t 
(v) qf^gr-qf^wr (degree o accuracy) 



(Object & Scope of the Inquiry) 

t |t 



( 
(collection of data) 

iw ^rr ^f^i 1 1 qfwiT 
ft 



(accuracy) 



ft ^TW T% fqiSR % ^ *T ^TPof t 

% T%Q: ^rg^^TR ft STPJ i 



TfcFT 1 

(Planning of the Investigation) 
% WT 



^ff RR^cT ^^cTT I T% 
(universe) % sr^F ^^ % ^ 

^ ^ST % sriai^rqt ^ ^ % ^T ^ 5 
^ ^^ % srft t ^R^ w^ R^ ^fi ^^ 1 



T% ST^J 'Bra^T % ^K % W^f^T-^T^f^ ; IPRT 
T% ^5^R ^fTRnqT-^rg^ra (census- inquiry) 
TRT I I IW ; OT^T ^|cT ^H ftcfT I, <R 

^T ^Tfwr 1 1 i^r% ^rtta ^ ^ff ^ 

^ ^f% 5Bt ft^^-Wg^R (sample-enquiry) 
Jf 5RP3Fn (enumeration) ^Rgrarj^^ WR R^^ 
I ^Rft^ m^T: l?3t ^m ^t ^RFIT ^TcfT I t ^ ^W ^R% (bias) ^T 
i%% W % W ^RrfT ^TR^rf I I 

T^ ^f RR^ f%^T ^cfT I T% 4i%^ ^THsft (original 
data) ^i mm 5 R3T t *n ^r CRJ ^Ti%cf ^rr 3q<^*f ^TH^ % 



% 

f%;qpp; 



ft it ft 
ft i f^r 
% ^rg^fR f%^ rr ^TOT 1 1 ^ OT5W swsft 



(primary inquiry) 

fR ^t erf fg^^r^-^oNrR (secondary 
inquiry) ^^TcTT | | 

W^^; i%^t wgw^r t qfft^cn (accuracy) 
ft ^t 5R^-^rg?rair[ (direct investigation ) 






(intensive) ^ps^PH W?n ft <sfo ^fff iwr-r^g ^t ^ ^ % ^T 
U^T ; 5fRI WI ^% I W^ ml gjRqr ^ i%^cf (extensive) 
ft ^t ^Tf 5fFT: WXft ^Tff ft ^^TT f% 

?r 

5KT 



1% (hearsay) qj f^PT-^R3 "K W^c^ ^q % 
M % qitFfk cRT 3ft ^f TWT % 'OTRjft-^JH^I FFTT ^TcfT 



^Tf T^^TcTT % ftcFI 

% ftcfi 3 

^R (indirect inquiry) ^fl ^f!cfT | | 
(survey) ^T f 



% ^^ff % qjg ^ER ^% ^xr^ JITH T%^ ^if ^rr ^r%^ ( inves- 
tigators ) $ ^fRcTT ^ 5fPT I T5?ft R^I% ^ WT % 5ER[K ^ 

ft 



f 

% 

ft 3t 



( initial ) | gt 

^?r % 
i3; n 5 



% ^^^ % 



, ^ 

( Statistical Units ) 



R^fT T%^ f^t % R^ ^ d IrfSFT ^t ^4 ^Tff ffcn I 



? 

% ^ 3" f WIT ^ ^T % Hft^cf ^ ^ % 
| | ^RTS^ ^T^FT imw 5R^ % ^ I^TI^t spt ^T 



cfii% ^ ^ 1%^ JT^R ^T ^rrr ^T ^1 srk ^irpft % 



% 



rT^npr^ t : 

( It should be specific 



and unmistakable ) 



, 
I 

^f ^STT^I^ ft^t ^ifft? ( It should be homogeneous ) 
(uniformity) WTRfFl I i ?3T ^f T% t3?p $ ?^if ^i 
RRvT ^ r 4Q?f ^TT ^pqt t RR^ M % fell ^PT | ^ ^^^ % 

i firf5ra TOR^M % 
SRI ^TH TR^IH vrrf^r^ d ^ 1 



?*fl 5? 

( subdivide ) ^% ^TT I%IW f^W cpt ^pfl- ^| fcfrjf % ^q- Jj- 



(stable) ^fk snrrfer (standardised) 

^TT I ^t r%^ RR^CT ^^ % 



(appropriate) 
(ascertainable) 



f^T ^cfl" | I 
^TCHcRT ^t TFTf ^f 5JT2T ^FT ^8RTf | : (?) 

5?rf (simple statistical unit) ^fk (^) ^1!; 
(composite statistical unit) | 



133? 






( Degree of Accuracy ) 
R^cii sira ^?n wcq^r ^fer I 
i% fSrtNpp ^ ^f% SRI sq^fj sqr^R ( instruments } 



R=q[R SR %?fi ^lllo; T% 
SIR t wt |o; R^IT % &rw (sources of error) 

% 






I ? 



( Collection of Data ) 
% 3rr^Fr % q^^rra; ^TOft-^fjipj rnw FRT 



t % iTRTO) ^riH^ (primary data) ^l ^^ SR^T I ^ fcft$ 
(secondary data) ^T I Irar ^cfT^T ^ri 5^1 | 5 uinr% ^nwft ^ 
( original data ) ^T ^^r ftcfi | ^ ftcfk^ ^mifl' Jf 
( investigators ) 



! (Collection of Primary Data) 
R^j%f%cf "<tra^f % 1%^ ^r ^^cri | : 
(Direct Personal Investigation) 
R (Indirect Oral Investigation) 
5RT (By Schedule Questionnaires) | 
(v) ^^Ffk' Jri^q! ?RI (By Local Reports) I 



irect Personal Investigation) 

1 1 ^ 



^5 ^W^RPi ^g|^f SR- ^l^r | ^ q^- 

5t WR 
r % c^ 



(intensive investigation) .$ ^TH^^cTi ft 
{ extensive investigation ) ^rr it at 



ft at 

cjcr cqfrfiijf % s^r tj^pR sffa 3^% f%^5 (cross-examination) 
SIT Sffiat I, I q^ ?OT WfcT ^T^TFft ^Rt 

HT<f sf^TT ^sgt (informers) q^: R^ ^cFT I I 



t cfTI 
f^TT ^j&TTt % ^ ^ I 

fe^-^Rg^R (Indirect Oral Investigation) 
*f ^Tf f ^TTF ^f^ ^T OT^T ^"ff 



lT % ^R^ ^T ft 

1 ^ (ira[ 
q^ft 1 1 qfqft ^5 T% 

1R ITTHTf%^ ffRT ^T ^BclT I I 

T% 5^ R%q f|^t % 

I ^P $ 3RRI%^ R 



IT ^ I 

^q^M ^T ^q^TtTT ^MST^TRg^ ^|^nFr-^ftf^T (inquiry 
committees) 5RT ^%^M % 1%^ ^ITcfT | I 

f 5RT (By Schedule Questionnaire) 



, 



% 



1T t 



T^ ^ eft 



% qfzrffr ^RT fer ^r^ 1 1 wn?: wg^TR ate ^5 

^t J7M ft ?f^t | I ^t *ft 

if ^ ^R^cn ^Hcr in i 

R?T% ^Tf^ ? ^q? g?r ^^rr ^rr I ^t 

^T ^lf|^ %i% ^R^^I^^cFT 5f=ft ^1 t 
1% ^T3^WR ^T ^W ^^\ I 



RT % I?I% 
( error ) 

I 



( enumerator ) 
t 1TR ^3Tftcf ^t ^TTcft | | ^ JT^PIpF ^^f ^t ^xT^ ^ if 



I 

% qw ^ t 






i%tf^f ^ % OT^nl | 3Rtf% 



ft 

*r ^1 i i$ 



ft 

tft f^T- $1% % m 1% s^SfR ft 
ft SRW SET 3^ 5f5T ^gcTT I I 

'4t 



f% 



^ ? t ^ ^T W%^ %^[ 
% sre ^ ^ ^ % ??r ^ 

f % %^[ ft ft% ^T 3^ I | 
( By Local Reports ) 
T (correspondents) 



t 3?k ^?t ?r%^ % ^m ft^r 
( biassed ), 

It 
% W{ S ^T WOT I, I^T 

f%% 



sppft *^^r ft ^ ^H^n UWRM ^WHT ( representa- 



3 O ^ff fejcfft 

tive data ) ^ fW ^ ^ ffcft 



( Representative Data } 



( census investigation ) SRT IW 
( sample investigation ) mi \ sfrWFn 3rg*faR if ^fpr % 
% ^ 



T W t^ at tf 
It TOit I | T% 
flcft ^ | 3 mil ^% ^I^^isq-,, cpqffr 



I% qf^ R5RFT ( sample ) 
JT^R % 5?TT ^ Wfftl | : 
(?) ^gR-f^k ( Deliberate Sampling ) 

( Random or Chance Sampling ) 
f^T ( Deliberate Sampling ) 

w ?n^Rr WJT if % p 
sgqff % SWR CR qRsnr 1 1 



rr war I jik ^ qr%m *n% ^f ^ ^ %^ Cample ) 

^T WS^f^H if q& ?rPR f%^TT 

ft at ?^^r OT^R r%qT ^rr ^rwi 1 1 



1 ^flre: 



1 -r * ^r % . 




(Random Sampling) 






T 



^ff SRT 



( mechanical process ) 

t S3) 



t%*n 



I ^ST 






T^IT 



% 

( substitute ) =rff WT ^rf|^ I 
^ ^T% ^T ^% ^?T ^rw ^5 I ft ^T^^^T % toHf (errors of 
estimation) ^t TRFH sSk TO^TR % ^5^ (significance of result), 
(theory of probability) 5RT 
^TT 335T I 



1 ft 



f^ ft ^cfT I f% 
^^^ t i% 
ftWT W I I 
( selection of sample ) 



^TT R 1 5F 
(selection) 



(law of 



statistical regularity) wh f^ft Hfps a?3T fSww (law of 
inertia of large numbers) I ft^tfo 



( law of statistical regularity ) 



srfiw q^f ^ ^f ^T tf%^ t& % (at random) 
(selection) fw^cft%^| ^9fcT?r ^nw % 2^ c ^ ^ffe- 
t ^t Wf 3K 1 1 q^t 15 f^ R^^f ( IT 
(at random) f^T ^ | w^H H2T % 
% l^T ^ fl ?R^; (chance) ^R t | ^ gKi ? ?? T% 



iq% 5KT ^t ^t ^Tf5t ^^TI * H t 
(law of probability) 41 ^1 nc[T t W 

I 






;|%^T ?ooo ^R ^RfT ! fTi gt $FPRPT ^oo SIR |^ (head) 
Sfl^ ^ (tail) ^IWTT | 

t % 

^T=r R 

^i (law of inertia of large numbers) 
qf FFW ^tferaSk FRftcn JWJ (law of statistical regu- 
larity) 3tf ^TOM (corollary) | l 



T TRHT 3 ^ ^"^FT^ ftcfl t 1 

1333; KW ii" ft*iT ^ f ^ if ^^ i%^: ^^rr ?r 



a; f ^r qc ?r w* 
? 

w 

ff 1 1% 35 i%^ ?| wpTRfiw (periods of time) 



% ^f^R ^t mf^icr % 

^ ITT 



% 

133? ^r % 

% 



(Collection of Secondary Data) 

U^T ^^%cf ^Tfpft ^T OT^T fw 
% t^liws srf^^ff (private reports) % 



^TT ^^T I 3T SRirfef ^TT^t (published information) % I 

(chambers 



of commerce) m^i^ ^rrf^ % 3q^r 5 ^ ^ra" fiTRRr 



% 

(official publications) 

% f%W^r RWl! 5RT, ^K^f^I^t (municipalities) qi 
? ?RRf^T ^?ra! 8KT, ^SPR 5RT R3^ ^^^FT 
(inquiry committees or commissions) SKI 

I 

) 
% i 

) q 

) ^r^ 1^%^ w^^ ( i% 

(Using Secondary Data) 
at 






r 



% ^ r% sR3<r sroft ft^wfh dk siFRifti 5 !* I cm ^ fM 

T% ^ 

% ^rg^f 



^ R=SIT?; 
r ^ ^w fi ^^r ^HR TRqi r ^ 



t % ; 5Tf^ WR^T % sffe^T % ^^psfqF f 
^cTT | ?PTIT ^T?f I 

Tpif (Necessary Attributes of Data) 



) f^^Ffrqrar (reliability) 
) ^rg^r^r (suitability) | 
) ^?HW (adequacy) | 



(Reliability of Data) 



f ^ 



TOT ^HT 



TORT R^=^f f%qj Ty^T ?T 



% fm: ^npft ^r JFER ^ fMt =grrf iz; 
fsre; ^t 



(Adequacy of Data) 
^rf|ij: arf% ^ at 



i 



i SIN B^ R^ % 
fro R5Ei^ ^ ? 



