'

FUNDAMENTALS OF APPLIED STATISTICS

3.42
TABLE 3.34
A "AT THE END OF THE MONTH"
Weights (w) Iw -
Items Index (I)
252 Wt
252 w 1 -
Food W3
205 Wg
205
Clothing W3
195 w 3
Fuel and Lighting 195
Ws
150 Wg
Rent 150
W2
212 w 2
Miscellaneous 212
W1 + W2 + 3wa 252 W1 + 212 W2 + 550 W 3
Total -
From the given conditions, we have
4
. 1 5 225 ca· ) 12 = 225 and I 1 =5 x 225 = 180
12 := 11 + 4 I1 = 4 11 = iven
Finally, assuming that total of weights is 100, we get three equations in three unknowns
w1, w2, w3 as given below:
I - 198w1 + 138w2 + 510ws = 180 3w1 - 7w2- 5wa = 0 ...(1)
1- w 1 + w 2 + 3wa
I - 252w1 + 212w2 + 550ws = 225 27w1 - 13w2 - 125wa = 0 ... (2)
2- W1+W2+3W3
and w 1 + w 2 + 3w 3 = 100 ... (3)

Solving (1), (2) and (3), we obtain w 1 = 54, W2 = 16 and Wa = 10

Hence the exact weights are :
Item Food Clothing Fuel and Lig~ Rent Miscellaneous
Weight 54 10 10 . . . ._ 10 16

3·6. BASE SHIFTING, SPLICING AND DEFLATING OF INDEX NUMBERS

Base Shifting. Base shifting means the changing of the given base ,pena.d
(year) of a
series of index numbers and recasting them into a new series based on some recent new base
period. This step is quite often necessary under the following situations:
(i) When the base year is too old or too distant from the current period to make
meaningful and valid comparisons.
(ii) If we want to compare series of index numbers with different base periods, to make
qui~k and valid comparisons both the series must be expressed with a common base
period. ·
Base s h"fti ·
i ng requires t_h e_ recomputation of the entire series of the index nu1:11bers with
ch
t~e new base. Howeve~, this is a very difficult and time-consuming job. A relatively m:se
simple, th0ugh app~oximate ~ethod consists in taking the index number of the new b the
!ear as 100 and then ~xpress~ng the given series of index numbers as a percentage ~f dei
-index number of t_he time period selected as the new base year. Tilus, the series of in
numbers, recast with a new base, is obtained by the formula :
Recast Index number of any year = Old Index No. of the year x l00
Index No. of new base year
 l'.:X NUMBERS
D 3.43

. = ( Index No. o:~~ base year ) x (Old Index Number of the year)

In other words, the new series of index numbers is obtain d . . .

s with a common factor : e on mu1tiplymg the old mdex
nuillber
k_ 100
- Index Number of new base year
Example 3•27. From the index numbers given below find out · d b b . .
--baseJ!'om 1993 to 2002: ' in ex num ers Y shifting
1993 1994 2002 2003 2004 2005
Year · 1995
JndexNo. : lOO 76 68 50 60 70 75
Solution.
TABLE ~-35 : RECAST INDEX NUMBERS (BASE 2002)
Year Index No. Index Number with Base (2000 =100)
1993 100 100
X 100 = 200
50
1994 100
76 50 X 76 = 152
1995 100
68 50 X 68 = 136
. .
100
2002 50 50 X 50 = 100
100
2003
'
60 50 X 60 = 120
100
2004 : I
70 50 X 70 = 140
75
..
2005 75 50
X 100 = 150
Remark. Strictly speaking, the method explained above is applicable only if the given index
numbers satisfy the circular test. In other worlds, it gives accurate results .if the indices ·are based on :
(i) simple geometric mean of price relatives, or (ii) Kelly's fixed base method.
If arithmetic mean or median is used for averaging the price relatives then the usual method of
base shifting consists in calculating the price relatives for each individual item w.r.t.. the new base and
then averaging their totals, i.e., the whole of the series is to be reconstructed. However, in practice,
even in these cases the approximate method described above gives result which are fairly close to those
obtained otherwise.
Splicing Two Index Numbers Series. In order to obtain eontinuity in the comp·¥ ison
of two.or more overlapping series of index numbers, we combine or splice them into a single
continuous series. For example, suppose an index number series 'A' with base period 'a' is
discontinued in period 'b' due to certain reasons and a new series 'B' of index numbers';is,
0
; mputed with base period 'b''(and the same items). In order to comp~re the series 'B' with
A', we splice the series B to A to obtain a continuous series from 'a' onwards. The process is·
very much alike to that ofb~se shifting and is illustrated below in Table 3·36.
 I

