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)

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 = \(\sqrt {\frac {\sum p_{1}q_{0}}{\sum p_{0}q_{0} }}{\times}{\frac {\sum p_{1}q_{1}}{\sum p_{0}q_{1}} \times {100}}\)

(i) It is an important tool for the formulating decision and management policies
(ii) It helps in studying the trends and tendencies
(iii) It determines the inflation and deflation in an economy

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

It is a general tendency of time series to increase or decrease or stagnates during a long period of time. An upward tendency is usually observed in population of a country, production, sales, prices in industries, income of individuals etc., A downward tendency is observed in deaths,epidemics, prices of electronic gadgets, water sources, mortality rate etc. It is not necessarily that the increase or decrease should be in the same direction throughout the given period of time. This feature is known as secular trend.

Time series has an important objective to identify the variations and try to eliminate the variations and also helps us to estimate or predict the future values.

A time series consists of a set of observations arranged in chronological order (either ascending or descending). It is a statistical data which relates to successive intervals or point of time.

Since the number of years is even(eight), we can equally divide the given data it two equal parts and obtain the averages of first four years and last four years.

Year	Production	Average
1990	15	\(\frac{15+11+20+10}{4}=14\)
1991	11
1992	20
1993	10
1994	15	\(\frac{15+25+35+30}{4}=26.25\)
1995	25
1996	35
1997	30

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

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.

Year	Sales	4-yearly centered moving total	4-yearly moving Average	4-yearly centered moving Average
2001	124	---	--	--
2002	120	--	--	--
		519	129.75
2003	135	--		139.37
		540	135
2004	140	--		139.75
		578	144.50
2005	145	--		147.87
		605	151.25
2006	158	--		162.50
		635	158.75
2007	162	--		162.50
		665	166.25
2008	170	--	--	-
2009	175	--	--	-

Year (X)	No. of Salesmen Y	X = x -1994	X2	XY
1992	46	-2	4	-92
1993	48	-1	1	-48
1994	42	0	0	0
1995	56	1	1	56
1996	52	2	4	104
	244	0	10	20

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

Year	Sales	X=x- 2002	X2	XY
2000	35	-2	4	-70
2001	36	-1	1	-36
2002	79	0	0	0
2003	80	1	1	80
2004	40	2	4	80
	270	0	10	54

 Since \(\sum X = 0, a = \frac{\sum Y}{n}\) = \(\frac {270}{5}\) = 54
b = \(\frac {\sum XY}{\sum X^2} = \frac {54}{10}\) = 5.4
∴  The required equation of the straight line trend is given by Y  = a +bX  
\(\Rightarrow \) Y = 54 + 5.4 (x - 2002)
The trend values can be obtained as follows: 
When X = 2000, Y_t = 54 + 5.4 (2000 - 2002)
= 54 + 5.4 (-2) = 43.2
When X = 2001, Y_t = 54 + 5.4 (2001 - 2002)
= 54 + 5.4(-1) = 48.6
When X = 2002, Y_t = 54 + 5.4 (2002 - 2002)
= 54
When X = 2003, Y_t = 54 + 5.4 (2003 - 2002) 
= 54 + 5.4 = 59.4
When X = 2004, Y_t = 54 + 5.4 (2004 - 2002)
= 54 + 5.4 (2) = 64.8

Sample	1	2	3	4	5	6	7	8	9	10	Total
Mean	383	508	505	582	557	337	514	614	707	753	5460
Range	95	128	100	91	68	65	148	28	37	80	840

\(\overset { = }{ X } =\frac { \sum { \bar { X } } }{ 10 } =\frac { 5460 }{ 10 } =546\)
The control limits for \(\overset{-}{X}\) chart is
\(UCL=\overset { = }{ X } +{ A }_{ 2 }\bar { R } =546+0.483(84)=586.57\)
\(CL=\overset { = }{ X } =546\)
\(LCL=\overset { = }{ X } -{ A }_{ 2 }\bar { R } =546-0.483(84)=505.43\)
\(\bar { R } =\frac { \sum { R } }{ n } =\frac { 840 }{ 10 } =84\)
The control limits for Range chart is
\(UCL={ D }_{ 4 }\bar { R } =2.004(84)=168.336\)
\(CL=\bar { R } =84\)
\(LCL={ D }_{ 3 }\bar { R } =0(84)=0\)

Commodities Price Weights
(V) \(P=\frac { { P }_{ 1 } }{ { P }_{ 0 } } \times 100\) PV 2007
(P₀) 2011
(P₁) A 350 400 40 114.285 4571.4 B 175 250 35 142.857 4999.99 C 100 115 15 115 1725 D 75 105 20 114 2280 E 60 80 25 133.333 3333.33     Total 135   17429 76 Value of Cost of Living Index Number\(\frac { \sum { PV } }{ \sum { V } } =\frac { 17429 76 }{ 135 } =129.1093\)
Hence, the Cost of Living Index Number for a particular class of people for the year 2011 is increased by 129.1093 % as compared to the year 2007.