( ? ) 
CO ^ 



() fiste 



( v ) fag- sRrar if gFpft-^ir fcm TRT n ^FTT 3% WIPT 

T ^3T I ? ST*if?J. ^ Friers ^f I ^fto; f% 
SRT ^RlMpTf R%*r ^7 % IRTRcr at ^ \ 

(Suitability of Data) 



mm 
at 



' 5.? ) 



(=0 



) srfwste 
t 
i 



% 



i ( 



;e%qr If 
( sft 



if 



(Editing of Collected Data) 



ft ^) | fiR 3ft ^ffeq^r ^qf % 



ft ^dt I f% 
ft ^TPT, ^ ^rr ^iipit % ^^rrspr % 



f I ( ? ) qftgfssrr-crRfflTO (degree of accuracy), ( ^ ) 
T3^pr (approximation) cf^ ( \ ) g|^^ RWI (statistical 
error) | ^T dHt ^rat ^t f^-f^r ^H^i^T ?n^RJ I I Wcf: 



(Accuracy) 

R^^T (perfect accuracy) ^T W'4 qf | f% 
I^TT ft SRH^T SIH I ^1 ^RI^R 1 I I f q 

% m =Tff s 



% %r fFT ^T^W ^t Sf^TcfT |, % ^ ^fff fit I 
% gJ^T HTH ^t ^r| ^Ifpft 3ft TJ^T: qfigs ?[ff ft 

51^1^ 1 1 



T ( physical sciences ) ^ Jj- ^f f^f j%<r 



at 



ft g^ f 3 q- fft g5f 5[j|f ^yq^ % sp 
*Tff t^frCT WRJcf Hcf 3T ^jcT ^ 5f| 



f 



TT 



f% 









% 
f I, dt ^T 



TOT ft ?TN 



( crude ) 

( tolas ) e?5T 



^Fpft 



it ^i5?r ^fk fq^sfqj ft 



( estimation ) 
able ) qrftgsar ft ^ f q 



1 1 ?PK ^ WRFT 



| 



T 



^cft I 



it 1 1 

q?cft | 

t ^t| 

we 



( reason- 



f^r-gfl: 






ff 



I "^tfWfft $ i%^f qR^cfi eft 
tfcft ^ft^r | 
( Approximation ) 



); 

( 1%, ^R ^Tf ^ ^^o % 



ft 

% 



, 



f%?i% ^rggp; i% TOT 



, l% 3 3PR spit 

^W?TT 

| T 



, 1%, ( ^ ^ra $ ) 



Vo 



JT^R ^T" Sflflft ^ f^ 
; ^T^5ft % 

t f^f^rar 



KT % 

% 






i 



^IFFTT 

STFH ^fiit^ I 






1 1 1% : 

*M V 00 ^ STFFIT \ 



5RT 



, o 
fPOT I f^ 



WPJt 



SKI 



| v^-o So- ifto- 

v o< So- 



f% 



dJ % 



fer 



f% 



(lower and upper) 



I %%, 



at f^% *wt ^ ^<> o f^rr 



iwr snr ^^ciT 1 1 WCT: ^ f^r^t ?FR tftar 

I ^T ^qt % ^ ^T^IF^cT W^ Rn^ % 

^rg^^; % 
% 



cf (approximated) ^f ^T &pfri S^T, : ^ : rFr; 
% m sron 1 1 gt ^r^THt ^cenft ^rft 

% WWT ^ t ^f f%WT I ^f SMcT, JWT%[, 



rt ^Nt % ^raMt ^T 3 ^=r ^ 
1 5^r srfcT ^t q^r f u^^; % 

t TORT 



(Statistical Error) 
ft t i%iR (error) 3rk ^ra^t (mistake) T^ 



(mistake) m ^r4 5tcn I, W^: (wrong) 
31 WTT ^ "^T ^^T I 3! ^T^T^Ti ^ ^ft ^ 
I ^ 1% ^ 1%^ ^nr ^cfF | I ^ ^ T%FT (error) 



T%RT ^rra 1 ^ ^TT w 1 1 
i^nsw d^ 



(Errors of Origin) l/T% ^ 
^3^3^ ^m\m ^\ ^rmcT (biassed) 
(inherent) ^ C 4^cn^ 1 1 
(R) STI^r^-Rraff (E-rrors of Manipulation) 



(Errors of Inadequacy)^ 



r i 5&T 
R 

3f 



% 



(Absolute & Relative errors) 
cl": (absolutely) qr ^r^fcl": (relatively) 
^ f^^n (absolute error) ^ f 
(relative error) | 






value) ^o" 
I 



(estimate) 



(approximated 



iFf 



{Percentage Error) 



ffTT 



(u) ft 



% 



" % S 



WW it 

% ^ 

inn 



^TIcIT I 

% 



eft 



c, fao^^ 

Absolute error 

and relative error 

_ ^ 



C, (U-L), q* (u) % 
3FR Tc, (uj, q 1 (u) % 



( positive ) 
( negative ) 



l%1nT (Biassed & Unbiassed Errors) 

SRJR % ftt | I 133* t ^fJRcf i%'R (Biassed Errors) 
(Unbiassed Errors) I 

f % q^ror ^ 
% ^TW i 



(cumulative) ^ | s ^ 



133? 



f^RFft W^ 5[R 



^FFTT HcFTT 

f%cFft 



i%o; 



ff ^IFFTT I 



?jt 



% 

Rcl^ft 



, err % 



i% 



ROT 



f^PRf ft^" f 

( chance) ^R^f % 
(compensating) 
; 3cRi ft 

" f , 
ftir ^r^r^^ 3?RT ft f^rwr 

ft ^fT^F 

f%q ft, 
ft 






123? 



ft, 



3?t 



ssrr ft ^t f^ 
(cancellation) 



sating ) 



% 



i% 



RWI 






( ^t ^RFra 1 rm 
I 



| ) 



I f^ tcT, ^TPT 



% faf ftq"% go; 



% ^ 



rm 






*Et 



Prsr 1 1 



(^5) g 



tl 



() 



finsm, 



ffc 



q, 



*f 



If ^i*^^: 
i ^ra w w % ^ ir 

? (quo q, 



$T wk 

( Classification & Tabulation of Data ) 



% HTH BFflft ^fcT ^?1 TJT% 

I/ET ^i *r I%^T SJHT w^r ^rff ^rar ^pnf% arscnr % 

f% 'WR SRg ^reOTT ^3^T % WT ^% ^ I 
% 1%^; ^ ^IT^RJ I f% W? ^ff^cT ^7T 

f 7f?T eft giftr % ^ ^pSt ^ ^T^PT (tabulation) 

\ m I^% ^5% 1% 
(arrange) 



(classification) t 

% ^^K 3RTT ^ ^ i^TF ^M | I ^f H^T^ % 

cr (arranged) 



( W^R ) ^ ^ ^rpfr SE^TT 1 1 ^cr^r ^ ^^^ % 

Jf W STRfr t i% ^ ^^ ^^RFT 1 ^^ ^^^1 ^liTI ^T ^5?n 
SRFR Jf^gcT ^ it W$t ^R^fl^ ^ff deft, q^ ?^% ^t 1RWI ffdt I 



t % 

T % ^rg^R (by attributes) | 
) ^nfrxTO % ^pft (by class-intervals) I 



( Classification according to Attributes ) 

^{% *r ^ 
fR*r ft 



JR5R efiT ^RffsR^ R^ffl ^ETRt t t 3T5RTT , 

(simple classification) ^T gTS-^SR-^fi^W (classifica- 
tion according to dichotomy ) ^?TRT ^TcTI I I 
E^ % ^[^ TQjf ^R ]%xfK 1%^ ^IcTF ft ^t 



r%qr ^FT ^f i ft ?FrnT=T ?rk ^ ^i S^^f f ifflf ft ^t q-f ^ 

s^r f f ^ I ^f w^w ^ 9^ ( ft. ^i 5^ ) 

I 3T5T ^ft OT^T % ?RP? cT WFt ^Toft ^ 5P^FT % 



I, 



( manifold classification ) 
| I W ST5RR ^1 ^1 5^ W^^T ^spFniFTT % lrf^%^ft t TFH ^fRTT | [ 

T 1 1? ^r^^; % 

WT 
5f?pTT ^T 



% w^W ^, ( W % ^"3^R f%o; 5713; ^fN^r ) 

^FPPW ^Tf f I ^ ^T ^ ^ ^ w ^ ^1%^ ^ ^-^F Fri^cT ft 
5TFT; ^Tf ^feW ^f^cT ft^II 

w^r tt ^t fwt 

ft 



% ^rg^R ^ siit f, HR^cicFT m 

ft ^3xTT f | ^T ^TWf ^ ^fff 133* 
I 1% 



gqjt % 

% ^ifN^pjr ^ t ^ qicf ^T nq- ^?^TT ^nft^; f% 
^IT dt %&> ^" % ?Rpfer 



( Classification according to Class-intervals ) 
^if^cR; % 

T%T 3^f ^ ^q^T flxTT I, 

5TFT, ^=1% ^T^R WTK I ^^ ^F ? % ^Tf 



^ff % 

5% ^TH^" w f^^rr ^sncfi 1 1 
f^ir ^fT^T 1 1 



(precision) | | ?*r ^ ^npff ^t wnfscR' (class- 
interval ) ^ 1 1 WR % ^TT V-v', v'-y,' ?qk y.'-^' ^f t ^% wfe 
I 3Plfas ^t f^RT ^ ^TT^t ^fTTC ^-^\?TI^ ( class-limits ) 
I ^3^ ^15^5 ^ ^^ ^ $ ^f-^HT^ V ^ v ; I 1 ^RT ^ 
% ^ % W^f^ ^t 3PT-i%^FK ( class-magnitude ) ^ | | 

?' 1 1 Hc%^ ^ % 
( class-frequency ) 



% 

iw ^icfr I f% xr^> ^rofcrc ^ T^-B\^T (upper limit) 
(lower-limit) ^ ^T^R: f%^FTT K^T ^rq 1 ^ I i^r SIRT 



sffc 



vg. 



% 






% 

t % 



3T1H ffft I 
- ft 



I 



% <fNr 

^r ^r^Fn ^ 

( integer ) 



( mid-value ) 



% 

% 

trf 
1 



T% 



f % 



( origin ) 



f^R^cT 



ft 
% 



sit 



irk 



WIT | 

OT m % 



% qr % 



% ^>F^ f^ wn ^m\ 1 1 

t 1% T3J 



% 



1% 
1 SFFRU 

(exclusive) 
(inclusive) 



S.-V 
?o-5. 

?"^ 
Jo'o 
>o'o 5,-c; ?o % S v^s ^"^ 



f^ra; 



(continuity) ^ i% qicft i f%*^iw ^T^W % ^ 



^ \7 *| \J > \J ^ *^ \ V \ ^"> 

?o'o ^.-c; ^dL 



" i i pn I ^T^K ^-reiR ? 
I ^ ^f ^flnr^ RR^cr ^^t f i W*R ^^r-q 
WO": Vi-V^ V*,-^ ??'^-?V^L C^ I 

% 



( Class-limits ) 



(Mid-value) 



(Frequency) 



5/y-s o-^ 



(Tabulation) 






1% 



(tabulation) 

^W^B^ 
(interpre- 



tation) WT?ft % FFTT ^fT ^% 1 *!H?r (Bowley) % 






I 5 ^K^^T ^ ^4 WFR ^ I 

% i%^ pj ^FT^IR^T (precaution) 



W^ft 



notes) 



(accurate) 



i 



(foot- 






ft gwr 1 1 
, 



i% 



q^f I f% 133? 



