Measurement of Seasonal Trend:
(a) Simple Average Method,
(b) Link Relative Method,
(c) Ratio to Moving Average Method, and
(d) Ratio to Trend Method.
(a) Simple Average Method: Under this method, the average ( ) of all the monthly or quarterly values for each year are found out. Each monthly or quarterly value (y_{i}) is divided by the corresponding average and the results are expressed as percentage:
Then the mean index or seasonal index (S_{i}) is calculated for each month or quarter. If the mean of all seasonal indices is not equal to 100, then they will be adjusted.
Example 8:
Take data from Example 1, and calculate
the four seasonal indices by the ‘simpleaverage method’.
Solution:
Year/Quarter 
y_{ } 
Mean 

I 
II 
III 
IV 

2003 
219 
357 
645 
513 
433.5 
2004 
549 
640 
701 
590 
620 
2005 
657 
394 
543 
600 
548.5 
Now the above observed values are converted to indices using the following formula:
Year/Quarter 
I 
II 
III 
IV 
Total 
2003 
50.52% 
82.35% 
148.79% 
118.34% 

2004 
88.55 
103.23 
113.06 
95.16 

2005 
119.78 
71.83 
99.00 
109.39 

Season Index (S_{i}) 
86.28% 
85.80% 
120.28% 
107.63% 
400.00 
(b) Link Relative Method: Under this method, the data for each month or quarter are expressed in percentage, known as ‘Link Relatives’. An appropriate average of the link relatives is taken, usually a median is taken. Convert these averages into a series of chain indices. The chain indices are adjusted for the fraction of the effect of the trend. The adjusted chain indices are further reduced to the same level as the first month or quarter.
Example 9:
Take the data from Example 1, and
calculate seasonal indices by using ‘link relative method’.
Solution:
The observed values are converted into price relatives or link relatives using the following formula:

Where P_{n} is the value of current year P_{o} is the value of base year 
and then the link relatives are converted into chain indices (chaining process) using the following formula:
= (L.R. × C.I. of preceding year) ÷ 100 
Where L.R. is the link relative C.I. is the chain index 
Year 
Quarter 
Total 

I 
II 
III 
IV 

2003 
– 
163.01% 
180.67% 
79.53% 

2004 
107.02% 
116.58 
109.53 
84.17 

2005 
111.36 
59.97 
137.82 
110.50 

Median (link relative) 
109.19 
116.58 
137.82 
84.17 

Chain Index 
100 
116.58 
160.67 
135.24 
512.49 
Adj. C. I. 
78.05%^{* } 
90.99% 
125.40% 
105.56% 
400.00 
^{*}
Adjusted chain index for QI: 100 ÷ 512.49 × 400 = 78.05, and so for
other quarters.
(c) Ratio to Moving Average Method: A 12month or 4quarter moving average centred is computed. The observed values are divided by the corresponding centred moving average and the results are expressed in percentage:
The monthly or quarterly averages of these percentages are found out. The adjusted values are the indices of the seasonal variations.
Example 10:
Take data from Example 1 and calculate
seasonal indices using ‘ratio to moving average method’.
Solution:
Quarter 
y 
4Quarter Moving Total 
8Quarter Moving Total 
4Quarter Moving Average (Centred) 
Ratio
to Moving Average



