Cost of Living Index Number is constructed to study the effect of changes in the price of goods and services of consumers for a current period as compared with the base period. The change in the cost of living index number between any two periods means the change in income which will be necessary to maintain the same standard of living in both the periods. Therefore the cost of living index number measures the average increase in the cost to maintain the same standard of life.
Further, the consumption habits of people differ widely from class to class (rich, poor, middle class) and even with the region. The changes in the price level affect the different classes of people, consequently, the general price index numbers fail to reflect the effect of changes in their cost of living in different classes of people. Therefore, the cost of living index number measures the general price movement of the commodities consumed by different classes of people.

The ratio between the total value of current period and total value of the base period is known as true value ratio.
(\( \frac{\sum P_{1} q_{1}}{\sum p_{0} q_{0}}\) is a true value tario)

Factor reversal test is another test for testing the consistency of a good index number. The product of price index number and quantity index number from the base year to the current year should be equal to the true value ratio. That is ratio between the total value of current period and total value of the base period is known as true value ratio. Factor reversal test is given by,
\(\text { i.e. } P_{01} \times Q_{01}=\frac{\sum P_{1} q_{1}}{\sum p_{0} q_{0}}=V_{01}(\text { Except the factor 100) }\)

Time reversal test is an important test for testing the consistency of a good index number. This test maintains time consistency by working both forward and backward with respect to time (here time refers to base year and current year). Symbolically the following Relationship should be satisfied, P₀₁ \(\times\) P₁₀ = 1 Fisher's index number formula satisfies the above relationship
\(\text { i.e. } P_{01}=\frac{1}{P_{10}} \text { or } P_{01} \times P_{10}=1(\text { Except the factor } 100 \text { ) }\)

Index numbers are studied to know the relative changes in price and quantity for any two years compared. There are two tests which are used to test the adequacy for an index number. The two tests are as follows.
(i) Time reversal test
(ii) Factor reversal test
The criterion for a good index number is to satisfy the above two tests.

Fisher's price index number is the geometric mean of Laspeyre's and Paasche's price index number. Hence it is weighted index number.
Fisher's price index number = \(\sqrt {\frac {\sum p_{1}q_{0}}{\sum p_{0}q_{0} }}{\times}{\frac {\sum p_{1}q_{1}}{\sum p_{0}q_{1}} \times {100}}\)

An index numberis a device which shows by its variations the changes in a magnitude which is not capable of accurate measurements in itself or of direct valuation in practice

The two normal equations are
\(\sum\) Y = n a + b \(\sum\) X
\(\sum\) XY  = a  \(\sum\) X + b  \(\sum\)X²  where n is the number of years given in the data.

 Seasonal index is a measure of how a particular season compares with the average season.

Irregular variations do not have particular pattern and there is no regular period of time of their occurrences. Normally they are short terms variations but its occurrence sometimes has its effect so intense that they may give rise to new cyclic or other movements of variations.
For example floods, wars, earthquakes, Tsunami, strikes, lockouts etc.

Cyclic uniformly periodic in nature. They may or may not follow exactly similar patterns after equal intervals of time. Generally one cyclic period ranges from 7 to.9 years and there is no hard and fast rule in the fixation of years for a cyclic period. For example, every business cycle has a Start-Boom-Depression- Recover maintenance during booms and depressions, changes in government monetary policies, changes in interestrates.

The different methods of measurements of  trends are
(i) Free hand or graphic method
(ii) Method of semi averages
(iii) Method of moving averages
(iv) Method of least squares

Smce \(\sum\)X = 0, a =\(\frac {\sum Y}{n}\) = \(\frac {244}{5}\) = 48.8
b = \(\frac {\sum XY}{\sum X^2}\) = \(\frac {20}{10}\) = 2
∴ The required equation of the straight line trend is given by
Y = a + bX  \(\Rightarrow \) Y = 48.8 + 2X
\(\Rightarrow \) Y = 48.8 +2 (X - 1994) .... (1)
∴ Number of salesmen in 1997 is put X = 1997 in (1)
∴ Y = 48.8 + 2 (1997 - 1994) 
= 48.8 + 2 (3) 
= 48.8 + 6 = 54.8
∴  Number of salesmen in 1997 is 54.8