Cost of living index number = \(\frac {\sum PV}{\sum V}\) = \(\frac {12610}{100}\) = 126.10
∴  The cost ofliving has increased upto 26.10% in 2011 as compared to 2010.

Cost of living index number = \({ \frac {\sum p_{1}q_{0}}{\sum p_{0}q_{0}}} \times100\)
C.L.I = \(\frac {174}{146.5} \times 100\)  = 118.77

The treand values is using fourly moving average  given below 

Using aggregate expenditure method,
Cost of living index number 
C.L.I = \(\sqrt \frac {\sum p_{1}q_{0}}{{\sum p_{0}q_{0}}}\times100\)
= \(\sqrt \frac {8600}{6510}\times100\)

= 132.10

Using family budget method,
  C.L.I =  \(\frac {\sum PV}{\sum V}\) = \(\frac {3284.72}{28}\) = 117.31

Using family budget method, cost of living index number
 C.L.I = \(\frac {\sum PV}{\sum V}\) = \(\frac {279550}{105}\) = 2662.38

Laspeyre's price index number
\(P^{L}_{01}\) = \({\frac {\sum p_{1}q_{0}}{\sum p_{0}q_{0}}} \times 100\)
= \(\frac {1532}{1058} \times 100\)  = 144.8
Paasche's pirce index number
\(P^{L}_{01}\)   =  \({\frac {\sum p_{1}q_{1}}{\sum p_{0}q_{1}}} \times 100\)
= \(\frac {1436}{994} \times 100\) = 144.4

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

The line of least fit is a line from which the sum of the deviations of various points is zero. This is the best method for obtaining the trend values.
(i) The straight line trend is represented by
 Y = a + b X ...(1)
where Y is the actual value and X is time
(ii)  The constants 'a' and 'b' are estimated by solving the following two normal Equations
 \(\sum\) 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.
(iii) By substituting the values of 'a' and 'b' in the trend equation (1), we get the line of best fit.

Tendency movements are due to nature, which repeat themselves periodically in every seasons. These variations repeat themselves in less than one year time. It is measured in an interval of time.
Seasonal variations may be influenced by natural force, social customs and traditions.

\(\overset { = }{ X } =\frac { \sum { \overset { - }{ X } } }{ number\ of\ samples } =\frac { 1971 }{ 6 } =328.5\quad \quad \overset { - }{ R } =\frac { \sum { R } }{ n } =\frac { 162 }{ 6 } =27\)
The control limits for \(\overset {-}{X}\) chart is
\(UCL=\overset { = }{ X } +{ A }_{ 2 }\overset { - }{ R } =328.5+0.483(27)=341.54\)
\(CL=\overset { = }{ X } =328.5\)
\(LCL=\overset { = }{ X } -{ A }_{ 2 }\overset { - }{ R } =328.5+0.483(27)=315.45\)

Given \(\bar { X } \) = 0532., σ = 0.002, n = 5
The control limits for \(\overset{-} {X}\) chart is 
\(UCL=\overset { = }{ X } +3\frac { \sigma }{ \sqrt { n } } =0.532+3\frac { 0.002 }{ \sqrt { 5 } } =0.5346\)
\(CL=\overset { = }{ X } =0.532\)
\(UCL=\overset { = }{ X } -3\frac { \sigma }{ \sqrt { n } } =0.532-3\frac { 0.002 }{ \sqrt { 5 } } =0.5293\)

Computation of four- yearly moving averages.

Computation of three- yearly moving averages.

Since the number of years is odd(seven), we will leave the middle year’s production value and obtain the averages of first three years and last three years.