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

Sample	1	2	3	4	5	6	Total
Mean	300	342	351	319	326	333	1971
Range	25	37	20	28	30	22	162

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

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.

Year	Production	Average
2000	105	\(\frac{105+115+120}{3}=113.33\)
2001	115
2002	120
2003	100(left out)
2004	110	\(\frac{110+125+135}{3}=123.33\)
2005	125
2006	135

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
\(UCL={ D }_{ 4 }\bar { R } =2.004(84)=168.336\)
\(CL=\bar { R } =84\)
\(LCL={ D }_{ 3 }\bar { R } =0(84)=0\)

Sample	1	2	3	4	5	6	7	8	9	10	Total
Mean	21	26	23	18	19	14	14	20	16	10	182
Range	5	6	9	7	4	6	8	9	4	7	65

\(\overset { = }{ X } =\frac { \sum { \overset { - }{ X } } }{ numberofsamples } =\frac { 182 }{ 10 } =18.2\quad \overset { - }{ R } =\frac { \sum { R } }{ n } =\frac { 65 }{ 10 } =6.5\)
The control limits for \(\overset{-}{X}\) chart is
\(UCL=\overset { = }{ X } +{ A }_{ 2 }\bar { R } =18.2+0.577(6.5)=21.95\)
\(CL=\overset { = }{ X } =18.2\)
\(LCL=\overset { = }{ X } -{ A }_{ 2 }\bar { R } =18.2-0.577(6.5)=14.5795\)
The control limits for Range chart is
\(UCL={ D }_{ 4 }\bar { R } =2.114(6.5)=13.741\)
\(CL=\bar { R } =65\)
\(LCL={ D }_{ 3 }\bar { R } =0(6.5)=0\)

Computation of Seasonal Index by the method of simple averages.

Year	I Quarter	II Quarter	III Quarter	IV Quarter
2005	255	351	425	400
2006	269	310	396	410
2007	291	332	358	395
2008	198	289	310	357
2009	200	290	331	359
2010	250	300	350	400
Quarterly Total	1463	1872	2170	2321
Quarterly Averages	243.83	312	361.67	386.83

S.I for I Quarter = \(\frac{Average\ of \ I\ quarter }{Grand\ average} \times100\)
Grand Average = \(\frac{1304.333}{4}=326.0833\)
S.I for I Q = \(\frac{243.8333}{326.0833} \times100= 74.77;\) 
S.I for II Q = \(\frac{312}{326.0833} \times100=95.68;\)
S.I for III Q = \(\frac{361.6667}{326.0833} \times 100=110.91; \) 
S.I for IV Q = \(\frac{386.833}{326.0833} \times 100=118.63\)

Computation of seasonal Indices by method of simple averages.
Months Jan Feb Mar Apr May June July Aug Sep Oct Nov Dec Year 2001 15 41 25 31 29 47 41 19 35 38 40 30 2002 20 21 27 19 17 25 29 31 35 39 30 44 2003 18 16 20 28 24 25 30 34 30 38 37 39 Monthly Total 53  78 72 78 70 97 100 84 100 115 107 113 Monthly Averages 17.67 26 24 26 23.33 32.33 33.33 28 33.33 38.33 35.67 37.67 S.I for Jan = \(\frac{Monthly\ Average\ (for\ Jan)}{Grand\ average}\times100\)
Grand Average = \(\frac{355.582}{12}=29.63\)
S.I for Jan = \(\frac{17.666}{29.361}\times100=59.62; \) 
S.I for Feb = \(\frac{26}{29.361}\times 100=87.72\)
Similarly other seasonal index values can be obtained.
 
\(\begin{array}{|c|c|} \hline \text { Months } & \text { Seasonal Index } \\ \hline \text { Jan } & 59.62 \\ \hline \text { Feb } & 87.72 \\ \hline \text { Mar } & 80.97 \\ \hline \text { Apr } & 87.72 \\ \hline \text { May } & 78.71 \\ \hline \text { June } & 109.08 \\ \hline \text { July } & 112.45 \\ \hline \text { Aug } & 94.47 \\ \hline \text { Sep } & 112.45 \\ \hline \text { Oct } & 129.32 \\ \hline \text { Nov } & 120.34 \\ \hline \text { Dec } & 127.09 \\ \hline \end{array}\)

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.

Computation of trend values by the method of least squares (ODD Years).

Year(x)	Production of Sugarcane(Y)	X=(x–2003)	X2	XY	Trend values(Yt)
2000	40	-3	9	-120	42.04
2001	45	-2	4	-90	43.07
2002	46	-1	1	-46	44.11
2003	42	0	0	0	45.14
2004	47	1	1	47	46.18
2005	50	2	4	100	47.22
2006	46	3	9	138	48.25
N=7	\(\sum Y\)=316	\(\sum X\)=0	\(\sum X\)2=8	\(\sum XY\)=29	\(\sum Yt\)=316

\(a=\frac { \sum { Y } }{ n } =\frac { 316 }{ 7 } =45.143;\quad b=\frac { \sum { XY } }{ { \sum { X } }^{ 2 } } =\frac { 29 }{ 28 } =1.036\)
Therefore, the required equation of the straight line trend is given by
Y = a + bX
Y = 45.143 + 1.036 (x - 2003)
The trend values can be obtained by
When X = 2000 , Yt = 45.143 + 1.036(2000–2003) = 42.035
When X = 2001, Yt = 45.143 + 1.036(2001–2003) = 43.071,
similarly other values can be obtained.