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SEASONAL VARIATIONS 1. 2. 3. 4.

The method of simple average Ratio to trend method Ratio to moving average method Link relative method

Method of Simple Average 1. Average the data for each month or quarter for all the years 2. Find the totals of each month or quarter 3. Divide each total by the number of years for which data are given. If we are given monthly data for 4 years, we must first get the total for each month for 4 years and divide each total by to 4 to get an average 4. We can get an average of monthly averages by dividing the total of monthly average by 12 5. We must take the average of month-averages as 100 and get the Seasonal Index as follows:

Exercise: 1. Compute the average seasonal movement for the following series by the method of simple average Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov 2005 318 281 278 250 231 216 223 245 269 302 325 2006 342 309 299 268 249 236 242 262 288 321 342 2007 367 328 320 287 269 251 259 284 309 345 367 2008 392 349 342 311 290 273 282 305 328 364 389 2009 420 378 370 334 314 296 305 330 356 396 322 2. Compute the average seasonal movement for the following series by the method of simple average Year Quarterly Production I II III IV 1998 3.5 3.9 3.4 3.6 1999 3.5 4.1 3.7 4.0 2000 3.5 3.9 3.7 4.2 2001 4.1 4.6 3.8 4.5 2002 4.1 4.4 4.2 4.5

Dec 347 364 394 417 452

3. Compute the average seasonal movement Bombay Stock Exchange closing data for the last five years by the method of simple average Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 2006 2007 2008 2009 2010 Ratio to Trend Steps (i) (ii)

By applying the method of least squares, obtain the trend values Divide the original data of time series for each season (month/quarter) by the corresponding trend values and multiply these ratio by 100

(iii)

In order to eliminate the irregular and cyclical movements, the seasonal figures are averaged with any one of the measures of central tendency, mean or median. These indices are adjusted to a total of 1,200 for monthly data or 400 for quarterly data by multiplying each index by a suitable factor

(iv)

(

)

4. Find the seasonal variation by the ratio to trend method from the given data below Year 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter 2001 86 95 96 99 2002 96 102 104 110 2003 103 108 106 107 5. Find seasonal variations by the ratio-to-trend method from the data given below Year 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter 2005 30 40 36 34 2006 34 52 50 44 2007 40 58 54 48 2008 54 76 68 62 2009 80 92 86 82

Ratio to moving average Steps (i)

(ii)

(iii) (v)

Calculate 12 months moving average (in case of quarterly data, calculate 4 quarter moving average) which eliminates seasonal and irregular fluctuations and represents trend and cycles i.e. T.C. Express original values as a percentage of centers moving average values for all months i.e.

Arrange these percentages according to years and months to eliminate irregular factors The sum of these indices should be 1200 (400) for monthly or quarterly data. If it is not so, an adjustment is made to eliminate the discrepancy i.e (

)

6. Calculate seasonal indices by the ratio to moving average method, from the following data Quarter 1998 1999 2000 2001 2002 I 40 42 41 45 44 II 35 37 35 36 38 III 38 39 38 36 38 IV 40 38 42 41 7. Calculate seasonal indices by the ratio to moving average method, from the following data Year 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter 2007 68 62 61 63 2008 65 58 66 61 2009 68 63 63 67 Link Relative Method Steps (i)

Calculate

the

link

relatives

of

the

seasonal

figures.

The

formula

is

(ii)

For calculating the average of the ling relatives for each season, take the arithmetic mean or the median.

(iii)

Convert these link relatives into chain relatives on the basis of the first season, the formula is

(iv)

(v)

(vi)

Find some difference between the chain relatives of the first season and the chain relative calculated by the previous method. This is due to the long term changes. To correct these chain relatives The chain relative of the first season calculated by the first method is deducted from the chain relative (of the first season) calculated by the second method. The difference is divided by the number of seasons. The resulting figures multiplied by1,2, are ducted respectively from the chain relatives of the 2nd, 3rd, or 4th season. These are corrected chain relatives. Theses corrected chain relatives are expressed as percentage of their averages. These are known as seasonal relations.

8. Apply the method of link relatives to the following data and calculate the seasonal indices Year 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter 1999 60 65 62 69 2000 62 68 65 68 2001 65 70 64 62 2002 70 75 68 67 2003 72 80 70 78 9. Apply the method of link relatives to the following data and calculate the seasonal indices Quarter 2005 2006 2007 2008 2009 I 6.0 5.4 6.6 7.2 6.6 II 6.5 7.9 6.5 5.8 7.3 III 7.8 8.4 9.3 7.5 8.0 IV 8.7 7.3 6.4 8.5 7.1 Deseasonalisation of data: The objective of studying seasonal variation is to measure them and to eliminate them from the given series. Elimination of the seasonal effects from the given values is termed as deseasonaliation of the data. It helps to adjust the given time series for seasonal variations, thus leaving with trend component, cyclic and irregular movements. Assuming multiplicative model of the time series, the deseasonalised values are obtained as:

Deseasonalisation is need for the study of cyclic component. It helps the management for planning future production programme, for forecasting and control. In additive model of time series, the deseasonalised values obtained on subtracting the seasonal variation from the given data. Thus: Deseasonalised Data = (T + S+ C + I) – S = T + C + I

