UNIT 6
Index numbers
The value of money does not remain same over the time. A rise in the price levels means a fall in the value of money and a fall in the price level means a rise in the value of money. Thus index number is a statistical device that measures the relative change in the level of price from one time period to another.
Definition
“Index numbers are quantitative measures of growth of prices, production, inventory and other quantities of economic interest” ………Ronold
An index number measures how much a variable changes over the time. Index number is calculated by finding the ratio of current value to a base value.
Features of index number
 Index numbers are specialized averages.
 Index numbers measures the change in one variable or a group of variable
 Index numbers measures the effect of changes over a period of time.
 Index numbers are meant to study the changes in the effects of such factors which cannot be measured directly
 Simple aggregative method – in this method, the index number is equal to the sum of prices in the current year as a percentage of the sum of prices in the base year.
Where, P01 = Index number
P 1= Total of the current year’s prices of all commodities
P 0= Total of the base year’s prices of all commodities
Examples 1–
Commodity  Price in base year 2005  Price in current year 2010 
A  10  20 
B  15  25 
C  40  60 
D  25  40 
Solution
Commodity  Price in base year 2005  Price in current year 2010 
A  10  20 
B  15  25 
C  40  60 
D  25  40 
 = 90  = 145 
Index number ( P01 ) =
P01 = (145/90)*100 = 161.11
It means the price in 2010 were 61% more than the price in 2005
Example 2
Find the index number from the data given below
Commodities  Units  Price in 2007  Price in 2008 
Sugar  Quintal  2200  3200 
Milk  Quintal  18  20 
Oil  Liter  68  71 
Wheat  Quintal  900  1000 
Clothing  Meter  50  60 




Solution
Commodities  Units  Price in 2007  Price in 2008 
Sugar  Quintal  2200  3200 
Milk  Quintal  18  20 
Oil  Liter  68  71 
Wheat  Quintal  900  1000 
Clothing  Meter  50  60 

 = 3236  = 4351 
Index number ( P01 ) =
P01 = (4351/3236)*100 = 134.45
It means the price in 2008 were 34% more than the price in 2007
Example 3 –
Construct the price index for 2003, taking the year 2000 as base year
Commodities  Price in 2000  Price in 2003 
A  60  80 
B  50  60 
C  70  100 
D  120  160 
E  100  150 



Solution
Commodities  Price in 2000  P 0  Price in 2003  P 1 
A  60  80 
B  50  60 
C  70  100 
D  120  160 
E  100  150 
 = 400  = 550 
Index number ( P01 ) =
P01 = (550/400)*100 = 137.5
Therefore there is an increase of 37.5% in the prices in 2003 as against 2000.
Example 4
Compute the price index for the years 2001, 2002, 2003, 2004 taking 2000 as base year
Year  2000  2001  2002  2003  2004 
Price  120  144  168  204  216 
Solution
Price index for different years
2000  (120/120)*100 = 100 
2001  (144/120)*100 = 120 
2002  (168/120)*100 = 140 
2003  (204/120)*100 = 170 
2004  (216/120)*100 = 180 
Example 5 –
Prepare simple aggregative price index
Commodities  Price in 1995  P 0  Price in 2003  P 1 
Wheat  100  140 
Rice  200  250 
Pulses  250  350 
Sugar  14  20 
Oil  40  50 
Solution
Commodities  Price in 1995  P 0  Price in 2003  P 1 
Wheat  100  140 
Rice  200  250 
Pulses  250  350 
Sugar  14  20 
Oil  40  50 
 = 604  = 810 
Simple aggregative index number = (810/604)*100 = 134.1
2. Simple average of relative method  in this method, index number is equal to the sum of price relatives divided by the number of items.
Where, N= number of items
Example 1 –
Commodity  Base year  Current year 
A  10  20 
B  15  25 
C  40  60 
D  25  40 



