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**The ratio-to-trend method :** The ratio-to-trend method is similar to ratio-to-moving-average method.

The only difference is the way of obtaining the trend values. Whereas in the ratio-to-moving-average method, the trend values are obtained by the method of moving averages, in the ratio-to-trend method, the corresponding trend is obtained by the method of least sequares.

The steps in the calculation of seasonal variation are as follows :

(i) Arrange the unadjusted data by years and months.

(ii) Compute the trend values for each month with the help of least squares equation.

(iii) Express the data for each month as a percentage ratio of the corresponding trend value.

(iv) Aggregate the January’s ratios, February’s ratios, etc., computed previously

(v) Find the average ratio for each month.

(vi) Adjust the average ratios found in step (v) so that they will themselves average 100 per cent.

The last step gives us the seasonal index for each month.

Sometimes the median is used in place of the arithmetic average of the ratios-to-trend. The choice depends upon circumstances but there is a preference for the median if several erratic ratios are found. In fact, if a fairly large number of years, say, 20 or 15, are used in the computation, it is not uncommon to omit extremely erratic ratios from the computation of average of monthly ratios. Only the arithmetic average should be used for small number of years.

This method has the advantage of simplicity and case of interpretation. Although it makes allowance for the trend, it may be influenced by errors in the calculation of the trend. The method may also be influenced by cyclical and erratic influences. This source of possible error is eliminated by the selection of a period of time in which depression is offset by prosperity.

**Illustration :** Find seasonal variations by the ratio-to-trend method from the following data :

**Year 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter**

2000 30 40 36 34

2001 34 52 40 44

2002 40 58 54 48

2003 54 76 68 62

2004 80 92 86 82

**Solution :** For finding out seasonal variations by ratio-to-trend method, first the trend for yearly data will be obtained and convert them into quarterly data.

Average 92.78 118.28 102.92 89.12

The average of quarterly average of trend figures :

Quarterly seasonal Index for 1st Quarter :

Quarterly seasonal Index for 2rd Quarter :

Quarterly seasonal Index for 3rd Quarter :

Quarterly seasonal Index for 4th Quarter :

The total of seasonal indices should be equal to 400 and that for monthly indices should be 1200.

**Merits**

(i) This method is based on a logical procedure for measuring seasonal variations. This procedure has an advantage over the moving average method for it has a ratio to trend value for each month for which data is available. So this method avoids loss of data which is inherent in the case of moving averages. If the period of time series is very short then the advantage becomes more prominent.

(ii) It is a simple method.

(iii) It is easy to understand.

**Limitations :**

If the cyclical changes are very wide in the time series, the trend can never follow the actual data, as closely as a 12-month moving average will follow, under the ratio-to-trend method. There will be more bias in a seasonal index computed by ratio to trend method.

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