5RT ?t snrft xrrit% I wwit ^FR % ww; % 



sn 



% 3F5pffa wi% rrdt ^fTH^fl" ^ ^RT % 
^?wr ^ % wz. 

f%*TT^RT ^ ^"T^fi" 



PT (heading) ^ ^^1 Wf ^ ^ ^ OTjftW (sub- 
headings) sft ^^rr *f$ ^ i ^a% ^^ifcf ^rflfo a^ 

% |t^ ^j{|% 1% 



% 

I ^f cf^ ft ^a% ^TOF[?T (approximation) 
"^ % ^T % ^^ ^T IT^oT qW?TI 

( details ) w f%% ^r ^j 1 1 



53% ^t % ^ ^^ ^cTT ^t f I 'QTR^fl 1 if M% ^ % 



(Different Types of Tabulation) 
I % % {%cFt 



I ^f 2TW; ^-3^T QK^R (Single-Tabulation) 

1 1 ^r pi % ^ft $ 



f^FT%f5$ET srwjt % ^r%^ff eft 



f *fT^ft % 
t % ^nfl % 



I T% vsM, cifra^f % t 






cz 



^rr nr ^cfi , 123* 






(Double-Tabulation) 
f)" ^ % *fK ^ ^t JT^ ^T ^rl^ ^TT^TT 



% 



V 

s. 



f^ f^FT 



1 1 



?% 



I, 
Jj^ff % ^xf^; ^tTT t I ^ ^t TWT % 

f , 



% 



^TcfT ^ I33J 



srpff 



% 




f 



( Manifold Tabulation ) % 



^^^ 



tft 



<E 



m m 



i^r 



4 



\ 

raf^^t 



(Simple Tabulation) ^ %^ ^ 5^ % fw? t" 
fe^r ^ETR^R (Complex Tabulation) % ^ % 

% ^ t ^cTT^T ^TcTT I ! 3^^ ^ ^t TT| 



SRrT 



f%q 
If f%?r srat ^ 

I 



() ^T^^it^^ t ^H ^m- ^rasrT^raf ^^ ? ^ 

srre q^^jt ^rfCf If 
' *f 



l%?ir 



(y>) srRr sil 



5TTP3I 






t? 



& 



{) 
pr 

(=;) 



(ir) 
(i) 



IT 



(a.) 



t swift 
^tnf 



i 5 ' 



( Statistical Averages ) 
( collection of data ) m 3^ Mt fWT % 



^FTT 5T|cf ^f^T ft ^cTT I I ^, ^ ^ *ft ^FT^IT 
( collected data ) t WI ?RR I I 



(v) 



( gtoup ) 

% ^^T WT ^^T I I 

f^a; p; ^sg ^T RM ( average ) ^ 1 1 
% t^ft ft ^fcT % 5fT^ 1 133) RT^cT ^^T 
fre% ^T^fTRr f%^t ^^r ( variable ), ( iRf% 

) 



(?) ^? 133, ^rf^ ^^T ff ? w^^r ^ % frq; 



f%*fr ^r5 ^T nf^RmccT ^sn I ^t ^ WFTH^ I i% 



; *rrt ^ ^ 
( items ) f 

W^ WT5 ^T qi^ 'TCf H^JT^fT ^Tf ^cfl | T%^ft 'SR^ % 
( unit ) 33% q^f ^ ^| ^t^j- | | |% ^^f ^ qpfcf) 1 q| 

t,!Ff : 



(%qr 

(?) ^%3^ ( Mode ) 
( Median ) 

(Arithmetic average or mean) 



(v) 3^^ HT^ ^T[ jqpsqqj ( Geometric average or mean ) 
( Harmonic average or mean ) 
( ^ v ^ y, ) ifiph qi^r 1 1 
^r^f FIT^T ^TT q^R ^ ^T ^rar 1 1 
( Mode ) 



I ^TcT: 5R ^ ^jf[ WRIT | T% TWt ^^ % 
( modal value ) \so ^o srf^q^r | 3t ^ 
f% ^9f ^T^ % ^TR^fl ^[^Tf ^t WT^T vso ^o jrfww t | 



( mode ) ^ qiwrc % 
ftir I Tff% i%^ WR^rar-^^1" ( frequency table ) 

( modal value ) 



t ^cTT I ^ ^qr^T-^T^ ( frequency table ) 
(irregularity) q- t I sTRcR H I^t 
I ^T% ^ ^JITO^ ( mode ) ^T 
( grouping method ) ^f ^ | I 
( grouping method ) t 



% 



% 



(?) ^ 



qt at 



aft 
^ ^9H%ar ( mark ) 

z ^ if f : 



: <t it 



( Size of item ) 



c; 
S. 



( Frequency ) 



(=5) 



J 



Y? 



i 



u 



* 









^Tcf 



WTcTT t I SIGNS' ^ f%cFJt 

% ^r Jf f^^rt ^ ; 
(Analysis-Table) 



*, 




?li%^fri WR^T^fl WT ^^T 'JpT (size 
of item containing mas. frequency 


? 




^ 


I 








^ 








V9 


^ 




i 


- 


^ 







c; 
c; 


s. 


^*,^ 


< 


V 




- 


' 


1 * 



% ^1 RT ft srTcrr I f% 

n^pgr ^ ^ff (group) t 

(mode) vs |^TT I ^^ft ^TR^ % 



f^TT 



(mode ) 



HFTT 



(modal value) 



(discrete) ft 



( continuous ) 

% 



I : 



gfrn* 



% 



= 1 -- - 

Zt j I o I 2 

where, Z stands for mode, 

I-L and 1 2 stand for the lower and upper limits of 
the modal group. 

x stands for frequencies in the modal group. 

f stands for frequencies in the group preceding 
the modal group. 

f 2 stands for frequencies in the group succeeding the 
modal group. 



z=1 >+ 



2"T0 



^FTT : 



v-c; 



"K c|>T 

(size of item) 



(frequency) 



(?) 



* 



(Analysis^Table) 



z=li+ -- (l2 - 



32-12-14 
= 12 + 1 

r= 1 4- 7 (approx.) 



$ 1 1 wii ?fr f^pa 
(irregularities) ^t w % 

(frequency distributions) 
I I 



% 



% 



(extreme) 



(unique) 



(curve fitting) 



% 



(distribution) 



i% 



( regular ) 



(symmetrical) 



( regular ) 



ding order ) If 

$^r 

q^; % 



= (25) 



=^jt % 



1 1 



ff I W&ft 



^FTT 



f%f% 3 J! 5t ^rff 



(MEDIAN) 



( ascending or descen- 
1 q 



% ^ 



I * 



M= Si 2e 



th item 



where, M= Median 
n= Number of items. 



(ascending order) ^ f5|FT ^? % 



\ 

V 



V9 

e; 



I = size 



i2e 6f(Hf) th em 

/7+i\ th or 
V~T~/ 



item 



= 6. 



SIR grrctr a*? If 



YO 



Vi, 



Vi. 



= /!il\ qi 






f%cFfl" 



item 



=Rs. 53 



(odd) *ft I 



( even ) 
I ^"J 



V5 o 



% fflRTRT 



( discrete series ) 
tinuous series ) 



= Size of 






( - J 



item 



3 -5th item 

Size of 3rd item -|- Size 
of 4th item 



62 " + 64 



= 63". 



( con- 



| I ftft 



frequency ) 



(cumulative 
( f^T ^K^TR^H- ? ) ^ 



( size of item ) 


qTRSTTCcTT 

( frequency ) 


( cumulative 
frequency ) 


V 


J 


j 


<i 


V 


e. 


* 


\* 





t; 


* 


?=; 



=-. t 



e 



= 6 



th . 
item 

th 
or 

io'5 th item 



f 



(interpolation) ^RFit ft^t I I 



% qsqcftf j^rf%t3; | 



(size of item) 



(frequency) 



Yo 



Vo 



Vo - 



Vo 



3RT= 



f ^ 



item 



(^7-7 or 

83rd item 



ITcT 



TOR 







% 



ft ^KfT 1 



where, 1 1 = lower limit of the 

median group. 
1 2 = upper limit of 

' median group. 
f= frequency of the 

median group. 
c= cumulative freq- 
uency of the 
preceding group- 
m== middle item. 

-\jf 2 . ( 4o 3o /02 

M=3o-M . (83 

(. 58 

-32.76 



( grouped ) 



f% 



% ^W q$ ^f IFf ft^TT KT^^^N' ^Tff t I 

f% *M T^ ^r ^ 
% f%ir 5feT I f^^t 
1 



i% rpa^r ^Pg w nflf%j% ^r ft I ^ w^fcRr ^2^r (discontinuous 
series) % %r qq^r ^T ^TH f^r^k^ ^?TT ft ^ ^| SIR f^^ 
WTciT | I fHt" IT^K ^R <3TO2ft ^TfcT (grouped) , ft ^t 
(location) SftrSft ^f fw ^fT ^JcfT I I 

*%& 
% 



1% ^f^r siH4l<i, fw^r ^5 % jrf^rf^r-^ ^IT^ %, ^r?q" ^HPTT^! % 

f%lW t ^xfJ ffr 5n% ^T%Tf 
ftcft f H^f^T ^T^ ^RR t I 



( Quartiles, Deciles & Percentiles ) 
% ^ t ^T ^TT l^T | f% ^f *HI^ 

(median item) 
ffeft 1 1 ^r % ^ 



: 3, . 
(quartiles), ^^TF (deciles) ^: Wcltf*' (percentile) 



WTT 



q^f 



1 3% 



% 



% 



(. 



= { Y (^7 ? ) } 



% 



1 






Q^Sizeof 



D 4 = 






item 
item 



| ij Vh i 
V 10 / 

4 (S.> i 

V ioo / 

( n lLl\th item 
V ioo / 

9o/2, x \th i 

. \ IOO / 



tem 



p 

J. 1 *AV* *. C) 

respectively stand 
for the first 
quartile, third 
quartile, first 
decile, fourth 
decile, first 
percentile and 
nintieth percentile. 



( ascending } 



V9V9 



Yo 
VV 

"w 

V* 



vs-\s^ f 



= vs f 



(qf 



-vs f 



(l+i) th 



or 7.75th item 



= Size of 7th item 4- 
I (size of 8th item 
size of 7th item) 



23.z5 th item 

=Size of 23rd item+ 
J (Size of 24th item 
-Size of 23rd item) 

==48+| (49-48) 






)}, b 



= 48. 25. 

Size of | 4 ( 

v. V 10 

or 12* 4 th item. 
:Size of 1 2th item + 
f (size of 1 3th item 
size of 12 th item) 

(37-35) 
35. 8. 



or 6.2 nd item 
Size of 6 th item+ 
(size of 7 th item 
- size of 6 th item) 




\s 
^ 



= size of 

io4'5 th item. 
= 7-5" 



3 io tb item 
9" 



SRR 



cRT 



where, q a and q 3 stand for first and third quartile 
numbers respectively, and the other symbols stand for same 
things for which they stood in the formula of median. 



qo 



5. c; 



?o 



?=; 



3T 



K^size of ( itl) or 
V 4 / 

i zth item. 

: 10 + 



IO 

io'4 rupees. 



th 



or 36th item. 
= I 4+{ '1^1(36- 

= 1 5. 3 rupees. 



% ^ff ^frc ^T % 

(dispersion) ?f 



t I 



( Arithmetic Average ) 

% crcrf % 



ITcT 



) % 



% 



? ? 



pnr i 



% 



m 



(variable) % 



U 



% 



where, a = arithmetic average 
2x= summation of 

individual values of x 
n= number of items 



% 






= 5 Io 6 inches. 
= 575*6 inches. 



ft 



PIT 



( assumed average ) 
^! % 



' ( deviations of the variable form 



the assumed mean ) 



% 



(***) 



dx 



-s. 
-* 

+5. 



o 

e; 



~ 



where, x= assumed average 
2dx= summation of the 
deviation from assu- 
med average. 
n= number of items 
In the above table 

a=576-f- inches 



= 575*6 inches 



(frequency) 



I . 



% 



% ^RT^ H^T^feira^; I 
(short cut method) ^frff 



(direct method) 



^^T ^T ^^ 
(size of item) 


V 


H, 


q 


^ 


?> 


^ 


\3 


e. 


^ 


? 


sn^cTTC^T 
(frequency) 


? 


V 


a 




i. 


^ 


^ 

i , 


^ 


^ 


? 



ttf% (direct method) 
(short method) % 



(size of 
item 
* (m) 


cfKWKcfT 
(frequency) 


=g^f ^T^if x 
^T^RcTT 

^T(mf) 


(deviation 
from assu- 
med av,(6) 


total dev- 
iation 
SRRT (f dx) 








=3R (dx) 




Y 


? 


Y 


_^ 


^ 


"i 


V 


^0 


_ ^ 


Y 


c; 


^ 


^v 


-K 


+ ^ 


R 


^ 


Y 


~Y 


" c; 


? 


