Statistics Chapter 5 (Index number)

Q.1) Define index number. Discuss problems in the construction of index numbers? 

Answer:
Index numbers are devices for measuring differences in the magnitude of group so related variables. The differences many have to do with the price of commodity, the physical quantities of goods produced consumed or such concepts as “intelligence”, beauty or efficiency. The comparison may be between periods of time, between places; between like categories, such as persons, schools or objects. Thus, we may have index numbers comparing the cost of living at different times or in different countries or localities, the physical value of production in different years or the efficiency of different school systems.
Problems in the construction of price numbers
The construction of index numbers requires a careful study of following points which may be termed as preliminaries to the construction of index numbers.
1. The purpose and scope of index number
2. Selection of commodity
3. Selection of the sources of data
4. Selection of base period
5. Selection of average
6. System of weighting
1. Purpose and scope of index number
It is important step in the construction of index numbers is to define in clear and concrete terms the objective or the purpose for which the index number is required.
2. Selection of commodities
When selecting the commodities, following points must be considered.
· Representative of the habits, tastes, customs and necessities of the people.
· The commodities should be such as remain uniform in quality from year to year.
· Select the most popular varieties for inclusion.
3. Selection of the sources of data
It is comparatively difficult to collect suitable price quotations for the commodities selected. The next job is to appoint any authority, who will supply the price quotations from time to time on regular basis, since price indices are often computed yearly, monthly, weekly and sometimes even daily. One of the usual practices for obtaining the data is through the data supplied by a suitable price-producing agency.
4. Selection of base period
A base period is the reference period and the prices for a given period are expressed as percentages of these for the same year.
In the construction of price index numbers selection of two base periods is an important step. There are two methods for selecting the base period. They are,
a) Fixed base method, b) Chain base method
5. Selection of average
In case of more than one commodity are taken into account and it is required to study the changes in the general price level, then price relatives are averaged into a single relative index for the each period. The averages, which are generally used for this purpose in view of their relative advantages and disadvantages.
i) Arithmetic mean (A.M)
ii) Median
iii) Geometric mean (G.M)
6. System of weighting
The commodities included for the construction of index numbers as food clothing etc. are not of equal importance.
Thus, in practice we may have two types of index numbers:
· Un-weighted index numbers
· Weighted index numbers

Read more: Statistics Class 11 Notes Cha 1 (Introduction To Statistics)

Q.2) Distinguish between weighted and un-weighted index number, and define four basic weighted index numbers.

Answer:
Thus, in practice we may have two types of index numbers
· Un-weighted index numbers
· Weighted index numbers
Un-weighted index numbers:
There are two methods of constructing un-weighted index numbers.
· Average of relatives method
· Simple Aggregative index number
Average of relative method
For computing this index number, we first obtain the simple relatives by using the ratio P_n / P₀ x 100 for each commodity in a composite index and then we obtain average of these simple relatives by dividing them by the number of commodities involved in the composite index.
Weighted index number:
A weighted index number is constructed by assigning weights to the commodities involved, weight could be quantities of the commodities consumed or purchased during the base year as well as in the given year.
Weight index numbers involve the following four basic weighted index numbers.
i) Laspeyre’s price index number or Laspeyre’s weighted aggregative price index number:
When the base year quantities are used as weights, the method is called Laspeyre’s price index number.
In the Laspeyre’s price index number formula, the prices vary from p_o to p_n, and base year quantity weights q_o are kept constant.

ii)  Paasche’s price index number or Paasche’s weighted aggregative index number
When the current year quantities are used as weights, the method is called Paasche’s index number and is as under:

iii)  Fisher’s ideal index number
This index number is the geometric mean of the Laspeyer’s and Paasche’s index numbers. The figure is obtained Fisher’s ideal index number, because it satisfies the theoretical Tests.

Q.3) Describe the various method of averaging that can be used in construction an index number.

Answer:
Methods of averaging:
In case of more than one commodity are taken into account and it is required to study the changes in the general price level, then price relatives are averaged into a single relative index for the each period. The averages, which are generally used for this purpose in view of their relative advantages and disadvantages.
· Arithmetic mean (A.M)
· Median
· Geometric mean (G.M)
1)Arithmetic mean is not recommended because it is an absolute measure of changes while in the construction of index number we have to deal with relative changes.
2)Median is the easiest of all three to calculate but it is ignores the extreme values like arithmetic mean it also measures absolute changes so median is also not a suitable average.
3)The geometric mean may be considered as one of the best averages in the construction of index number, because the main advantage of G.M is that it measures relative changes. As we deal with ratios and relative changes in the construction of index numbers. In addition, the index number constructed by geometric mean is reversible, due to which base shifting in easily possible.
Hence, geometric mean is the best average in the construction of index numbers.

Read more: Statistics Chapter 2 (Collection and presentation of Data) | Class 11 Notes

Q.4) Explain the process of construction index number of prices with Fixed base and Chain base giving examples.

Answer:
Construction of price index numbers selection of two base periods is an important step. There are two methods for selecting the base period. They are
ii) Fixed base method
iii) Chain base method
Fixed base method
In this method a particular normal year (may be the average of several year) is selected as base period, which does not change in the construction of index numbers for all other subsequent years. A base year is assigned an index 100 to indicate that the base year selected is a normal year. The prices of the other subsequent years are expressed as percentage relatives (also called price relatives) of the prices of the base year.
If pn denotes the price of a commodity during a given year (non-base year) and p0 denotes the price of the same commodity during the base year, then
Price relative, for the given year = pn / p0 x 100
Price relative is independent of unit of measurement.
Example Compute the price relative for the following data using 1989 as a base.