Here the base year quantities are given, therefore we can apply Aggregate Expenditure Method.

Commodities	Number of Units q0(2010)	Price (2010) p0	Price (2015) p1	p0q0	p1q0
Rice	5	1500	1750	7500	8750
Sugar	3.5	1100	1200	3850	4200
Pulses	3	800	950	2400	2850
Cloth	2	1200	1550	2400	3100
Ghee	0.75	550	700	412.5	525
Rent	12	2500	3000	30000	36000
Fuel	8	750	600	6000	4800
Misc	110	3200	3500	32000	35000
			Total	84562.5	95225

Cost of Living Index Number \(=\frac { \sum { { p }_{ 1 }{ q }_{ 0 } } }{ \sum { { p }_{ 0 }{ q }_{ 0 } } } \times 100=\frac { 95225 }{ 84562.5 } \times 100=112.609\)
Hence, the Cost of Living Index Number for a particular class of people for the year 2015 is increased by 12.61 % as compared to the year 2010.

Commodities	Price		Quandity		p0q0	p0q1	p1q0	p1q1
	2000 (p0)	2010 q1	2000 (p0)	2010 (q1)
Rice	38	35	6	7	228	266	210	245
Wheat	12	18	7	10	84	120	126	180
Rent	10	15	10	15	100	150	150	225
Fuel	25	30	12	16	300	400	630	480
Miscellaneous	30	33	8	10	240	300	264	330
				Total	952	1236	1110	1460

Laspeyre’s price index number
\({ P }_{ 01 }^{ L }=\frac { \sum { { p }_{ 1 }{ q }_{ 0 } } }{ \sum { { p }_{ 0 }{ q }_{ 0 } } } \times 100=\frac { 1110 }{ 952 } \times 100=116.60\)
On an average, there is an increase of 16.60 % in the price of the commodities when the year 2000 compared with the year 2010.
Paasche’s price index number
\({ P }_{ 01 }^{ P }=\frac { \sum { { p }_{ 1 }{ q }_{ 1 } } }{ \sum { { p }_{ 0 }{ q }_{ 1 } } } \times 100=\frac { 1460 }{ 1236 } \times 100=118.12\)
On an average, there is an increase of 18.12 % in the price of the commodities when the year 2000 compared with the year 2010.
Fisher’s price index number
\({ P }_{ 01 }^{ F }=\sqrt { \frac { \sum { { p }_{ 1 }{ q }_{ 0 } } \times \sum { { p }_{ 1 }{ q }_{ 1 } } }{ \sum { { p }_{ 0 }{ q }_{ 0 } } \times \sum { { p }_{ 0 }{ q }_{ 1 } } } } \times 100=\sqrt { \frac { 1110\times 1460 }{ 952\times 1236 } } \times 100=117.36\)
On an average, there is an increase of 17.36 % in the price of the commodities when the year 2000 compared with the year 2010.

Computation of trend values by the method of least squares.
In case of EVEN number of years, let us consider
\(X=\frac{\text{(x-Arithimetic mean of two middle years)}}{0.5}\)

Year(x)	Sales(Y)	X=\(\frac{(x-1998.5)}{0.5}\)	XY	X2	Trend Values (Yt)
1995	6.7	-7	-46.9	49	5.6166
1996	5.3	-5	-26.5	25	5.7190
1997	4.3	-3	-12.9	9	5.8214
1998	6.1	-1	-6.1	1	5.9238
1999	5.6	1	5.6	1	6.0261
2000	7.9	3	23.7	9	6.1285
2001	5.8	5	29.0	25	6.2309
2002	6.1	7	42.7	49	6.3333
N = 8	47.8	\(\sum X\) = 0	8.6	168

\(a=\frac { \sum { Y } }{ n } =\frac { 47.8 }{ 8 } =5.975;\quad b=\frac { \sum { XY } }{ { \sum { X } }^{ 2 } } =\frac { 8.6 }{ 168 } =0.05119\)
Therefore, the required equation of the straight line trend is given by
Y = a + bX; Y = 5.975 + 0.05119 X.
When X = 1995, Y_t = 5.975 + 0.05119\(\left( \frac { 1995-1998.5 }{ 0.5 } \right) =5.6166\)
When X = 1996, Y_t = 5.975 + 0.05119\(\left( \frac { 1996-1998.5 }{ 0.5 } \right) =5.7190\)
similarly other values can be obtained.