FUNDAMENTALS OF APPLIED STATISTICS

3.44
SPLICING OF TWO INDEX NUMBERS SERIES
TABLE 3-36 :
Series II spliced to Series I spliced to-
Year Series I, base 'a' Series II, base 'b' Series I (Base 'a') Series II (Base 'b')-
100
100 X 100
a 100 ak
100
a1 -akx a 1
a+l a1
' 100
a2 -akx a 2
a+2 a2
.
. 100
ak-1 -akx ak-1
b-1 ak-1
100 ak 100
b ak
ak
b1
I
100 X b1 b1
b+l
ak
b2 100 b2 b2
b+2 X
ak
b+3 ba 100 x ba ba
' \

Explan ation. When series II is spliced to series I to get a continu ous series with base 'a'.
100 or II series becomes ak => b 1 of II series becomes (ailOO) x b1,
with a
and b 2 of II series becomes (a,/100) x b 2 , and so on. Thus,-mu ltiplying each index of the series II
spliced to series J (Base 'a'). In this
constant factor (a,/100), we get the new series. of index numbers
case, series I is also said to be spliced forward.
If we splice series I to series II to get a new continuo us series with base 'b', then
ak of first series becomes 100,
ak-1 of first series becomes (100/ak) x ak-l,
=>
Ia2 offirst.seri~~ beco~es (100/a.k) x a 2 , and so on.

:hus, the ne_w s_eries of ~Iidex num~ers with series I spliced to -series II (Base 'b') is
obtamed on multiply mg each mdex of senes I by new constan t factor (100/a k ). In this case we
say that series is spliced ba~kward.
E:ampl e 3·28. Given in 7:able 3·3_7 are two price index series. Splice them on the base
2004 - 100. By what per cent·dzd the price of the commod ity rise between- 2000 and 2005 ?
TABLE 3 37
Old price index for the commod ity
-
-
f'
Year

2000
2001
Base (1995 = 100)
141·5,
163·7
New price index for the commodity
Base (2004 100)
-
2002 158•2
2003 156•8
2004 99·8
157•1 1oq-d
2005 -
-
- 102·3
INDEX NUMBERS
3.45
- Solution.
OLD PRICE INDEX TO N EW PRICE INDEX
TABLE 3•3 7A : SPLICING OF
Year / Ol d price ind ex for the Commodity New price index for th~ commodit
y
Ba se (1995 = 100) Base (2004 - 100)
100
2000 141•5 157•1 X 141 ·5 =99· 06
100
2001 163·7 157 •l X 163 •7 = 104 ·19

100
2002 158·2 157 •l X 158•2 = 100•69

' 100
156·8 = 99·80
156·8 157.1
X
2003
I ,'· 100·00
2004 157·1
102·30
2005 ' -

rea se in the pri ce of the commodity between 2000 an d 2005 is :

Th e pe rce nta ge inc
102·30 - 90·06 100 = 0· 135 9 X 100 = 13· 59
, 90· 06 X
.
, the req uir ed ~n cre ase in the price of the commodity is 13·59%
Hence
nu mb er ser ies wa s sta rte d wit h 1992 as base. By 1996 it rose
Ex am ple 3·2 9. Aprj,ce ind ex s 95. In thi s yea r a new series was started. Th is
new
ve for 19 97 wa rise was no t
by 25%. Th e lin k relati r. Bu t du rin g the subsequent four years the
po int s in the nex t yea were
series rose by 15
pri ce lev el wa s on ly 5% hig her tha n in 2000, an d in 2000 these
rapid. Du rin g 20 02 the two series an d cal cul ate the ind ex numbers for the various
8% higher tha n in 19 98. Splice the
years by shi ftin g the base to 1998.
So lut ion . BASE INDICES
E 3-38 · CONVERSION OF LINK RELATIVES INTO FIXED
TABL II Series
Series I - Ind ex Nu mb er
Nu mb er Year Lin k Relative
Lin k Re lat ive Ind ex
Year
100 -
1992
I