Solution:

Chain base method
Situation may arise in which we need to compare the prices of one year with the next, due to the change in tastes, habits and customs or requirement of the people. Here fixed base method will not be any more useful. It is suitable to change the base year frequently and index numbers should then be constructed by taking each previous year as a base. The method in which the base year is not fixed but changes from year to year is called chain base method, where the relatives computed by this method are called link relatives. Thus we have
Link relative = Price during a given year / Price during a previous (as a base) year x 100
Example 5.3. Compute the chain relative of 1992 and link relatives.

solution:

Hence 1992 chain relative=111x120x133/100×100= 177, indicates prices have increased by 77% from 1989 to 1992. A process by means of which we obtain link relatives and converting them back to a fixed base is called the chaining process; the indices thus determined are called chain indices.

Q.6) Explain the two methods of un-weighted index number.

Answer:
There are two methods of constructing un-weighted index numbers.
(1) Simple Aggregative Method (2) Simple Average of Relative Method
Simple Aggregative Method:
In this method, the total of the prices of commodities in a given (current) years is divided by the total of the prices of commodities in a base year and expressed as percentage.
Pon=∑Pn / ∑Po×100
Simple Average of Relatives Method:
In this method, we compute price relative or link relatives of the given commodities and then use one of the averages such as arithmetic mean, geometric mean, median etc. If we use arithmetic, mean as average, then
Pon=1/n∑ (Pn/Po) ×100
The simple average of relative method is very simple and easy to apply is superior to simple aggregative method. This method has only disadvantage that it gives equal weight to all items.
Example:
The following are the prices of four different commodities for 1990 and1991. Compute a price index by (1) Simple aggregative method and (2) Average of price relative method by using both arithmetic mean and geometric mean, taking1990 as base.

 Solution:
The necessary calculations are given below:

 Simple Aggregative Method:                  
Pon=∑Pn / ∑Po×100=2662/2623×100=101.49
(2) Average of Price Relative Method (using arithmetic mean):
Pon=1/n∑ (Pn/Po) ×100=14(407.64) =101.91
Average of Price Relative Method (using geometric mean)
Pon=antilog (∑log/n) =antilog (8.02134) =101.23

Q.7) Explain the difference between price relatives and Link relatives giving examples.

Answer:
PRICE RELATIVE
The prices of the other subsequent years are expressed as percentage relatives (also called price relatives) of the prices of the base year.
If p_n denotes the price of a commodity during a given year (non-base year) and po denotes the price of the same commodity during the base year, then
FORMULA:
Price relative, for the given year =p_n / p _0 x 100
LINK RELATIVE
The Link relative method in which the base year is not fixed but changes from year to year is called chain base method, where the relatives computed by this method are called link relatives. Thus we have
FORMULA:
Link relative = Price during a given year / Price during a previous (as a base) year x 100

Q.14) Explain the meaning of cost of living index number. Describe the method of construction adopted.

Answer:
Cost of living index:
The cost-of-living index, or general index, shows the difference in living costs between cities. The cost of living in the base city is always expressed as 100. The cost of living in the destination is then indexed against this number. Therefore, to take a simple example, if London is the base (100) and New York is the destination, and the New York index is 120, then New York is 20% more expensive than London. Similarly, if London is the base, Budapest is the destination, and the Budapest index is 80, than the cost of living in Budapest is 80% of London.
A cost-of-living index is a theoretical price index that measures relative cost of living over time or regions. An index measures differences in the price of goods and services, and allows for substitutions to other items as prices vary. As different people, consume different kinds of commodities and the same commodities in different proportions. The consumer price index helps us in determining the effect of size. Fall in price index helps us in determining the effect of rise and fall in prices on different classes of consumers living in different areas. The consumer price index number is significant because the demand of a higher wage is based on the cost of living index and the wages and salaries in most nations are adjusted according to this index number. The ‘Cost of living index’, also known as ‘consumer price index or Cost of living price index’ is the country’s principal measure of price change.
Methods of Construction Cost of Living Index –
The Living Index or Consumer Price Index has been described as a basket of goods and services, which is nationally purchased each quarter. As prices change from one quarter to the next, so does the total cost (or price) of the basket. Of the various ways in which a CPI could be described, this description conforms closely with the procedures actually followed. Traditionally, there are two methods to construct a CPI, namely, aggregate expenditure method, and the family budget method.
1.       Aggregate Expenditure Method – The quantities of commodities consumed by the particular group in the base year are estimated and these figures or their proportions are used as weights. Then the total expenditure on each commodity for each year is calculated. The price of the current year is multiplied by the quantity or weight of the base year. These products are added. Similarly, for the base year total expenditure on each commodity is calculated by multiplying the quantity consumed by its price in the base year. These products are also added. The total expenditure of the current year is divided by the total expenditure of the base year and 100 to get the required index numbers multiply the resulting figure. In this method, the current period quantities are not used as weights because these quantities change from year to year.
2. Family Budget Method – The family budgets of a large number of people are carefully studied and the aggregate expenditure of the average family on various items is estimated. These values are used as weights. Current year’s price are converted into price relatives on the basis of base year’s prices and these prices relatives are multiplied by the respective values of the commodities, in the base year. The total of these products is divided by the sum of the weights .

Read more: Statistics-Chapter 3 (Measures of Location)