^ 


^o 


+ Y 


+ ^o 


V 


^ 





^ 


-.^ 


V9 


^ 


Y^ 


+ ? 


+ ^ 


5. 


^ 


^ 


+ ^ 


+^ 


*' 


J 


? ^ 










*=^o 


(2m) 




(2dx) 






^TTcTT 






(short method) ^Iqt % 



Direct Method 

a=^? 
n 

_i9o 
__ 

= 6.33 
Short-cut Method 



dx 



n 



3o 



= 6-33 



% 



% 



5J3H ^f% ( direct method ) 



Vo 



*. 



r 

& 

^ 

,r 





E 

3T 
P 



<tf 



._. 

IF 



tr ff fr 

hr 



hr 



-BE 



o 
o 

0** 



oooooooo 
7^ o y ooo 
o > > v* itv nv < 



f I 1 



oooooooo 
> v (V <v* <v <v nv 

Mil 



o 
o 
a/ 



o 
cv 



oooooooo 

TTTTTTT f 

oooooooo 

<v (V av > y ur 5> 



Direct Method 




= 49-58 
Short-cut Method 

'2 




T -r T*T? ^xR 



where, 2fdx stands for 
deviations from the assu- 
med average ~- magnitude 
of the class interval x fre- 
quency, totalled r together 

c = magnitude of class ia 

terval 



% 



;>+? ? ~M 



*W?T: - V, 

sffa gR'- 



SPRRR; 



cf if 



? ? ^o I <=rr 
WE ft rrat 1 



^TT fM *ft ^ % fifcffr 

^ c; $t , ^ 3 c; ? ^ 3 v, 
=^ ffrnr i 

^ 
% 



49-58 



cf^TT 



sfipr ^f^fcf it^t t 



1% R3> 5 



? ? ^ % 



^3 x 



RfcIT I 



% %tar ^^f % 



1 



ft *ndt I 



i%?rft3? *r^ 3i 

ft 31 






" TR^TK 



31 
I 3? 



T 



sft 



ft Y 3fe f, % 



eft 



? 

v 



R?F 



wail 1 



yo 



HFFH 



TO^KFT 

R^f 



\V 



^TT ^cTT I 



sft ^T ^cft I I^T ^^T^f ft ^^ % 
I 1% ^FT ^ftf%% ft 



% 
( WH ft ) 



$00 



^ V 



SRJSPC I 

(Weighted Arithmetic Average) 



f M % 9RT 
{ index numbers ) % \ 



f^^TT SHcO; t I 

I ^t TRfi^icf t fH^T ffl^R ^ WT 



) g^T ^ I ^c ^f 5^1*1*31! % %r ^t *TR! ^t %r % 



% 

( ^r ) ^r ( ) % ^^fR TO *IR ( *r ) qi ( w ) 

^o 3 w. a. 



W ...... w 



w. a. = - 



or 



2 w x 
- 

% *re 



% ^ % 



%(*) 


q( W ) 


( xx w ) 




V, 


jj 


. 






^-v 


( 2wx ) 




mo HO = _~ 


N^ vifv* 


iS; ,?.*. 


i36o 
= -? annas 
3o 

= Rs. 2. 1 3 as. 4 pies 
per Ib. 



^ITcfT I *TR 



mated ) *IKf ^T 
WK % ^RfFFT ^ ff 

( negative ) ffrrr ^k f ^ 



( esti- 

^ I f 



SJ ^ % 
( positive ) % 
( cancellation ) 



cflf 



Rf % JTlHfaJ "FFT 



ft 



57% 






o/ 
/o 






% 



V9o 



qo 



vt. 



it 



o o 



x 



" 



(V (V 

> aJ ^ \S 

(V" ex^ 1 <v 







x x 



(V 



*s 

9 W 






B? 



tr 



o > 



o 

kf 



. 



o 

tl 



& 



S? 

U 

?S 



% 



Simple arithmetic ave- 
rage of the marks o x, y and 



20 



250 



65 and 62' J respectively. 
Weighted arithmetic average 

of the marks of x, y and z 



63_3 

10 ' 



IO 



648 

IO 



JRR 



at 



63-3 62-4 and 
64* 8 respectively 
% 



% 



- ( x ) 



w, a.= 



% 



% 123? 



Vo 



%^ I|t 



^00 



Voo 



TO 



(deviations 

from 
(as. av. 28) 



(dxw) 



Yo 



^00 

YOO 



^ 

(W 



+ 
+ 



o 
+ Voo 

-j- ^00 



(2dw) 



% ST 
f IcFT 



% 



, % 



HTRcT 



w. a = 



= 28 2*6 annas 
= 25 "4 annas 



ft it I f% 
1 1% 



*ft 
?t?rr 



% 






tf ( component series ) % *Rt ft 



% 



% T^f ^ 
g-^q|^ f 







= 








(rates) 3i ^rgrot (ratios) ^ 

if, 



1 



% f^rc; f% 

( component groups ) ^t ^ ^f qpit I 
% 



*GI 



/.V" % W ^fTt ^T% sqlxiRf *$l TOW 



I 

( weighting ) 



q~s[! 

*rrfer 



( GEOMETRIC MEAN ) 

(nth root) 



ifaift ^5T Q^Tf^ ff^^ ^$TT^[ I I (^*rf% ^T ^^T ^^ ^T^^ % 
|) I 3RK ft# 3*1$ % S ( n) q^T,, ^, ^[3, ...... ^ 

x s ...... xa ) f Wk ^ TT ( g ) ^ 



isg Xn 



^ v, ? 

( cube root ) 5t*TT I ^T^fc^ JJPjtxR 



%* 



ft ^^ncft I I f^rc ifcT ^rr ^fgn^r^f ( logarithms ) 
^TcfT t I H^lxix, 3R[^W f^cf^f^- % f^r ^ jj-^q- ^j| 
%2[T ( logarithm ) $m ^ ^ i%^T ^KfT t ^ ffift ^^ 58 
^^n % f%*nfrar ^% ^t ^rfs^r *n ^rn^r ( quotient ) 



I 3W #RT%^T ( anti-logarithm ) 



*T= 



g=Antilog.i 

r ?%: 



% 



- - log 



(size of item) 



(logarithm) 



(a) 



%=^? 

(2 logs) 



*uftH*l' % 



g= Antilog (Io8*i+lo8 *,+ ...... log.*) 



A T 2I ' 7z56 

fl* Ju 

= A.L. 3*io37 

= 1 2? I 

;P?KR *M3> ^3% WORR JPBRJ % ^9% w ftar I, 
% ^ q^t ^r 3j^r 'SWT I ^t 



( Weighted Geometric Mean ) 



% ^f % y^Hsw ^ ^ ^f ^ (nth toot) 



, 



W ...... Wn 



XnWn 



%?rl%cr w qt ai^r wr deft I, 
( Antilogrithm ) ^ ^r ^ ^r ^rrfer 

% ^T ^ ? 



f (Iogx 1 xw 1 ) 



w.g. = . 





I^, v, \s, 



xwo) 



, va, c;, g,, ?o 



(size of item) 
' 



(weight) 
r(w) 



(log x) 



x 



(log. x weight) 



c; 
S. 



V 

\9 



(2w) 



^'OOOO 



(r3r*) 



= A. L. 
= 8*002. 



% 

% SIPT, ^fe 

CT I 



q"c[ ^iw , ^^ JRSP ^r ^ f^f^ra ^OT % ?RT 
q^f ^r ^w^ ^^iiKf w licrr I 

f ^f ~$tf%t% q^ ^TT 5tcfT 

(ratios) qr mr (rates) 

1 f^f^ i^ran 1 OT%T ^^FP! (Index 
numbers) *f 4 j%qj ^cfT I I q? ^ qfipk ^f^lf % 
^ 1 1 <rc ^R fM ^ m $$v%w**n ^ITOR^ PIT 



(imaginary) 

5f *ft ?^f T 

^rk wr^ft ^r^f CFT % 

I q? T33> ^ *fW ft 



(Harmonic Mean) 
(reciprocals) 



^"TO (x x 
t 9t (h) ^ 



"1J ^T., ^3- 



f % ffim 3R^T I cf^TT ^ (n) 
(Harmonic Mean) ^ M 



h=. 



or 






xn 



where h=harmomic mean 



individual values of 



n= number of items 



(size of item) 



(reciprocal) 



'005. 


^0000 
^0*0000 
? 0*^00 

00 0c; 


..)-,. 


ww < 


? o 


Fkst Method 
h- 10 - 


m*tl ?* T*"" 9 
** >*^ 

^ i^iz ^u^ypu -il ., m __, ., r ,._, ,,,,,^ n 


i45-568i 
= -o6849 
Second Method 

h-reciptocal l45 ' 5681 




r 10 
= reciprocal i4'J568i 



(Weighted Harmonic Mean) 



% 



% 



ten I; i ^ 



% 



HlRcf 



(item) 


(weight) 


(teciprocal of items) 


(weight xreci.) 


< 


^ 


^"o oo o 


*^' O O O O 


"JL 


? 


J* o o o o 


^Jo'o oo o 


vi'o 


To 


"? o o o 


^*O O O 


m ':< 





^00*0000 


^ o o'o o o o 


Y*o 


?^v 


"^oo 


3*^00 


* 


c; 


* 


*^?YY 








R^?"^^?. 



First Method 
85 



b=- 



23r77i9 
= 3663. 
Second Method 

h= reciprocal 



85 

= reciprocal 2-7z7 
= 3663 



cI^T 



(rates) 



t 3 



(speed) 






fer % 

( A| 



( ^f^ ) = 



??T 



% 



JT|% ^ 



; n?fcf 



% % ^5 ^T ^jft % 3R q<%\ q^C ft^^ ^^TT Well | I 

fa; ^5 qsr ^t ^ q- d I ?^ WI 
% ^ri%^ ^fe^r ftaT I ^ ^Rt ^ri^^T (abstract 
ness) % w$$ ^r^t WFTT ^t ^fe^T t I T^ i?r ^ff % 

1 1 



^ o V9 



( rates ) *n ^TSTT^f ( ratios ) ^T JIM 

{ smallest ) SIR grtr <K! ^ wfqw* t^ f^qr STRT ft, 



I, fff% ^ tfwsi! ^T sg^r ^ ^n?ff % SFFT ftar I, 

% 



( Limitatipns of Averages ) 

R ^1 ^cfT 1% I {% ^ W% q?aT ^ 
t I ^ *ft ^ff^TT ^fT 5^1 1 f% ftfe qpajf t ^5 ?JQI 



^Tff f^^T^FTT ^rf|^ i% 



f% 

% 



% 

% 
| 1% I %^^T ^I^^f t I % fifcft ^ ^TT ^ % 

% 



Jf 

f% ^^T fw % SI^J rR^j ^T ^cR f cRT ^ | I 
T^R^Ff: f^ FR^f ^T ^cR ^<> ^WT % 

% ^q ftirr i 5^55 ^ ^>^RT ^ri| 1% ** 



\ 



( Standardized Death and Birth Rates ) 

% M 



1*3 **P*r (Crude birth or death rates) 
IJ 

% 



% ^n^TR <FC ^t ^^fHf ^IT 



g^RT ?Tf ^TT ^T^cit | ^T ^^Fff % ^13-^13^ (age 

composition) ( 



5fT ^ ft % S3) JTSFT^C % 
C1OUS results) ^T fWTT 



f%^TT ^fTcTT | I WlRci 1 W^f ^ WTT 

I i% ^ **R ^r ^rg-^TB^r ^ % ^rrrR 1 1 

% ^F3R ^JT i%^^r?r ^ f^rr ^M 1 1 1%^" sj^n^qr % 
(normal) HRT ^TCTT I, H% RW 1 ? ^r^^^r (standard 



population) ^'t I W^ ftf xwFT-^ti^wr % ^^T (distribution) 
% ^ ^r w^-w^f ^i tfH^Vi ^:% JWTOT 

*n& 1 1 ^ ^ ^ ; ^n;^j! 

i 

=T*Kf, ^ ^Ih isT, % 

1 ^?T% % 

t * 




o 

(V 



ur' 



o 

CV 



O 
O 



O 

o 




o 
tf 



o 
o 

{V 



o 
ixr* 



o 
o 
o 



E 



o 



t 



/h? 



o 
(X 



^oo 0+^000+^000+ ^ 



X %o o) 



WI^TOIT qv* | | 



^nf % 



^00 



Voo 



o 

y 



o 
o 
o 
U 



o 

(V 




(V 



(V 
CV* 



o 
o 

^ 

o 

CV 









o 

CV 



o 





hr 

o 



hr 



for 



I i 



RT 

^ 



ftcfll I 



% 



I; 



% flro; 



% 



% 



^n ITcf ^T It 



SfTrIT 



(H.) fFrf5f%cr ^T^f ^T 5J^T8^ ITRr 



(size) 



(ftequency) 



(size) 



(frequency) 



\3 

e; 
8. 