125 X 100
= 125
1996 100
95 X 125 100;0 ,
= 118·75 19~7
1997 95 100
115 X 100
1998 115 = 115·0
100
108 X 115
2000 = 124·2
100
- 105 X 124 ·2
2002 '
' = 130·4
100

. I
3-46 FUNDAMENTALS OF APPLIED STATISTICS

SHIFTING (1978 = 100)

TABLE 3-38A: SPLICING OF SERIES I TO SERIES II AND BASE
Year Series I spliced to (1998 = 100) series II Index Numb er (1998 = 100)
100
1992
100
100 X 118.75 = 84•21 84·21 115X= 73·23
100
100
125 X 118.75 = 105•26 105•26 X 115 = 91·.5 3
1996
100
100·00 100 X 115 = 86•96
1997
115·00 100
1998
100
2000 124·20 124·20 x 115 = 108
100
130·41 130·41 x - = 113·4
2002 115

Defla ting the Index Numb ers. Deflat ing means 'makin g allowa nce
for the effect of
of people over
chang ing price levels'. The increa se in the prices of consum er goods for a class
class. For examp le, the
a period of years means a reduct ion in the purcha sing power for the
increa se in price of a partic ular commodity from Rs. x in base year 'a' to
Rs. 2x in the year 'b'
with Rs. x which
implie s that in 'b' a person can buy only half the amoun t of the comm odity
only 50 paise in 'b' as
he was spend ing on it in 'a'. Thus, the purcha sing power of a rupee is
compa red to 'a'.
e' for a class
The idea of 'the purcha sing power of money' or 'a measu re of the real incom
by an approx imate
of people is obtain ed on deflati ng the wage series by dividin g each item
be conve rted into
price index e.g., the cost of living index. The real wages so obtain ed may
index numb er if desira ble. More precisely,
R al _ Money or Nomin al Wages lOO
e wage - Price Index x
ively used to
The real incom e is also known as deflated income . This techni que is extens
and so on.
deflat e value series or value indices, rupee sales, invent ories, incom e wages
The following examp le illustr ates the techni que of 'deflat ing':
TABLE 3.39
Year Average wage of Consu mer Price Indices
workers in Rs. per houi' • (1987-89)as base period
Exam ple 3·30. The adjoin ing 1987 119 95.5
Table 3·39 s.hows the averag e 1988 133 102·8
wages in rupees per hour of 1989 144 101·8
worke rs in a factory dur~ the 1990 157 102·8
year 1987 to 1998. So also are 1991 175 111·0
given the Consu mer Price Indices 1992 184 113·5
for these years with 1987 to 1989 1993 189 114·4
as the base period. 1994 194 114·8
(a) Determ ine the real wages 1995 197 114·5
\qf t'?-~ . rail road workers during 1996 213 116·2
the years 1987-1998 as compared 1997 228 I
120·2
to their wages in ·1987. 1998 245 I
123·5
 INDEX NUMBERS
' -_3.47
.(b) Use the Consu. mer Price Index t o d etermin
. e th . . -,
various years assum ing that in 198. 7, one rupee wa t . ie purcha sing power of a rupee ,or t h- e ·-
power. s s net Y worth rupee one iri purcha sing

Soluti on.
TABLE 3-40 : CALCULATIONS FOR
PURCHASING POWER OF RUPEE
Averag e Consum er Price Consumer Price Defiated Real Purchas~ng p~wer of a
Year Hourly Wage Index bu:lex Wage rupee
(1987-8 8 =100 ( 1987-89 =100) (1987- 100) (1987 -100) (1987 =100)
(2) (5) =
(1)
- (3) (4)
{(2)/ (4)) X 10o (6) =[(5) I (2)) - [11 (4))
1987 119 95·5 100 119 1·00
102·8
1988 133 102·8 95.5 X 100 =107·6 124 0·93
I