Solution
Commodity  Base year  Current year  Price relatives 
A  10  20  (20/10)*100 = 200 
B  15  25  (25/15)*100 =166.7 
C  40  60  (60/40)*100 =150 
D  25  40  (40/25)*100 =160 
N = 4 

 = 676.7 
Index number = 676.7/4 = 169.2
Example 2 –
Construct the index number for the year 2010
Commodities  Price (2009)  Price(2010) 
P  6  10 
Q  12  2 
R  4  6 
S  10  12 
T  8  12 



Solution
Commodities  Price (2009)  Price(2010)  Price relative 
P  6  10  166.67 
Q  12  2  16.67 
R  4  6  150 
S  10  12  120 
T  8  12  150 
N = 5 

 603.34 
Index number = 603.34/4 = 120.68
Example 3 –
Using simple average of price relative method find price index for 2001, taking 1996 as base year for the following data
Commodity  Wheat  Rice  Sugar  Ghee  Tea 
Price in 1996  12  20  12  40  80 
Price in 2001  16  25  16  60  96 
Solution
Commodities  Price (2009)  Price(2010)  Price relative 
Wheat  12  16  (16/12)*100 = 133.33 
Rice  20  25  (25/20)*100 = 125 
Sugar  12  16  133.33 
Ghee  40  60  150 
Tea  80  96  120 
N = 5 

 661.66 
=661.66 = 132.33
5
Therefore Price Index for 2001, taking 1996 as base year, = 132.33
Example 4 –
Using simple average of price relative method find price index for 2010, taking 2009 as base year for the following data
Commodities  Price (2009)  Price(2010) 
A  60  80 
B  50  60 
C  60  72 
D  50  75 
E  25  37 .5 
F  20  30 
Solution
Commodities  Price (2009)  Price(2010)  Price relatives 
A  60  80  133.33 
B  50  60  120 
C  60  72  120 
D  50  75  150 
E  25  37 .5  150 
F  20  30  150 
N = 6 

 823.33 
= 823.33/6 = 137.22
3. Weighted aggregative method – in this method, according to the relative importance, different weights are assigned to the items. Many formulas developed to estimate index numbers on the basis of weights.
Some of the formulas given below
 Laspeyre’s formula  in this method, the quantities of the base year are accepted as weight
 Paasche’s formula – in this method, the quantities of the current year are accepted as weight
 Dorbish and Bowley’s formula – this method is the combination of Laspeyre’s formula and Paasche’s formula
 Fisher’s ideal formula – this method is the geometric mean of Laspeyre’s formula and Paasche’s formula
 Marshall – edgeworth method  In this method also both the current year as well as base year prices and quantities are considered.
 Kelly’s method –
Where q refers to quantity of some period, not necessarily of the mean of the base year and current year.
Example 1 –
Commodity  Base year  Current year  
PO  QO  P1  Q1  
A  10  5  20  2 
B  15  4  25  8 
C  40  2  60  6 
D  25  3  40  4 
Solution
Commodity  Base year  Current year 



 
PO  QO  P1  Q1  Poqo  P1qo  Poq1  P1q1  
A  10  5  20  2  50  100  20  40 
B  15  4  25  8  60  100  120  200 
C  40  2  60  6  80  120  240  360 
D  25  3  40  4  75  120  100  160 




 265  440  480  760 
 Laspeyre’s formula
P 01 = (440/265)*100 = 166.04
 Paasche’s formula
P 01 = (760/480)*100 = 158.33
 Dorbish and Bowley’s formula
P 01 = ((440/265) + (760/480)) *100 = 162
2
 Fisher’s ideal formula
P 01 = √ ((440/265) + (760/480)) *100 = 162.1
Example 2
Commodity  Base year  Current year  
PO  QO  P1  Q1  
A  15  500  20  600 
B  18  590  23  640 
C  22  450  24  500 
Solution
Commodity  Base year  Current year 