U.VS 



?=; 



Vo 



( Piactical Problems in Statistics No. 5 1 ) 



? ? v 



9? 3- 



X 



X 



X 



Vi. 



ft ft 1 1 



(Practical Problems in Statistics No. 53) 



(frequency) 



j^o o o 55 


%ffetti 


Ff Vooo 


^o %i 


W 


^ 


YOO o 


?3 55 


\G O 


55 


33 


^V3 


^o o o 


35 J5 


^0 O O 


n 


33 


RK 


c o o o 


33 55 


V9o o o 


15 


3y 


K 


V9 o o o 


)5 33 


c;o o o 


53 


33 


^ 


c;o o o 


35 35 


.00 o 


53 


33 


^ 


5.00 


33 n 


^OjO 


>J 


)3 


& 



(Practical Problems in Statistics No. 55) 



H % 



3? J> 

3f J> 

^1 

>J J> 



(s.) 



( Practical Problems in Statistics No. 57 ) 



V 

* 



VH. 



3o 



IS. 



*fc 

Vo 



YY 



Yo 



( Practical Problems in Statistics No. 34 ) 



( size of shoes ) 



( frequency ) 



y 

K 



vv 



?:r 



fh: gSta ^3^^, 
?TIT x ^f TOITO tft 



, ^ srf 



( Practical Problems ia Statistics No. 36 ) 



\So 



- ( Practical Problems in Statistics No. 37 ) 



(class-ititerval) 



(frequency) 



(class-interval) 



(frequency) 



?- 



Y 

V 



( Practical Problems in Statistics No. 38) 



o o ? oY 



m-m 



^00 



. (Practical Problems in Statistics No. 4o) 



o V 

\~ S. 



3\oo 
\3"3o o 



(Practical Problems in Statistics No. 4i) 



heads qft ^i 



( ^t f% 
heads 



^ffift I ) 



heads ft ^f 1 



heads 



( Practical Problems in Statistics No, 6 ) 



KT *W|TK 



V"*. 

*. 



e; 
=;*, 

5. 



u 



( Practical Problems in Statistics No. 7 ) 



( Practical Problems in Statistics No. 8 ) 



( Practical Problems in Statistics No. 12 ) 

fsr^f 3 f%f*r5r ^gf % f^q:, ?jf % SCTT^T ^ ( 

fst^r 



s^ ift% f^ur f ^r 1 1 



u-n 



f%^T 

(*) 



( Ptactical Problems in Statistics No. 10 ) 



% 



Yo 



eft 



srig 



(Practical Problems in Statistics No. i3) 
^t ^ f%?ITft % 



Yo " " 

v 

E ~ " 33 



Vo 



33 
33 



(Praptical Problems in Statistics No. 16) 



wif 



% 



Yo 



33 33 

33 -33 



(Practical Problems in Statistics No. 18) 
) fvF^fef^T TC^ ^ Q UR8- I ^ S^R^r |f 



sqfeqf 



J^TI ( 5TM 



? % 
? % 



?- ^ 
3K- V 

> o o^ ooo 



5.0 



( Practical Problems in Statistics No. i9 ) 



qfte 



Voo 



( Practical Problems in Statistics No. 2 5 ) 



tf o ^oT R sR ?ra ^ 1 

I ? (Practical Problems in Statistics No. 29) 



f series) 



CO 



c; 
oc; 



'CO 



ooc;'? 



(Practical Problems in Statistics No. 63) 



W ( index numbers ) f%q f <? \ 5*T Iff 



^IWT (f*FT) 



?=; 



(Ptactical Problems in Statistics No. 66) 



sr 



(Practical Problems in Statistics No. 7o) 



o? 



(Practical Problems in Statistics No. 7i) 



BT T, 

, fa^r 



( Dispersion & Skewness ) 



5jz?r (distribution) *T jrfcrf%fr^r 3R3T t I 
% ^T^T % TOR ^ it% i w*n; %w 

ft t 35T % ^ ^ ^ ; 5fR^ q^f %Rit | fR ^f ^t ifFRT 



% ru^T % f^RW ( variations ) f%cf^" f I ^t 
( dispersion ) 



(dispersion) ^% frT^ % ^% f%firT R^f ^>T 1%^^ (variation) 

pr qf ^cfRT | f% rfT'^T 

1 WR f%^ ^F^ w 
xrfgfrtf% 7$ FIHT ^T ^Rnf I 

?nft^t 2ft qrff ^r OT^T ^rf%o: f%qr 
(frequency distributions) t TT 
4^ ^t TO3; % U3> ^ % ^ ft ^^* f 
(?) 3R% ^T^T W^FI-^R ff CR f?T^f % ^f% ^ % 

(deviation) TZ?P % ft I 1^ WFR ^t P?5RfT ^r% q^t % 



ftfiftr ( 5FRr: K ^c ?^ ) f ^ wsit % frftra t^t % 

ff | 



ft SfT^t | CR ^ 5?pft ^TTf f^T 

^ \ 



(deviations) ^T ^ HR ^T TfcTT | I 5$ TR=%^ ^ f=T HTTT 

% l%f^ 5JN 



f, 



) fWK (range) 

) ^5*N> R^FT (quattile deviation) 
R^cTr^; ( semi-inter-quartile range ) I 

) JRpar R^^FT ( mean deviation ) 



( ^ ) 
( *T ) " 

) jjrn^r R^FT ( standard deviation ) 1 



( Range ) 

^tt sfer ( distribution ) f^TT t gt ^OT% qj^ % 
i 



I srof^faftste* SPT ^T^R (tange) 

I f%^ % 



n 



1 1 



(range) 



(dispersion) 
1 1 



( accuracy ) 



f%qr 



I 



(dispersion) % 1%o; ^F^ 
WRIT 1 



(?) 



r % *3*f{ *$ ( extreme items ) % 



q??r % 



(fluctuations ) % 



( inquiry ) 
% fins; ^ 



, S., *t?TT, 

^r % 
?TPBfiH^<if % 



( range ) ?v 
(extreme values) w f%^" 
(range) 



) 



q^ % f%=^f?r (deviation) 

% =qRlT-q^f % 



( symmetrical ), 



ft 

( asymmetrical ) 
1 1 



i% g^% l^t % srcfaw ( dispersions ) ^ % I, 



( Quartile Deviation ) 
( dispersion ) $ ^9" wt f*wi spftr 



1 1 ??r q^f % 

^cfT 1 1% 



) 



f at OT ^t % 



Quartile Deviation 



WRIT I 
quartile range) 

^f % 



(semi-inter- 



I I 



^t ^ % 



vs-z; 



5.-? 



c; 
5. 



V 1 *. 



?V 

3V 



= size of 5oth item 



= 8-38 



item 



= size of iJothitem 

= 10+ 

: IO*44 



Q. D.= 



xo-44 -8-3 8 



= ro3 



( relative ) 
co-efficient of dispersion) 
of quartile deviation ) 



(absolute measure) | \ 
ijT^ (quartile 
( co-efficient 



ITcT 



Coefficient of quartile dis- 
persion 



. _ 

Qa+Qx 

z 

10*44 -8-3 8 
38 



(range) ^ *rf fcf ?s% Jjpr <rc ^ q^f % q^ff 



1 1 qf sre ^ % ^rrar t f% % IM $ RRT % 

q^ 
t iw 



IR q^f ^cfT ! ^ ^^T ^^(Hcftq 1 (asymmetrical) 



( Mean Deviation ) 

(range) dk ^3% f%^l?r ^t ^HRT ^^" Jj 1 OT^ % ^ 
% f^sRRt ^ i^ 1 ^ ^ i%^T ; 5fTcfT 1 1 ^r^ crft^cfT (accuracy) 



% ^ ^ ^ FRK lW ^^fTclT |) ^ SP^RJ 1%^R ^t 'FIRT 
^r^t ^^TT 
fT % q^f % 

( absolute ) 



I 1 1 

(OTPcR *T^^, *T*W ^fT ^ftfS^) % f^'^f^ff % 

1 i^r ncf 



1 



4L 



(0 






ft. 



% 



I: iP 5Z r 



% ^ FTR" 



(l) Sa = 

n 

where, $ = mean deviation 
2da = Summation 

of deviations from 
mean. 

8 a = mean deviation 
from mean 

/ \ j2jdQ 

n 
where, 2dm = Summation 

of deviations from 
median. 
8m = mean - 
deviation from median 



where, 2dz= Summation 

of deviations from 
mode 



deviation from mode 



% 



*n at ^5 



*TM 

if I ^ 5WRR 

^^T ^ % 
( coefficient of mean deviation ) 



f^rr 



?f 



X 

%= - 



Coefficient of Mean Deviation 

1 i) from arithmetic average 

a 

(2) from median = 8m - 

m 

(3) from mode = s ~? 

z 



m 







Rqrft^% ^^I^T(U) 


f^n ^ % ijq^ 




3^ 


% ^^pj- (deviations 


flP*^ (?v) % Rf^^R 


HVR ^^n 


(values) 


from median (i3). 


(deviations from a. 




T(x) 


signs ignored) 


a. signs ignored) 






m (dm) 


^ (da) 


\ 


Y 


I 


? 


^ 


^ 


Y 


<^ 


^ 


s. 




^ 


Y 


?? 


^ 


^ 


U 


?^ 


o 





s 


K 


* 


Y 


* 


5 


\ 


<: 


^ 


^v 


u 




* 


S =w 


(2dm)" 


Sr 


(^) qcqcgf (i) Median 


HT^A+Af m ^ t q^ m =size of ( 9 + T> \ th 



%) 



-=-T-=- 



or 5th item=i3 

Mean dev. (from median) 



n 









%) 



(size of item) 

3K3RffT 
(frequency) 



Coefficient of m . d 

_8m^_2jL 
m "~ x3 *""" 

(2) Arithmetic average 
3_ 2x 126 _ 
~"n 9 

mean dev. (from a. a.) 



n 

54 



= 6 



Coefficient of mean dev. 



i4 



-= -43 



v v 



X a 

u *rj 



_ 

T3 



* 



: O 

: J2 g 

: or 1 



hr 





u 









IT 






*\t 2, t 



Aiithmetic average 



180 
36 : 



Median = size of ( Xil 
or i8'5th item=5 



Mean deviation : 
since a=m=z 



Coefficient of mean deviation 



% ) 



tr *- 



o 

OS 



> 



w 'a' 
8.2 



o o 

(V <v 



8 



rr o 






tr 



\) a/ 
2) cJ 



U" }J 
ixJ a/ 



If 

ffV 

11 

tr 



aJ 



5? 



S? 






U 



fV a*' 



ooooo 



o o o o o 

cv* <V W > 



=. 






*r_ 

MM. Z 

3 



(i) By mtefpolation 
Median=z5'5 

Mean deviation ( from 
median) 

2d m 359-5 

sm -~-""4T 

= 7-9 

Coefficient of mean dev. 
(from median) 

_d m 7-9 _ 

m z5-5~' 3 
(z) Arithmetic average 

= 25-2 

Mean deviation (from a. a.) 

2da^ 353-8 
8a ~" n ~ 45 

= 7-8 

Coefficient of mean dev. 
(from a. a.) 

da__ 7-8 
a~I?-2 

= 3 



TOT* 



% f^rt 

RlisiK ^TT 



( Standard Deviation ) 
% JTR! qjr cRpr f^ *nit I 

^TOFlt % WWI ?faT | I fST 3TcT ^ 

% IIT^T %^^r OTRPT (tools) f, 
I i% B^T ^^tT ^^di % 



qf|f 1 1 
deviation ) 



(standard 
Jf 



% 



( Direct Method ) 

(standard deviation) 
% 



(square root) | | 






...... ^ . ( d 15 d 2 ...... d n ) f 



^: 



=7 



3T 


standard Deviation or 


=v+=v ^ 


__ /dV+dV-i- da 2 


tf 


V n 


"V 

s 


/2d_\ 
= v n 



( Short-cut Method ) 



% fsre; ^ 1 1 



% f*na; 



% 



*TT*I 












/ 

1= J 



or, 



/ 

V 



or, 



or, 



XC 



where, 

a^ Standard dev. 