101·8 -
1989 144 101·8 95.5 X 100 =106·5 135 0-9-4 ·
'
102·8
1990 157 102·8 95.5 X 100 =107·6 146 0·93

111·0
1991 175 111·0 95.5 X 100 =116·2 151 0·86-

113·5 -
1992 184 113·5 .
95 5
X 100 =118·8 155 0·84

114·4
1993 189 114·4 .
95 5
X 100 =119•7 1·58 0·83

114·8
1994 194 114·8 X 100 =120·2 161
J
0·83

114·5
1995 197 114·5 X 100 =119·5 164 0·83

116·2
1996 213 116·2 X 100 =121•7 175 0·82

120·2
1997 228 120·2 X 100 =125·9 181 0·79
I
-

123•5
1998 245 123·5 95-6 X 100 =129·3 189 0·77

The real wage of the worker s in any year as compar ed to 1987 will be the wage in
that
year multip lied by the purcha sing power of one rupee in that year as compared to 1987
Exam ple 3.31. Your are \given the inventory position of a company for si:x: years. Find
out
index numbe r of physic al volume of inventory at constant prices. w.r.t. base 2000. Comme
nt '
upon the nature'-of the invento ry position : '
Year ·. 2000 2003 2004 2005
2001 2002
I

Inventory (in '000 Rs.) 472·4 492·6 . 524·7 54·8

425·6 447·8
Wholesale Price Index 1 158·0 173·9 162·6 _i81·5
108 2 121 ·5
(1995 = 100)
3.49 FUNDAMENTALS OF APPLIED ST~TISTICS

Solution.
TABLE 3.41 : COMPUTATION OF DEFLATED INVENTORY
Year Inventory Wholesale Price Physical Volume Index
('000 Rs.) Index (1995 = 100) Defiated Inventory
(Base 2000 = 100)
(1) (2) (3) (4) = [(2) I (3)] x 100 (5)
2000 425·6
425·6
. X 100 = 393•34
100
108·2 108 2
447·8 100
20.01 447·8 121·5 . X 100 = 368•56
121 5 393 .34 X 368·56 = 93•7 I
472 4
2002 472·4 ' X 100 = 298•98
158·0 158 3 :~.~ 4 X 298·98 :::: 76•02
492·6 100
X 283•27 = 72·02
173 9 100 = 283·27
2003 492·6 173·9 . X ,
393 34
524·7 · 100
2004 524·7 X 322•69 = 82·04
162 6 100 = 322·69
162·6 . X ,
393 34
540·8 100
181 .5 X 100 = 297·96
2005 540·8 181·5 393 34 X 297 ·96 = 75·73
.

3-7. INDEX OF INDUSTRIAL PRODUCTION (IIP)

The index of industrial production is aimed at reflecting changes (increase or decrease) in
-the volume of industria l productio n (i.e., production of non-agric ultural commodit ies) in a
given period compared to some base period. These indices measure, at regular intervals, the
general move,men t in quantum of industrial production . Such indices are useful for studying :
(i) the progress of general industrial isation of a coq:ntry, .and
(ii) the effect of tariff on the developme nt of particular industries .
These indices of industrial activity are of great importanc e in the formulati on and
implemen tation of industria l plans. For the constructi on of the indices of industria l
productio n, the data,about production, of various industries are usually collected under the
following heads :
(i) Textile Industries : Cotton, silk, woollen, etc.
(ii) Metallurg ical Industries : Iron and steel, etc.
(iii) Mining Industries : Coal, pig-ion and ferro-alloy s, petrol, kerosene, copper (virgin
metal), etc.
(iv) Mechanic al Industries : Locomotives, sewing machines, aeroplane s, etc.
(v) Industries subject to excise duty : Tea, sugar, cigarettes and tobacco, distillerie s and
brevre;r;ies , etc.
(vi) !p_le'ctricity, gas and steam : Electric lamps, electric fans, lrtectrical apparatus and
appliaiu;e s, etc. ,;.,
(vii) Miscellan eous: Glass, paints' and varnish, paper and paperboar d, cement, chemicals
etc.
Usually, the data (figures of output) a;re obtained for various industries on monthly basis
and the indices of ~ndustrial production are obtained as the weighted arithmetic mean (or
sometime s geo~etnc meanYof the production (quantity) relatives by the formula ·:
. _ "f.Qiw; ·
1OJ - LW ·
' J
.. .(3·22)
where Qi= Production relative= Qij I Qoj, and Wj is the weight assiged tojth industry.
I .