 
PO  QO  P1  Q1  Poqo  P1qo  Poq1  P1q1  
A  15  500  20  600  7500  10000  9000  12000 
B  18  590  23  640  10620  13570  11520  14720 
C  22  450  24  500  9900  10800  11000  12000 













 28020  34370  31520  38720 
 Laspeyre’s formula
P 01 = (34370/28020)*100 = 122.66
 Paasche’s formula
P 01 = (38720/31520)*100 = 122.84
 Dorbish and Bowley’s formula
P 01 = ((34370/28020) + (38720/31520)) *100 = 122.66
2
 Fisher’s ideal formula
P 01 = √ = ((34370/28020) + (38720/31520)) *100 = 122.69
Example 3
Commodity  Base year  Current year  
PO  QO  P1  Q1  
A  2  8  4  6 
B  5  10  6  5 
C  4  14  5  10 
D  2  19  2  13 
Solution
Commodity  Base year  Current year 



 
PO  QO  P1  Q1  Poqo  P1qo  Poq1  P1q1  
A  2  8  4  6  16  32  12  24 
B  5  10  6  5  50  60  25  30 
C  4  14  5  10  56  70  40  50 
D  2  19  2  13  38  38  26  26 




 160  200  103  130 
 Laspeyre’s formula
P 01 = (200/160)*100 = 125
 Paasche’s formula
P 01 = (130/103)*100 = 126.21
 Dorbish and Bowley’s formula
P 01 = ((200/160) + (130/103)) *100 = 125.6
2
 Fisher’s ideal formula
P 01 = √ = ((200/160) + (130/103)) *100 = 125.61
 MarshallEdgeworth method
= (200+130)/(160+103) *100 = 125.48
Example 4 – Calculate the price indices from the following data by applying (1) Laspeyre’s method (2) Paasche’s method and (3) Fisher ideal number by taking 2010 as the base year.
Commodity  2010  2011  
PO  QO  P1  Q1  
A  20  10  25  13 
B  50  8  60  7 
C  35  7  40  6 
D  25  5  35  4 
Solution
Commodity  2010  2011 



 
PO  QO  P1  Q1  Poqo  P1qo  Poq1  P1q1  
A  20  10  25  13  200  250  260  325 
B  50  8  60  7  400  480  350  420 
C  35  7  40  6  245  280  210  240 
D  25  5  35  4  125  175  100  140 




 970  1185  920  1125 
 Laspeyre’s formula
P 01 = (1185/970)*100 = 122.16
 Paasche’s formula
P 01 = (1125/920)*100 = 122.28
 Fisher’s ideal formula
P 01 = √ = ((1185/970) + (1125/920)) *100 = 120.55
Example 5  Calculate the Dorbish and Bowley’s price index number for the following data taking 2014 as base year.
Item  2010  2011  
PO  QO  P1  Q1  
Oil  80  3  100  4 
Pulses  35  2  45  3 
Sugar  25  2  30  3 
Rice  50  30  54  35 
Solution
Item  2010  2011 



 
PO  QO  P1  Q1  Poqo  P1qo  Poq1  P1q1  
Oil  80  3  100  4  240  300  320  400 
Pulses  35  2  45  3  70  90  105  135 
Sugar  25  2  30  3  50  60  75  90 
Rice  50  30  54  35  1500  1620  1750  1890 




 1860  2070  2250  2515 
 Dorbish and Bowley’s formula
P 01 = ((2070/1860) + (2515/2250)) *100 = 111.38
2
Example 6 – calculate a suitable price index from the following data
Commodity  Quantity  Price  

 2007  2010 
X  25  3  4 
Y  12  5  7 
Z  10  6  5 
Solution
Commodity  Q  P0  P1  P0Q  P1Q 
X  25  3  4  75  100 
Y  12  5  7  60  84 
Z  10  6  5  60  50 