2d 2 x = summation of sq- 
uares of deviations from the 
as. av. 

a = arithmetic av. 
x=as. av. 



n = no. of items 
2~j 2 = deviations from 

the as. av. divided by the 
magnitude of class-interval^, 
squared and to tailed 
together. 

c = magnitude of class- 
interval 



It 




fr 

dE 

8 
F 
tr 



cr > 

eo o 



8 * 



CO 

S 



fl 



V JJ, 



o 

IT*' 



i i i i-t- + 



cv* o ^ jr* 

os o*- (V 



b- 
(r 



IT 



tF 



^ 






-j- 



Direct Method 
Arithmetic Average 

= _=_ = ir 5 



Standard Dev. 



= / 2 ^1 
V n 



= 7 



_ 289-5o 



% 



= 6-9 

Short-cut Method 
Standard Dev. 



_ /2d 2 x / 2dx \ 
V n \ n / 



_ z9i 



=^48-25 
= 6-9 



% 



snTFT 

% wr 



^ ^tfif (direct method) 



o 



IBS- 

i K O M-4 

i ^F 3 ^ 

f? 5^ fe" 

"^ w 



(xxf) 



/ft? 



a 

o 



h- 
[jr 



* (d) 



(fxd 2 ) 



t; 
S. 



5. 

=; 

Y 



. 

O 

c; 



2d 2 ) 



5 - 



Arithmetic Average = = i = 9 
5 n 48 



Standard Dev. = 

n 



iz4 



II 

Kr ><j 
<v cq 
IF -O 

? W 



/ft? 



I* 
f 



3 

CT*^ ft- 

^ IF 



* x 



s 

o 



fc 

fU <D 

g a 



<v o ov > oj or* 



-h 



(XT' tW W yf US 

<V* (OV* CV <Kx* 



II 



i I +4- + + 



It*' JJ yt > 

0V* 



2> U cu o v 



= 5. 



y 

= V' 



-y 



C 



^ / 

Vq V 



/m 

/ MIUMCUU, 

V vq 



Shott-cut formula no. i 



= 9. 



48 



Short-cut formula no. 2 



= /2d 2 x-n (a-x) a 
v n 

_ / 172-48 (9-8) 2 

=7^ 



48 



48 



Short-cut formula no. 3 



/2d 2 x_/Sdx_\ 2 
' ~ ^ n ^ 

= / I Z?_/i 

V 4 \ 45 

~V~48~ 



SRN 



RH*w ?gg $fcf 3^1 ^r $% 



v 

c; 



rr 



IF 



tx 
/ft? 



a 



fc 



04 






Lg t i . 

c/ t l ^I.^' 


O O JT* jy- O O O 

o o 5> 2> o 3^ o 




O <* x^x 


^fr ^ *** 


c^ (V (V oV V cv cv 


jr* IF TI 


is is? D 
tr f#> fr" 

ds 


IT y t*j <v (V <v 

5* O (jj m f yf 0) 


^ "^ 


<*T 


&&>& 




v_*- 


O O O O O 




IT 


S> U" a/ o^ jv nv 




> tr "^ 


* 3- *- * * * 3- 






a/ a/ / o o o 




T!^ ^ 


(V v <v (V* nv* 




' l*^ 






x-x 




o 


tr [5 ^2 U^^^ 







X ^ ? X 1^ 


O Q -yC <y< Q O O 
(X G^ fj> (V >> nV to' 


*n 


a- ^3 ^^ 


s fV 3< jr* rtV (X 


* 6 I 


u 




o 
a^ 


t^ <D ' i i 

^ rj ' 


> U" <^* 3^ f^ a?' > 


II 


9 X^N 


cv (V* nv > yf {& 




rt 


o o o o o o o 
<v fK nrV > 3< yy 5> 

I 1 I 1 i 1 1 




J, 


o o o o o o o 





=/ 



=7 



1 5 08 5 'oo 



=v / '25i.4i = i5"8 marks 



w 



4E 



K- <vfcT 



X 



." 



<v 

tW 



(V 

<*< 



> 

(*' 



ov 

Ov 



<s 



C3 

IIT IX 

ir "a 



I 

dE 
Ir 

1 



RT 

/ft? 
fe 

IS 



^ cj 

4~ 

F O 



hr 



(X 



+ + + 



o o o o o o o 

flV (V cv v (V fV 



+ + 



U 



(sonjBA-ptra) 



^^ 

.ili- w 



O 
ov* 



IF 
lUC/ 



O 
fX 



O 
<v* 



yf 
(V 



o 
m' 

o 
fV 



o 

f 

o 

> 



o 

T 

o 

2T< 



ft? 









S.D.= 



/ 
V n 



- /2d 



60 60 



XC 



X 10 



= vr5ix 10 
= i 58xio 
= 1 5*8 marks 
(Coefficient of standard deviation) 



JTTIT iw ^rrar 1 1 



wr 



% SKT 



1 1 f 
( coefficient of standard deviation ) ^^ I I 



?fo ^ Jj 1 

5$ 3m % 



prt I 



(Coefficient of Variation) 



( 

tj \ 



(percentage deviation) I ^ % 





i- , u 
.1 



(fluctuations) fJT JT^R *ft 



iWT (moderately skew) ^^ % 
1 1 



(fluctuations) 

I 

3RHT 



q$ f%^T ^fT ^f^cR I 



% 



WT: W ftcfl | | f^R^T^f'^t Rf^ftcT ^R^ % 
WTcfT I, ^RT^ ^RH-^t % R=^RT $t frf^P ^c^T %fcfT | 
% 5fTcf3g[ ^t ^f qfisg^n ^ ^nq 1 ^^T ^TfcfT "I, 



I f% {%^l^r T^U^n $ ^^IclT %, 
^^ % OT^T % 

( 

^IWW 



1 1 wr f^Fr FT 



gg? 1 1 



ff, 



( Skewness ) 



(irregular) 

% 



|t 



f j ^fT^^TT I 

( symmetrical ) | ?ik ^K % ^t WRcf ( asymmetrical ) 1 1 
^o ? ^ f^fr^i mi ^FfaRflT ^R U3> ^cTcf ^TO ^5R ( continuous 
symmetrical curve ) 1 1 f%^[ ^o ^ ^ ^ ^ f^^n3; TOT ^5B 
( continuous asymmetrical curves ) t I 

TO rwt 



vmrn % 



skewness 

t K?rr T^T ^^ i%w ( skew ) 

f | fwr ^it ( skew 



distributions ) ^f ^t wff 



1 1 

% iwr ^T ^fRTT | ( positively skew ) i 
% WJ^T ^ ^?TFfT ^t WIcfT | gt 

I ( negatively skew ) | f^o ^ ^ if 
% ftw ( positively skew ) | sik ftra ^ 



TO 




\ 



\' *> 



A 




ffTOT ( positive skewness ) 
(negative skewness ) *ft ^ WT ^TOcfT 1 1 

( Tests of skewness ) 



% ^ft qrc fip^ ^ 1 1 

TO % %o; ^R^, ff^T^r ^ ^RT?cR: riT^T % Jj^j q;^ ^ ( identi 
cal ) 5& 1 1 ^ 3&t mm VR iwi | gt ^T *$ ftar i 



( positive skewness ) | ^ 



I ^ ^ ^W: ipsq^T frszpp W^t | 
( negative skewness ) 
wcr: 



1 TO gqr^R; *TM % %o; ^ q^f % 



( identically ) ^ ^ I fw ^T mm 
% SRFR; 1 1 fira ^r % 



<TFT ( Measurement of skewness ) 

I WfcfT % qw 



srrar 1 ^ % 

.f5nS^ 

1 



( coefficient of skewness ) 
I : 



wrrwrar 



Pi Weil ^T 3RH *| ^(a - z) pTT 

t | ^ ^rf^r ^_q (z - a) 

% %3T m$m % 
% 

^RFCR rpss^ % 
( coefficient of skewness ) *, ^ ^ (2 -a) 



q-" 
*r = i_ 

=^T 

f IT W 

*=-v* 



^r= 



a z 



or = 



<r 

a 2, 



where j = Coefficient of 

Skewness 
2= mode 

a = arithmetic average 
o- = standard deviation 
8 = mean deviation 



_ 3 (arithmetic av. median) 

a 

_3(a-m) 



orj 



- _ (arithmetic av.-mediaq'), 



3(a-m) 

$ 



% 



iwr 



! ) 



. 



inter-quartile-r-ange) 



(Qa-Qx) 



(Qa-Qi) 



'= 



(Karl Pearson's Coefficient of Skewness) ^^ | 



iwrar 



( Wit 



% f 



In the above table 

a. a. =29 rupees 
2. =37.7 rupees 
m =32*6 rupees 
Qi =9*3 rupees 
Q 3 =42-8 rupees 
M. D, ( from a. a. ) = i6,5 rupees 
S. D.= i8-9 rupees 
Coefficient of Skewness 

,, a~z 29^37*7 
(i) j = .^ 



42'8+9-3--_2 
42' 8^> 
- i3-i 
33-f --'^ 



% 



I I 3f W I f% 



(negative) 
% W I I 



(positive) 
% 



i% 
(normal) 



% WTW flcFT 



% 



I i% 



% 



^rff 1 1 

% HR 5^ 35 



^TCIT 



I ^TT 



T%cRt 



31 ^fe 32^" (normal distribution) ^T TRI 

cf?TT WITOT Wg^^^ff ^ R^^cfT ^JT TT3T 



wsrrp?. fe 
( 



% 



SR5TC % 



f% 



(?o) 



l fee srerc qft 



SJRT 



fee ^ 



Y 
* 



(Practical Problems in Statistics No. 85) 



1.V3 



(Practical Problems in Statistics No. 86) 

mate 



f?^ f? 1 1 



V9o 



(Practical Problems in Statistics No. 87) 



i 
ifjj'o 3|r<> 


,. . 


J '. 


^ 
o 


^ Y 
^ o 


* 


Y 


8. 9 




c; 


Y c; 


\* 


Y 


Y o 


Y 


?^ 


^ 


Y 


o 


^ " ?^ 


^ 


o 


^ o 


Y 


c; 


^ o 



(?) 



inare 



(Piactical Problems in Statistics No. 77) 



(Practical Problems in Statistics No. 78) 



V 

? 



(Practical Problems in Statistics No. 80) 



o- y, 

*-?<> 



VVS. 



( Piactical Pioblems in Statistics No. 82 ) 



% 



3? 



% f 



J^oo 



( Practical Ptoblems in Statistics No. 84 ) 
SFTR 



?<>=; 



, ^^ ), 
( Practical Problems in Statistics No. 9i ) 



( Vo o 



( Practical Problems in Statistics No. 9 2 ) 



swn 



*f 



% ftftw 



KT 



ft *rf 1 1 



WRSJT 



(Practical Problems in Statistics No. 97) 
f^rerar 



V9 

c; 
S. 



\3\3 
V? 



(Practical Problems in Statistics No. 97) 



3>r 



f ^rr , SWR 



Yo 


1 




M N^ *? 



(Practical Problems in Statistics No. 99) 



% 



) 



V ^ 

=;-? 



s. 

o 



E. 
v 



(Practical Problems in Statistics No. 100) 



wft 



( UR[ Tt? 



|rr 1 



Vo? 



(Practical Problems in Statistics No. 101) 

o^? TR-^T| % 
JWTT 



o Yoo 
Yo? 



- ^000 



Yooo 
YYoo 



(Practical Problems in S 



( variance ) 



Vo 



33 
33 



33 
33 
33 
33 
33 
33 



Yo 





(Piactical Problems in Statistics No. 108) 



"dforc (Collar) 



f%cr 



(==;) 



4 



(B. Com. Raj. i949) 
(Practical Problems in Statistcs No. io9) 



VH 



s:* 
* 



53 
33 
33 
33 
33 
33 



V9 

8. 
v 



(Practical Problems in Statistics No. 



If % 



^ (?) 



. 

(Practical Problems ia Statistics No. 1 17) 



_(\o) 
moment of dispersion ) 



(second 1 



(Practical Problems in Statistics No. n9) 



% 



Vo 


33 


,7 


Vq 


33 


,3 


"^00 


5, 





w 





33 


-w 


3; 





Y ^ 


33 


33 



550 



V9 c; 

c; c; , y 

*L * 33 

? ^ J3 

? ? ^ 35 



3> 
33 
33 
;3 

33 



37 
33 
33 
3? 
33 
33 
33 



( Practical Problems in Statistics No. 120 ) 



H'Rt>4!' 



