Weighted index numbers

1. NADEEM UDDIN
ASSOCIATE PROFESSOR
OF STATISTICS

2. Weighted Index Numbers:
There are two methods of calculating weighted index numbers.
1- Weighted aggregative price index numbers.
2- Weighted average of relatives price index numbers.
Weighted Aggregative Price Index Numbers:
In this we calculate the total expenditure of the current year and the
base year. Price of each commodity is multiplied with the weight of the
commodity which is usually the quantity consumed or quantity
produced. The quantity of the base year or current year can be used as
weight. The current period expenditure is compared with the base
period expenditure.

3. 𝑃𝑟𝑖𝑐𝑒 𝐼𝑛𝑑𝑒𝑥 =
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑌𝑒𝑎𝑟 𝐸𝑥𝑝𝑒𝑛𝑑𝑖𝑡𝑒𝑟𝑒
𝐵𝑎𝑠𝑒 𝑌𝑒𝑎𝑟 𝐸𝑥𝑝𝑒𝑛𝑑𝑖𝑡𝑒𝑟𝑒
× 100
The general formula for computing a weighted aggregates price index is
100, 


qp
qp
I
o
n
no
100, 


Wp
Wp
I
o
n
no
OR

4. Example-1:
For the following data calculate weighted aggregative price index
number
Weight Price 2004 Price 2005
32 2.20 2.60
119 2.60 2.90
75 3.10 3.20
16 3.30 3.35
W po pn pnW poW
32 2.20 2.60 83.2 70.4
119 2.60 2.90 345.1 309.4
75 3.10 3.20 240 232.5
16 3.30 3.35 53.6 52.8
721.9 665.1
Solution:

5. 100, 


Wp
Wp
I
o
n
no
100
1.665
9.721
05,04 I
%54.10805,04 I

6. Items
Price
2003
Price
2004
Price
2005
Average
no.used
A 1.25 1.50 2.00 900
B 6.50 7.00 6.25 50
C 5.25 5.90 6.40 175
D 0.50 0.80 1.00 200
Example-2:
MR A is incharge of keeping in stock certain items that his company
needs in repairing its machines ,he arranged the data in the following
table. Calculate weighted aggregative price index.
Solution:
po pn q pn q po q
1.25 1.50 900 1350 1125
6.50 7.00 50 350 325
5.25 5.90 175 1032.5 918.75
0.50 0.80 200 160 100
2892.5 2468.75

7. 100, 


qp
qp
I
o
n
no
100
75.2468
5.2892
04,03 I
%16.11704,03 I
po pn q pn q po q
1.25 2.00 900 1800 1125
6.50 6.25 50 312.5 325
5.25 6.40 175 1120 918.75
0.50 1.00 200 200 100
3432.5 2468.75
100, 


qp
qp
I
o
n
no
100
75.2468
5.3432
05,03 I
%03.13905,03 I

8. 100, 


oo
on
no
qp
qp
I
100
qp
qp
I
no
nn
n,o 


There are various kinds of weighted aggregative Price index numbers.
(i) Laspeyre’s Index:
In Laspeyre`s method base year quantities are used as weights
for price index number.
This is also called base year quantity weighted method.
(ii) Paasche’s Index:
In Paasche`s method current year quantities are used as
weights for price index number.
This is also called current year quantity weighted method.

9. (iv) Marshall- Edgeworth price index:
In Marshall –Edgeworth the weights are taken as an
average of the respective quantities in the base period
and in the given period.
𝑃𝑜𝑛 =
𝑃𝑛(𝑞 𝑜 + 𝑞 𝑛)
𝑃𝑜(𝑞 𝑜 + 𝑞 𝑛)
× 100
(iii) Fisher’s Ideal Index:
Fisher Index is the geometric means of the Laspeyre’s and
Paasche’s Index.
This index lies between Laspeyre’s and Paasche’s indices.
IndexsPaascheIndexsLaspeyreI no '', 
100, 




no
nn
oo
on
no
qp
qp
qp
qp
I
OR

10. Example–3:
Construct index number of prices from the following data by using
Laspeyre’s method taking 2007 as base year.
Commod
ities
Price
2007
Quantity
2007
Price
2008
Quantity
2008
A 8 45 12 50
B 4 100 4 110
C 6 50 8 55
D 12 30 14 35
Commodities Po qo pn qn
pnqo poqo
A 8 45 12 50 540 360
B 4 100 4 110 400 400
C 6 50 8 55 400 300
D 12 30 14 35 420 360
 1760qp on  1420qp oo
Solution:

11. 100, 


oo
on
no
qp
qp
I
100
1420
1760
I 08,07 
%94.123I 08,07 
Comments:
The Prices increased by 23.94% in the year 2008 with respect to 2007.