 195  234 
Kelly price index
= 235/195*100 = 120
4. Weighted average of relative method – in this method different weights are used for the items according to their relative importance. If p = [p1/ p0] × 100 is the price relative index and w = p0q0 is attached to the commodity
Where, means sum of weights for different commodities
Sum of price relatives
Example 1 –
Commodity  Weight  Base price year  Current price year 
A  5  10  20 
B  4  15  25 
C  2  40  60 
D  3  25  40 
Solution
Commodity  Weight  Base price year  Current price year  Price relatives  RW 
A  5  10  20  20/10*100 = 200  1000 
B  4  15  25  25/15*100 =166.7  666.8 
C  2  40  60  60/40*100 = 150  300 
D  3  25  40  40/25*100 = 160  480 
 14 


 2446.8 
P01 = 2446.8/14 = 174.8
Example 2 – compute price index by applying weighted average of relative method
Commodity  Quantity  Base price year  Current price year 
Wheat  20  3  4 
Flour  40  1.5  1.6 
Milk  10  1  1.5 
Solution
Commodity  Quantity  Base price year  Current price year  Weight  Price relatives  RW 
Wheat  20  3  4  60  133.3  8000 
Flour  40  1.5  1.6  60  106.7  6400 
Milk  10  1  1.5  10  150.0  1500 










 130 
 15900 
P01 = 15900/130 = 122.30
Example 3 – calculate weighted average of relative method
Commodity  Base price year  Current price year  Weight 
x  3  4  7 
y  1.5  1.6  8 
z  1  1.5  9 
Solution
Commodity  Base price year  Current price year  Weight  Price relatives  RW 
x  3  4  7  133.3  933.33 
y  1.5  1.6  8  106.7  853.33 
z  1  1.5  9  150.0  1350 


 24 
 3136.66 
P01 = 3136.66/24 = 130.67
The construction of the price index involves the following problems
 Selection of base year – the first step in preparing index number is the selection of base year. The base year is that year with reference to which changes in the price in other years are compared and expressed in percentage. The base year should be normal year free from abnormal condition like war, flood, etc. Base year are selected in two ways
 Fixed base method in which the base year remains fixed
 Chain base method in which base year goes on changing. Ex 2000 base year for 2001, 2001 base year for 2002, and so on
2. Selection of commodities – selection of commodities is one of the problems in constructing index number. As all commodities are not included, only representative commodities are selected keeping in mind the purpose and type of index number.
The following points are considered while selecting commodities
 Items should be representative of taste, habits and custom of people
 Items should be recognizable
 Quality of the item should be stable over two different places and periods
 The economic and social importance of various items should be considered
 The items should be fairly large in number.
 All those varieties of a commodity which are in common use and are stable in character should be included.
3. Collection of prices – the next problem is the collection of prices. Problems are from where the prices to be collected, which price to select wholesale or retail, whether to include taxes in the price. The following points should be considered while collecting prices
 Where commodities are traded in large, prices should be collected from those places
 Published information regarding the prices should also be utilized
 Selection of price depends on the type of index number prepared
 Non biased individual and institution should be selected who supply price quotation
 Prices collected from various places should be averaged
4. Selection of averages – Fourth problem is to choose a suitable average. Theoretically, geometric mean is the best for this purpose. But, in practice, arithmetic mean is used because it is easier to follow.
5. Selection of weights – commodities included in the calculation of index numbers are not of equal importance. Therefore proper weight should be assigned to the commodities for accurate index numbers. Weight should be unbiased and be rationale. For ex – price of books should be given more weightage while preparing index numbers for teachers rather than for workers.
6. Purpose of index numbers – the important point in the construction of index numbers is the objective of index numbers. Before preparing index numbers, it is important to be clear about the purpose of the index numbers. Different index numbers are prepared with a specific purpose.
Sources
 B.N Gupta – Statistics
 S.P Singh – statistics
 Gupta and Kapoor – Statistics
 Yule and Kendall – Statistics method