"0 ^oo 
?oo ??o 



HO 



( Practical Problems in Statistics No, 122 ) 



fipir i^rr 



o ;o 

^o Vo 

YO v 

So \so 



( Piactical Problems in Statistics No. ia3 ) 

ft ftsr 



% 



^o Vo 
Yo 1o 



(Practical Problems in Statistics No. iz4) 



% fe^T' 



i FOTKR % $jr % sr^r 



(wft t) 






Yo 
V9o 



YY-YC; 



( Practical Problems in Statistics No. 



- c; 



( Pf actical Problems in Statistics No. 1 26) 



frwrar 



?o XT cKEJ" 
w \l T<~ 

A jf ?3 

^ 
Vo 



J) 5J 
J) 5J 



\So 



(Practical Problems in Statistics No. iz7) 



Y 

c; 



(Practical Problems in Statistics No. iz8 ) 



l^firff^RT 



, SWFT 



I srrcr 



( wit 



13 3* 

33 33 
33 33 



V 

o 



0?o) ft*qfe 



(Practical Problems in Statistics No. iz9) 

g 1 ^fRjft % ^i^ foroER ^i f^rq^rr 3^1^ urg 



i 



o % 
? 33 



3; 

11 

^ 
i 



^00 



\30 



(Practical Problems in Statistics No. i3o( 



(Index Numbers) 



ff I ^T^C ^3R?lf ^PT-^T^r^T ff ^TT 



^FTT^TT 

% 



SIFT: ^?n^ ^ ^qpr T ifr ^ ^ 5^ tinwai (simi- 
larity) ^^ 1 1 ^3 ^HMcTT % ^R 9 ^ 1? ^^TT^f ^t %IHPT ? (general) 
^T JPTc?T ^t f, il% fqfiFT ^R3^t % 5J 
% *ER*g ?t ^RFT-^-^R; (general price-level) 

f% 



(relative change) ^ fT^ 1 1 ^^T WPHJ^-^ % 

R (cost of 



living ), ^NfrffiR? 3cqT?=r (industrial production) 

|f 5! TOT 



3T ^FcR'Wi 1 qr wr, f, ^ 3 



WIPT ^ ^ R- 7 -^ 
3r?q: ^5cTI I F% % ^R^ 
^ff. | T% ^ ^! V TO^I f^ (common denominator) 



I wk ^srflna; 



-^t' % ^R r%o; M^ ^r 

% 3$ ^Fcf % 
% ^q- if ^ % ^R^ ^^ jjpr ^ ^ ^y^qt % 

WI ^ I W^C It!F%q; ^^' qp^Ti; 3fF?n SWFT d ^Icft I I riF 

\ 



ft ^i i ^ET mm $t ^RFB (index 

number ) ^lt | ^ ^ ^cHcfl | {% 




133? t ^g % 



% ^^R ft ^ 
t eft 



j f%% 



% $ 



% RRT ^tr% %?f t Tff WT SIT ^<rr i CR 'srrcfare 

t 1'ff 



% 

f 



( Construction o Ptice Index Numbers ) 

% ^f % ^ sn^P^^ ^rar ^^ 



I : 

% ^ 



grT ^ff (items) ^r e^T!^ q^?iT I sffa 

^t ^R^T % iM 

^^^^f^IRR 

% ^R^T f%R^r q^i % 



(-0 gq^qr % 

O; ^ ? f ^T^ (weight) 

WcfT I f% 
(V) 



base-yeat 

1% 






13" (Selection of items) 

ISfSTC Weft 
{% 



\ ^j 5PFT t ^ ^Kf ^T ^TR ^JsRT Weft I 

sRpff # HPT ^i 
(representative commodity) 
TfiK : 

(?) 3 



\ 

(graded) ^ Jrm% (standardized) 



, cTIJ% ^ 5J^f % ^ JJ ^^TT ^t ^T ^ ff 5t 

^r^^rr ;% ^ wr ^ ^t 

^ ^t 5TT ^^fl" | 

m\ (number of items) ^f ffi 3R3Wf ^ft 



I, 

CR WTT 



(sensitive) |^T^f4! q& ^T ^^ft ft dt 
^r% ^ w^^ 5f I f^ ^^TTOI 
TW ; 5fTclT I I ^^cf^ 1 Economic Adviser 



% 

Economic Adviser- 
^% ^R TO; 1 1 

t Board of Trade Wholesale Price Index $ ^q^r ^ R 
^T ^TW^T fw ^TciT 1 1 W^R^T $ The U. S. Bureau of Labour- 
Statistics' Index of Wholesale Prices vy,o sRg^rl ^ 

(quality of commodities) 3"*g^if % 

| ^jqr?^: ^ jj^Rf (varieties) % 



(varieties) ^ % wf^ IT 9t ^T m ^ 

% ^rfw ^r^R %^* % ^g ^t l%%q q|c^ ft<?r ^RIT 1 1 ^TT^T ^ Economic 

Adviser % ^RTf t ^^ ^^ (quotations) 



123? $" 3Rg % Rf*P5T sraflJ % lyT 33! 3" 1%^ 5fT 






35T T*ft3R*!J ( classification of commodities ) 

% ^ *t ^J^FT ^TT ^ % 

^; if^ ^ff % 

1 



TT ^^FT t I f^ JT^T^; ^jffecf ^^ % ^T^T^ranf (homoge- 

neity) *[f ^Tfcfi' I; i 3PK ^ R? ^t sTsnrf ^ ^i ^fT^r 9t ^ ^^TMCIT 

^qr ^f ^qcfif % f%qq- jj- |%%^ ^PT^ HTH ft" 
I (Economic Adviser) % ^Rpi ^ ^ft |t ^ ^S^rf s& <rNr 
I ^ ^ ft^%IW I : ( ?) ^ q^Tf, 

(?) ^ Rfe q^r^l, (v) nfe T^rsf, ^ (u) 
% ^T ^ R^ri%cr f^n TFTT t ^t (?) 



( selection of representative- 
places ) 



f% 133 ^3 % ^ ^F Wtf % 

^^iRf % 

CTRf 



(quotation of prices) 
% i^n^r % 



| I ^T^ ^ oT^Hr ^T ^TT ^Ic^^f ^N^^ I I 3! ^ f% f^R" 
(bias) ^ 



|, ^ % 

war I f% j%sr 



ft 

^ 1 1% SRS ^T TRHFO" JTfcT ST^T ^ f^lt ( quantity 
of commodity per unit of money ) ^t H^cT f%^TT ^n^T ^FR ^JSRT 
If n %z^ %1 %mi^ w 3Rg f^t wt (quantity of money per unit' 
of comnodity) % ^T if ^cFR ^"H 1 ?7R ^ ^t ^ ^T ^M | 
eft R#fq-.ij|>T (inverse price) I f^T% j^fq 1 t T%^ 5R5R ^T 



(inverse ratio) |tcfl I ! ^-TJ\ W^ ^ ^f ^1 fTO ^T I 
fe$t ^g qT ^JfT H ^o srf^f JR rjpn ^fR 3t 1% c; 
| Wll T|$fT ^^'^5"^; c; ^o ITM 3RFT ft 
%I TO ^o % ^TQTIT I ^CRt^t t 



cff 

(wholesale price) IJHT 
I F% %-^ i23> ^IH m JTR: ^HR ^ I, T* 5^^^ ^ (retail 
prices) qg> ff ^ ff 4R if o;^ ^ % .^i^ ^rff tt I f ^f 
SRT ^TI% ?to % 3T^WI fit I I 

j % MIT ^^ ggpprr^- fit I i $z^ jjjft ^ 
I. fERrfu; gq-Jt ^ 5[r^ qRci^qt 't ^ff^-R^^^fi (time-lag) 

Wife R W q& B>T-S[^ ^^T T|! ^ W | 
RR^cT W^ t ^Tcft | % lti% ^T^f?^ 
% IJ^T ^t 3T OT% ^P-T % 
(incidental sq^f ^ ^tf^ 3TIK ?t^ ^ f ^ ? TO 



5fRf qR FR^ ^1 T% 



(number o quotations) j 3TO STHlft^-^fif % 



l%cR7 f^t, *R"T fafflf ; 5|H37TT I f%cFf wf^ff f^ff % 

' c\. 

5T ^TOT ^Rg^I W^t I Economic Adviser % 
% f^r % *t?*T ^ ^it I I %^^T f^f RMcf 






% ^jRfa ^t ^^"i ^4 st 

V ^TT 



(Selection of Base) 

i%f ^r^r %, 
1 ?s R'f|^<r ^r^ ^ ^ 



41i% ( fixed base method ) 
( chain base method ) 



T%qf 

^TclT | | f g" ?TfrR ^t WRR^T ^H^? cRJ ^nT^TPTT ^TcTT 






% 



WWK 



tfr % 3fcT ^% ^ 5ff <J =ft% 



1 1 



(inclusioo) 



I US ^F^T ^R? ^Tf | ft =TC q^f q? 

(removal ) f%2TT ^rr 



( Calculation of Price-Relatives ) 



() fere ^ri^R ftfir If 

% ftter w % mt JWT 



5. 

e. 
s. 

u 



? 
$ 

o 

^ 

$ 



| I ^T ^ % 5TCT 313^^ HcT ft ^m I 



X?oo 



I? ^?T % 5TRT ^ 
% MO; HTH 



Price relative for the current 



current year's price 
""base year's price 

f I ^r 5[^K TRPI T *&& % 



XI 



(V) 



o 

c; 



f% 



% 



K % 



% 



% %3; ? v*? 
(link relatives) 
IW 



f^% 1% 



? ^ V ? 



% 



Link Relative 



previous year's price 
% 



X 100 



? 



?o 



? 



u 



?.*. 



?o 



? 5.1.0 



I, 



( Choice of Average ) 

% 



% Jjgaff 



at 



% 



% 



( *TO v ) 5} 



FT 



15 



^00 
^00 



(total) 



(a. a.) 



(median) 



(gcometiic mean) 



(chain relatives) 



^00 

?00 

^00 

00 



%r (Total) 

*W (a. a.) 
(median) 



o o 



(geometric mean)) 



*.=; 



we. 



% 



% 



^ % 



| f% 



% 



% 



1 



t f% 

(extreme items) % ^qt 



(reversible) srfftl Scsppw (reversibi- 
lity )$ 



f%*rT <TOT I I ^ gSfa I 



*ft sft^ifa ^?sr 1 1 ^ l^rr ^THT ^r i^t I, *r? ^^ ^ % 
Broiler fTcrr I ^ ^t ^rfe ^ ^ 1 1 w: SRR; 

^ ft t ^R 
Wf^ *?TRcr ft 

qnszj ^ ^ 5j^n; IPW ^rfir^ ^%ft 5to 1 1 iter 
I i% 



SIR ^i I f% 



( Methods of Weighting ) 



H! % ^r t ^ ^ra^r I f% 
% ^ ^t SRFR; 3qf^r %r ^rr 1 1 ?a^t ^ IT^T^: & ^ir ^r 
% fire; ^ ? | rt^r n^R: % ^Frt^ 






% 

( varieties ) % Jj*qf ^r ^HT^R W^fn-W^T R Hpn ^TRH 1 1 



( varieties ) 
JWR t % =^^r 

% m % f^TT ^ ^F?T I 

% *n% 
( implicit weights ) 



(Calcutta Wholesale Price Index Number ) 
1 1 



1 1 



t I 



i i*r IT^K % ^r 

% ^irRcT ^" % 



t : 



relatives ) 



( explicit weights ) 

^lfed ^ra^f efJT ^T 

( Weighted average of 



t I%^T ?t^f % 
( values ) ^t 1 1 ^ I TH! ^t 

r ^ % 
( value ) 1 1 

, s^t ^rf^r ( quantity ) wN; ^r% 3j^r % 
1 1 f ^f^; ^ pj^^r *IT ^9* 

^Rt^ ^t ^^T I%^T SIM I ! 

Jt % ^t w % ^ 

^t Tit I I f*T ^I^t % 5TT% HPKT ^[RWJ 



f% 



(unit) 



w* 



IR 



O'o 






t : 



m % ^ 






( value ) 



I I 



(commo- 
dity) 


F3%cf 33 s&T *Js?3T- 
3TO 
(price relative of 
the current year) 


(values or wei- 
ght) 




^K X ^jrtl^W 
(weight X price 
relative) 

W3 (IV) 


- 


>; :; 


U 


r 






* w 


(2IV) =^^^ 



(weighted index number of prices) 



^ 



if : 



weighted index number 



where, I == price relative 
V = value 






2V 



\ 

^/ 



(Weighted aggregative method) 



?oo % 



1 1 



% 



x 



o 

cu 
XX 

o c 





o . 

fcrO 

** 




rv 



O H 

^ cu 






- 



tf 



o 
O 



or 
a/ 



o 

^ 



X ? o o = ? ^. 