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The method of simple average Ratio to trend method Ratio to moving average method Link relative method

Method of Simple Average 1. Average the data for each month or quarter for all the years 2. Find the totals of each month or quarter 3. Divide each total by the number of years for which data are given. If we are given monthly data for 4 years, we must first get the total for each month for 4 years and divide each total by to 4 to get an average 4. We can get an average of monthly averages by dividing the total of monthly average by 12 5. We must take the average of month-averages as 100 and get the Seasonal Index as follows:

Exercise: 1. Compute the average seasonal movement for the following series by the method of simple average Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov 2005 318 281 278 250 231 216 223 245 269 302 325 2006 342 309 299 268 249 236 242 262 288 321 342 2007 367 328 320 287 269 251 259 284 309 345 367 2008 392 349 342 311 290 273 282 305 328 364 389 2009 420 378 370 334 314 296 305 330 356 396 322 2. Compute the average seasonal movement for the following series by the method of simple average Year Quarterly Production I II III IV 1998 3.5 3.9 3.4 3.6 1999 3.5 4.1 3.7 4.0 2000 3.5 3.9 3.7 4.2 2001 4.1 4.6 3.8 4.5 2002 4.1 4.4 4.2 4.5

Dec 347 364 394 417 452

3. Compute the average seasonal movement Bombay Stock Exchange closing data for the last five years by the method of simple average Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 2006 2007 2008 2009 2010 Ratio to Trend Steps (i) (ii)

By applying the method of least squares, obtain the trend values Divide the original data of time series for each season (month/quarter) by the corresponding trend values and multiply these ratio by 100

(iii)

In order to eliminate the irregular and cyclical movements, the seasonal figures are averaged with any one of the measures of central tendency, mean or median. These indices are adjusted to a total of 1,200 for monthly data or 400 for quarterly data by multiplying each index by a suitable factor

(iv)

(

)

4. Find the seasonal variation by the ratio to trend method from the given data below Year 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter 2001 86 95 96 99 2002 96 102 104 110 2003 103 108 106 107 5. Find seasonal variations by the ratio-to-trend method from the data given below Year 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter 2005 30 40 36 34 2006 34 52 50 44 2007 40 58 54 48 2008 54 76 68 62 2009 80 92 86 82

Ratio to moving average Steps (i)

(ii)

(iii) (v)

Calculate 12 months moving average (in case of quarterly data, calculate 4 quarter moving average) which eliminates seasonal and irregular fluctuations and represents trend and cycles i.e. T.C. Express original values as a percentage of centers moving average values for all months i.e.

Arrange these percentages according to years and months to eliminate irregular factors The sum of these indices should be 1200 (400) for monthly or quarterly data. If it is not so, an adjustment is made to eliminate the discrepancy i.e (

)

6. Calculate seasonal indices by the ratio to moving average method, from the following data Quarter 1998 1999 2000 2001 2002 I 40 42 41 45 44 II 35 37 35 36 38 III 38 39 38 36 38 IV 40 38 42 41 7. Calculate seasonal indices by the ratio to moving average method, from the following data Year 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter 2007 68 62 61 63 2008 65 58 66 61 2009 68 63 63 67 Link Relative Method Steps (i)

Calculate

the

link

relatives

of

the

seasonal

figures.

The

formula

is

(ii)

For calculating the average of the ling relatives for each season, take the arithmetic mean or the median.

(iii)

Convert these link relatives into chain relatives on the basis of the first season, the formula is

(iv)

(v)

(vi)

Find some difference between the chain relatives of the first season and the chain relative calculated by the previous method. This is due to the long term changes. To correct these chain relatives The chain relative of the first season calculated by the first method is deducted from the chain relative (of the first season) calculated by the second method. The difference is divided by the number of seasons. The resulting figures multiplied by1,2, are ducted respectively from the chain relatives of the 2nd, 3rd, or 4th season. These are corrected chain relatives. Theses corrected chain relatives are expressed as percentage of their averages. These are known as seasonal relations.

8. Apply the method of link relatives to the following data and calculate the seasonal indices Year 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter 1999 60 65 62 69 2000 62 68 65 68 2001 65 70 64 62 2002 70 75 68 67 2003 72 80 70 78 9. Apply the method of link relatives to the following data and calculate the seasonal indices Quarter 2005 2006 2007 2008 2009 I 6.0 5.4 6.6 7.2 6.6 II 6.5 7.9 6.5 5.8 7.3 III 7.8 8.4 9.3 7.5 8.0 IV 8.7 7.3 6.4 8.5 7.1 Deseasonalisation of data: The objective of studying seasonal variation is to measure them and to eliminate them from the given series. Elimination of the seasonal effects from the given values is termed as deseasonaliation of the data. It helps to adjust the given time series for seasonal variations, thus leaving with trend component, cyclic and irregular movements. Assuming multiplicative model of the time series, the deseasonalised values are obtained as:

Deseasonalisation is need for the study of cyclic component. It helps the management for planning future production programme, for forecasting and control. In additive model of time series, the deseasonalised values obtained on subtracting the seasonal variation from the given data. Thus: Deseasonalised Data = (T + S+ C + I) – S = T + C + I

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