I 
II 
III 
IV 

2003 
I 
219 



– 





II 
357 



– 



1734 


III 
645 
3798 
475 


135.8 


2064 


IV 
513 
4411 
551 



93.1 

2347 

2004 
I 
549 
4750 
594 
92.4 




2403 


II 
640 
4883 
610 

104.9 



2480 


III 
701 
5068 
634 


110.6 


2588 


IV 
590 
4930 
616 



95.8 

2342 

2005 
I 
657 
4526 
566 
116.1 




2184 


II 
394 
4378 
547 

72.0 



2194 


III 
543 




– 





IV 
600 





– 



Mean Seasonal Index (total = 410.5) 
104.3 
88.5 
123.2 
94.5 

Adjusted Seasonal Index (S_{i}) (total = 400) 
101.6%^{*} 
86.2% 
120.1% 
92.1% 
^{*}
Adjusted seasonal index for QI: 104.3 ÷ 410.5 × 400, and so on for
other three quarters.
(d) Ratio to Trend Method: An average for each year is found out and a straight line is fitted by least squares method. The trend values for each month or quarter are calculated on the assumption that the data correspond to the middle of the month or quarter. Each original value is divided by the corresponding calculated trend values and expressed in percentage. A mean of these percentages are calculated for each month or quarter. The adjusted values are the indices of seasonal variation.
Example 11:
Take data from Example 1 and calculate seasonal indices using ‘ratio to trend method’.
Solution:
Quarters 
y 
x 

x 

x^{2 } 




2003 
I 
219 
–11 




455 
48.13% 


II 
357 
–9 
469 
76.12 

433.5 
–8 
–3468 
64 


III 
645 
–7 
484 
133.26 







IV 
513 
–5 
498 
103.01 






2004 
I 
549 
–3 
512 
107.23 







II 
640 
–1 
527 
121.44 

620 
0 
0 
0 


III 
701 
1 
541 
129.57 







IV 
590 
3 
556 
106.12 






2005 
I 
657 
5 
570 
115.26 







II 
394 
7 
584 
67.47 

548.5 
8 
4388 
64 


III 
543 
9 
599 
90.65 







IV 
600 
11 
613 
97.88 




0 
1602 
0 
920 
128 


Now arranging the above calculated values in last column as follows:
Year/Quarter 
I 
II 
III 
IV 
Total 
2003 
48.13 
76.12 
133.26 
103.01 

2004 
107.23 
121.44 
129.57 
106.12 

2005 
115.26 
67.47 
90.65 
97.88 

Seasonal Index (S_{i}) 
90.21 
88.34 
117.83 
102.34 
398.72 
Adj. S.I. 
90.50%^{*} 
88.62% 
118.21% 
102.67% 
400.00 
^{*} Adjusted seasonal index for QI: 90.21 ÷ 398.72 × 400, and so on.
Measurement of Cyclical
Variation:
The cyclical and random components of a time series are first isolated from the time series using the multiplicative model:
y_{i}
= T_{i} + S_{i} + C_{i} + R_{i}
Where T_{i}: Secular trend
S_{i}: Seasonal variation
C_{i}: Cyclical variation
R_{i}: Random variation
This can be done by dividing y_{i} by the product of T_{i} and S_{i}:
The Random component R_{i} will now be separated from the time series by using the smoothing technique, moving average. These moving averages show the indices of cyclical variation.
Example 12:
Take data from Example 1 and isolate cyclical component from the time series.
Solution:
Quarters 
y 
^{* } 
S_{i}^{** } 

C_{i}
× Ri =
×100 
3Quarter Moving Average (C_{i}) 

2003 
I 
219 
455 
86.28 
392.57 
55.79% 
– 

II 
357 
469 
85.80 
402.40 
88.72 
85.10 

III 
645 
484 
120.28 
582.16 
110.79 
98.41 

IV 
513 
498 
107.63 
536.00 
95.71 
110.26 
2004 
I 
549 
512 
86.28 
441.75 
124.28 
120.51 

II 
640 
527 
85.80 
452.17 
141.54 
124.52 

III 
701 
541 
120.28 
650.71 
107.73 
115.95 

IV 
590 
556 
107.63 
598.42 
98.59 
113.30 
2005 
I 
657 
570 
86.28 
491.80 
133.59 
103.60 

II 
394 
584 
85.80 
501.07 
78.63 
95.86 

III 
543 
599 
120.28 
720.48 
75.37 
81.74 

IV 
600 
613 
107.63 
657.77 
91.22 
– 
^{*} As calculated in the previous example
** As calculated in Example