X 



1 1 



Index number = 

]p q 

where, p x = price of the 
current year. 
p =r price of the 

base year, 

q = quantity of the 
base year. 



^ % 



(in whole numbers) *RI*R 1 1 
t 

% 



(Relation between Price Relatives and Link Relatives) 



% 



% 



Voc; 



^00 



(year) 



(fixed base 

index 
numbers) 

CO 



srrarc 

(fixed base index numbers 

changed to chain base 

index numbers) 



(chain base 
index 

numbers) 
W 



Voo 



$00 



tf- t ^ 

""" G^ 
^ w 



III 



u 



tor' 



& 



o 
X 



o 

X 

olo 

X 



X 

Jlo 

cr[(r 

X 

o 

X 

* 



O 

OV 



X 

>0|0 
00 

<rcr 



ojo 

<r[<r 

X 



X 

>OJO 

ojo 

o-Jcr 

X 



X 

H: 

v 






<v (X > U <w ~ 

a/ o o t*y o o 



aj o 



uj cJ aJ aJ 



sibility ) 



3JT 



(Reversibility Tests) 

flcfi 1 1 <Tf *ft WT ^c^FIcfT ( time rever- 

( factor reversibility ) \ 
(Time Reversibility) 
ftctt t f 



( reciprocal ) ^ I 
RFIT 



; 3fFr^TT 



ft 



^o n ( Poi ) 
(pio) ^ ^ 



Pio 

or 



t I 




M" 

O 







*^ CO 

t. 






CO 

<L) 

u 



U" 



-d 

O 



n 



o 

o 




(reciprocal) 
( 



t I ) 



fw: ( Professor Fisher ) ^ 



( Fisher's Ideal Formula ) ^ 1 1 
1 1 3f ^ ^ 

Fisher's Id sal Formula, 



fa* *> * i 

Jlj!_ v .kJ-L X 



X 

X 
X 



where, 

PiQo = Current year's price 

X base year's quantity 
PiQi = Current year's price 

x current year's quan- 
tity 
Po Qo = Base year's price 

xbase year's quantity 
PoQ x =Base year's price 

X current year's quan- 



f% 



JRJK 



tity 
2= Summation 



( Factor Reversal Test ) 



% 3fcR-<TR^T ( inter-change ) w& % 
( inconsistent ) 



% ^RT^ % 



( total value ) 



(relative change) 



^ Star ^> 



01 (q i) 



( total value ) ^ ^ ntr ^^r % 



% ^tR 

I f^T {%{^T ^Tti^M % 
1% 35 3 



% f 



STOft % 



ft 



Vo 



ft 



e;'o 



U" 1 



E 



3^ ON 

aJ HI 







t 






^ o 

o v\ 

Z ^ 



hr 



C* 






o 

O' 



(V 



o 



o o o 



o 
m* 

jr- 





o iv 



V 



o 

10* 



o 
ur 



fV 



O 
O4 



>> u*' 5) 



u 



: / 



A^x'Wf 

V OWk ^Vo 






"^ 



x ^Co ? 



/ 

V 



1?T JRiR 

To satisfy the time reversal test 

PoaXp 10 j=i __ 

f>01= /Sl qOp! q x 



= /74J 64o 



Pio= 



/Sp_o_qo 



x 53 -? 
745 64o 



= /2p i _qo x 2p 1 _qi x 2 
* 2 i S 



2p q'o 2p qi spi q spi 

= /745 64cTTo5 53o 

X r-. A .-", - > " 



6^5 J 3o 745 64o 
To satisfy the factor reversal test 

P nT has been calculated above. 



01 



= /53o 64o 
v 6o3 745 

L ^ VOl 



2p qi 
2p q 



__ 

2p q 



2po q 

in the above example P 01 x Q 01 
,53o v 64o~ 



: /Til 
v 6ol 



6o5 53o 6o5 745 



: /6 4Q x 6 4o 
-v 6o5 6o5 



64O 
6 05 ^ 

-.ejtjo 

605 



Thus the factor reversal test has been satisfied. 



1 1 ^ WJT mnc ^ 
% %c ST^RT srt % 
SIFT: 



( Construction of Cost of Living Index Numbers ) 



?ff ^t ^cTI^ t I 



wf % c^f%^ff % f^fe-^pr if 
f % 






ft ^fMt f 



I ^% ^l! 5KT ^3K 5^^ 'J^ft (retail prices) 
I, ?^r%3; ^-^4t ^r ^3T^r ^r% ^f ^n I *rc 

% ^^=T t 3T 133? ^ ^sqpf ^ ^ ^75 % 



^T I, % 123* ^ ^T^T^ ^ ^5 % 1 1 

R^f ? ^^ ^^RT^T % ^^f^TcT | % % f f% 



1 I^R fr t^r^^t ^ ^w^r % 



.,.-, 

ft 



I 

% 
T^- % 



1 1 

ifr ft* 



sampling )m 

I f* IRJR JIRI 



1 iR ^ %rat 
w?r, 



|gT?T 



t TOPIR 

% 

% ^j^t % ffr wi% TR^rf ^r qfc^r 3i^rT-?t^T ttcir I, 

323* i^ 'n =3^% stgrof ^t T^tfer ^ % 
1 1 M *i?3 % J-Efr ^ ^ ^T% <jR*Ml 



% 
1 1 c?; ^ ^T^^ 5?W[ ft% ( aggregate 



expenditure method ) ^r^^gt ^fe" ( aggregative method) 
qftffK arsra: $1% (family budget method) 
T ftf% ( weighted relatives method ) 
1 1 



% OT^TT % ^R fw STIST I 



^rT^; % M t OT^T fw ^FTT 1 1 ffo Jic%^ ^ % 



^rftr ^r ^^ ^gwft ^rr % 
1 ^ ^Rgw! % {%% inar ??f jj^^f ^ % 3t*i ^t ^TNK ^ 

% 



^f^r ^ <t% t f^ irf^rl% Tf^Rf % 
% c ^c 

^TcfT | I 



^t % ^r ^t ^ % *?far % i%^TTO ^ f^rr ^rar 1 

JTRT ^FT ^J^T i%^ff -^R-^rTO $m I 1 



<u 

a 



s 

*3 



<u 

cs 
50 



faO 



RT 



n? 



|pr 



nr 



- 

la 






rtr 



tzr 



(V J > 



Lf 



ol/o o ouyrvo oo^o^tP" 

(X ^ Ow* <V fV' CV <V 



aJ \S 



> . 



fir 






tr 



Ji 
s 




budget method ) *n 
method ) % R^I-^ 



Index number for current 
year 



_ 453 

422 

= io7 



Xioo 



Clfit ( weighted relative 



x 



0QOOQOOO 

oooo ooo o 



o 
o 

2 * , 

^i rt i o. 



ooooooooooooo 
ooooooooooooo 

xxxxxxxxxxxxx 

> 




^ ct/ 

o 




? 

o 




Rr 






y(E; dp g g {r 



3$ % 



$5T ft 
WTcIT I I 



1 1 



1 1 

^rr 



l?qt 



5 | ft T5RT 

m i^iawf 
q* ^r 
(income-group) % 
^ftrr 



indexnumber foi current 
year 






% SEER! 



1 



I, 



2V 
453oo 

422 

= io7 



% 






^TcFT I I 



^tft 
R ftt 



% 



I 



1 



it TOJ 1 1 



1 ^ $T ^T tf^cTT 1 1% & ^Rffif 
Jt f ^ SJTTCR ^Hr % wfr% ffe ^T sHiHH ^f^TFTT ^RTT | | 

% ^-sf-r % ^n; ^ 3N ^K IIFT^ ^ ^t $TOI ^f | \ 
sfrfr-sffcr fj $ ^i ^^ t 

% 



% ^T^ft?' (Indices of Industrial Production) 
% ft^-R% ^RFfif % 



( industrial output ) % 
% ^ ^T ^ri^ I ^ff^T3; ^Ff^ ^cqi^T % 
% %R; ^^JT^W ^5 ^FRT 3IM?W I f% ^J % 



% ^dTT^T ^T TRqi^ ^^TT I I % ^FTf^ ^!% ^ f ^ 
% M t ^?Rl^ ^RT 5R?fr ft ^ ^ qjf^pqt % 

1 1 f 



^T^R^cf: l^fJT%i%cr tflfcX % 



(?) ^R ^gfrr (Mining industries) 
(ores) ^ ^PT ^FRI ^r ^n^r WM |, 1% 



^37 (Metallurgical industries)?^ 
dRt ^t ^rg^t % 

I 1% sftfT &k 



(Mechanical industries) 



(Textile industries) 1% 



ft (Industries 
subject to excise duties) ^frft, 



(other important industries) 



5ft 1 
% ^T 

I HKf ^ 



% *^ ^n f%^ ^r^r sRcr ^TITO % 
*rrfer 



(gross output) ^TT NltdR* 3&$3 (net- 
output) % fire; ^% ^ ^r^ 1 1 

t^ (Indices of Business Conditions) 



1 wft rp^t ^ |, wt t^t i ^ ^tm t ^jfe ^ | sih; 

^^T^^fT ^ ^cTT | I l^T TR^f^f % fijjtjr ^t 
I ?^ TRJ^TT ^R^* ^ ^ ^f R ^H *ft f f% ?/=[ % 3KT 
(forecast) ^TFTT ^T ^^RITl, T(f% 



(periodic) 5^ t I ^ \fo ^m^^^^l *$t *{R3X$ % 



i t^ri^ % 3Tt%^; <ft^ (Professor Pigou) 



(?) ^Rnf% iilcRlcldl (unemployment percentage) | 
(consumption of pig iron) I 
(prices in ngtand) I 



(v) |snf%3> f%q^f <R r| 3ft 3;? (rates of discount on three 
riionths'bills) | 

(1) ftfe <re[T*JT ^T <rftRTCT (volume of manufactured 
goods) I 

(^) fR % ^f^J^Tcr ^rr^T (agricultural production) | 

($) ^ 3^3 TO?jf $ JTRT t23?y sqsr (yield per acre of nine 
principal crops) | 

(^) ^FTT % ScTjcpr % ^Hi^ (index of production from 
mines) l 

(5.) W^^ $gftftv SOT % ^arcfR (clearings of London 
clearing house) | 

(?<>) f^-^JM ^ ff% (increase of bank-credit) I 

(??) wnff ^WC (credits outstanding) I 

()^R^srf^W H^^^i% if% (annual increase in 
the aggregate money wage) | 

(U) m^m rR^t $ ^ (rate of real wages) | 

(?v) ^TRFT ^FT%> sq-qfa (general aggregate consump- 
tion ) | 



(proportion of reserve to liabilities of the Bank of England) f 

% ^ 



(Uses of Index Numbers & their limitations ) 






ft 
1 



%, fcf5 ^cR %, sfNMSfaJ ^flcpT %, s^TCTCFT^T %, H^ %, 

% ^nf^; i ^ff % ^FTW! WT i^ff t f 
I f^% 5^T ^T HFT (valae of money) 



?ter ft gt ^T^-C^R^ Jt 



^FlT^f * ^5Prai % ^ it *^T 1 

^^t M-?rf% ^t ^T ^Rt^t STO 

STO ^T^fl%^ r^f^ (real wages ) 



t^t (indices o industrial activity) 

KT tot 



WfWl m ^RfT ! 
(fluctuations) 
pt ^ft ^ 



1 f^t SRR f%^ s ^ira^ % 

% J??qt ^TlR % ^Rfo 'ft ^^^t TO^f % ^t t 



(approximate indicators) f ( 

1 % 



^T (distribution 

of weights) ^ ^^it^f ilcft t I TC S?% ^T^g 4 ^ TO ^R f^W 
ft ^FW ftH R^T t ^ ^ ^K =^T (variable) 
(trend) ifrft I 



*PWT JEW fi^^T (interpretation) fto; 



^ ^ V 



% wfer ^wra ^t sg^qr w 
i 



(=0 



(M) fcrawftf WSJT^ t (sr) g^T^ ?rt^ ?rrr 

(claims) 5Krr5S I 



flfsrq 



HV 



ex 



=;=; 



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