12. Example-4:
Construct index number of prices from the following data by
using Paasche’s method for the year 2008.
Commo
dities
Price
2007
Quantity
2007
Price
2008
Quantity
2008
A 8 45 12 50
B 4 100 4 110
C 6 50 8 55
D 12 30 14 35
Commodities po qo pn qn
pnqn poqn
A 8 45 12 50 600 400
B 4 100 4 110 440 440
C 6 50 8 55 440 330
D 12 30 14 35 490 420
 1970qp nn  1590qp no
Solution:

13. 100
qp
qp
I
no
nn
n,o 


100
1590
1970
I 08,07 
%90.123I 08,07 
Comments:
The Prices increased by 23.90% in the year 2008 with respect to 2007.

14. Example–5:
Construct index number of prices from the following data by using
Fisher Ideal Index method.
Commo
dities
Price
2007
Quantity
2007
Price
2008
Quantity
2008
A 8 45 12 50
B 4 100 4 110
C 6 50 8 55
D 12 30 14 35
Solution:

15. 100, 




no
nn
oo
on
no
qp
qp
qp
qp
I
100
1590
1970
1420
1760
I 08,07 
1002389.12394.1I 08,07 
1005355.1I 08,07 
1005355.1I 08,07 
1002391.1I 08,07 
%91.123I 08,07 
Comments:
The Prices increased by 23.91% in the year 2008 with respect to 2007.

16. Example–6:
Construct index number of prices from the following data by using
Marshall- Edgeworth price index.
Commo
dities
Price
2007
Quantity
2007
Price
2008
Quantity
2008
A 8 45 12 50
B 4 100 4 110
C 6 50 8 55
D 12 30 14 35
Solution:
Commodit
ies
Price
2007( 𝑃𝑜)
Quantity
2007( 𝑞 𝑜)
Price
2008( 𝑃𝑛)
Quantity
2008( 𝑞 𝑛)
(𝑞 𝑜 + 𝑞 𝑛) 𝑃𝑛(𝑞 𝑜 + 𝑞 𝑛) 𝑃𝑜(𝑞 𝑜 + 𝑞 𝑛)
A 8 45 12 50 95 1140 760
B 4 100 4 110 210 840 840
C 6 50 8 55 105 840 630
D 12 30 14 35 65 910 780
𝑃𝑛(𝑞 𝑜 + 𝑞 𝑛)
3730
𝑃𝑜(𝑞 𝑜 + 𝑞 𝑛)
3010

17. 𝑃𝑜𝑛 =
𝑃𝑛(𝑞 𝑜 + 𝑞 𝑛)
𝑃𝑜(𝑞 𝑜 + 𝑞 𝑛)
× 100
𝑃07,08 =
3730
3010
× 100
𝑃07,08 = 123.92%
Comments:
The Prices increased by 23.92% in the year 2008 with respect to 2007.

18. Example–7:
Construct index number of prices from the following data by using
alternative formula of Marshall- Edgeworth price index.
Commo
dities
Price
2007
Quantity
2007
Price
2008
Quantity
2008
A 8 45 12 50
B 4 100 4 110
C 6 50 8 55
D 12 30 14 35
Solution:

19. 𝑃𝑜𝑛 =
𝑝 𝑛 𝑞 𝑜 + 𝑝 𝑛 𝑞 𝑛
𝑝 𝑜 𝑞 𝑜 + 𝑝 𝑜 𝑞 𝑛
× 100
𝑃07,08 =
1760+1970
1420+1590
× 100
𝑃07,08 =
3730
3010
× 100
𝑃07,08 = 123.92%
Comments:
The Prices increased by 23.92% in the year 2008 with respect to 2007.