We have to find a way of isolating and measuring the seasonal variations. There are two reasons for isolating and measuring the effect of seasonal variations.

- To study the changes brought by seasons in the values of the given variable in a time series
- To remove it from the time series to determine the value of the variable

Summing the values of a particular season for several years, the irregular variations will cancel each other, due to independent random disturbances. If we also eliminate the effect of trend and cyclical variations, the seasonal variations will be left out which are expressed as a percentage of their average.

### Seasonal Variations

A study of seasonal variation leads to more realistic planning of production and purchases etc.

**Seasonal Index** Method

When the effect of the trend has been eliminated, we can calculate a measure of seasonal variation known as the seasonal index. A seasonal index is simply an average of the monthly or quarterly value of different years expressed as a percentage of averages of all the monthly or quarterly values of the year.

The following methods are used to estimate seasonal variations.

- Average percentage method (simple average method)
- Link relative method
- Ratio to the trend of short-time values
- Ratio to the trend of long-time averages projected to short times
- Ratio to moving average

**The Simple Average Method**

Assume the series is expressed as

$$Y=TSCI$$

Consider the long-time averages as trend values and eliminate the trend element by expressing a short-time observed value as a percentage of the corresponding long-time average. In the multiplicative model, we obtain

\begin{align*}

\frac{\text{short time observed value} }{\text{long time average}}\times &= \frac{TSCI}{T}\times 100\\

&=SCI\times 100

\end{align*}

This percentage of the long-time average represents the seasonal (S), the cyclical (C), and the irregular (I) component.

Once $SCI$ is obtained, we try to remove $CI$ as much as possible from $SCI$. This is done by arranging these percentages season-wise for all the long times (say years) and taking the modified arithmetic mean for each season by ignoring both the smallest and the largest percentages. These would be seasonal indices.

If the average of these indices is not 100, then the adjustment can be made, by expressing these seasonal indices as the percentage of their arithmetic mean. The adjustment factor would be

\begin{align*}

\frac{100}{\text{Mean of Seasonal Indiex}} \rightarrow \frac{400}{\text{sums of quarterly index}} \,\, \text{ or } \frac{1200}{\text{sums of monthly indices}}

\end{align*}

### Example of Seasonal Variations

**Question:** The following data is about several automobiles sold.

Year | Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 |
---|---|---|---|---|

1981 | 250 | 278 | 315 | 288 |

1982 | 247 | 265 | 301 | 285 |

1983 | 261 | 285 | 353 | 373 |

1984 | 300 | 325 | 370 | 343 |

1985 | 281 | 317 | 381 | 374 |

Calculate the seasonal indices by the average percentage method.

**Solution:**

First, we obtain the yearly (long-term) averages

Year | 1981 | 1982 | 1983 | 1984 | 1985 |
---|---|---|---|---|---|

Year Total | 1131 | 1098 | 1272 | 1338 | 1353 |

Yearly Average | 1131/4=282.75 | 274.50 | 318.00 | 334.50 | 338.25 |

Next, we divide each quarterly value by the corresponding yearly average and express the results as percentages. That is,

Year | Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | |
---|---|---|---|---|---|

1981 | $\frac{250}{282.75}\times=88.42$ | $\frac{278}{282.75}\times=98.32^*$ | Total (modified) | $\frac{288}{282.75}\times=101.86^*$ | |

1982 | $\frac{247}{274.50}\times=89.98^*$ | $\frac{265}{274.50}\times=96.54$ | $\frac{301}{274.50}\times=109.65^*$ | $\frac{285}{274.50}\times=103.83$ | |

1983 | $\frac{261}{318.00}\times=82.08^*$ | $\frac{285}{318.00}\times=89.62^*$ | $\frac{353}{318.00}\times=111.01$ | $\frac{373}{318.00}\times=117.30^*$ | |

1984 | $\frac{300}{334.50}\times=89.69$ | $\frac{325}{334.50}\times=97.16$ | $\frac{370}{334.50}\times=110.61$ | $\frac{343}{334.50}\times=102.54$ | |

1985 | $\frac{281}{338.25}\times=83.07$ | $\frac{317}{338.25}\times=93.72$ | $\frac{381}{338.25}\times=112.64^*$ | $\frac{374}{338.25}\times=110.57$ | |

Total (modified) | 261.18 | 247.42 | 333.03 | 316.94 | Total |

Mean (modified) | $\frac{261.18}{3}=87.06$ | $\frac{247.42}{3}=95.81$ | $\frac{333.03}{3}=111.01$ | $\frac{316.94}{3}=105.65$ | 399.52 |

* on values represents the smallest and largest values in a quarter that are not included in the total.

### Statistical Software for Seasonal Variation

Several statistical software packages can automate these calculations for you. Popular options include:

- Python libraries like Pandas and Statsmodels
- R statistical computing environment
- Excel with add-in tools like Data Analysis ToolPak