What is seasonal variation?

Important Points to Remember in Chapter -1 - Time Series And Forecasting from Tamil Nadu Board Statistics Standard 12 Solutions

1. Definition:

According to Mooris Hamburg “A time series is a set of statistical observations arranged in chronological order”.

2. Additive approach:

$Y = T + C + S + R$ .

3. Multiplicative approach:

$Y = T \times C \times S \times R$ .

4. Graphical method:

Time series are plotted on a graph paper by taking time variable on the -axis and the values variable on the -axis.

5. Semi average method:

(i) Even number of years:

Find the average of the two parts of the series and place these values in the mid-year of each of the respective durations.

(ii) Odd number of years:

The middle year will be omitted. After dividing the data into two parts, find the arithmetic mean of each part.

6. Three yearly moving averages:

(i) Add up the values of the first years and place the yearly sum against the median year. [This sum is called moving total].

(ii) Leave the first year value, add up the values of the next three years and place it against its median year.

(iii) Each -yearly moving total must be divided by to get the -year moving averages, which is our required trend values.

7. In method of least squares, the trend line equation:

$y = a + b x$ , where $a = \frac{\sum_{i = 1}^{n} y_{i}}{n}$ and $b = \frac{\sum_{i = 1}^{n} x_{i} y_{i}}{\sum_{i = 1}^{n} x_{i}^{2}}$ .

8. Method of least squares when trend values $n$ is odd:

(i) Subtract the first year from all the years, say $x_{i}$

(ii) Take the middle value, say $A$ .

(iii) Find $u_{i} = x_{i} - A$

(iv) Find $u_{i}^{2}$ and $u_{i} y_{i}$ .

9. Normal equations in method of least squares:

$\sum_{i = 1}^{n} y_{i} = n a + b \sum_{i = 1}^{n} u_{i}$ and $\sum_{i = 1}^{n} u_{i} y_{i} = a \sum_{i = 1}^{n} u_{i} + b \sum_{i = 1}^{n} u_{i}^{2}$ .

10. Method to find the seasonal indices:

(i) The data is arranged season-wise

(ii) The data for all the seasons are added first for all the years and the seasonal averages for each year is computed.

(iii) The average of seasonal averages is calculated, $Grand average = \frac{Total of seasonal averages}{number of years}$ .

(iv) The seasonal average for each year is divided by the corresponding grand average and the results are expressed in percentages and these are called seasonal indices.

11. Irregular variation:

In a time series is composed of non-recurring sporadic (rare) form which is not attributed to trend, cyclical or seasonal factors.

12. Forecasting:

According to T.S. Lewis and R.A. Fox “Forecasting is to estimate what will happen at some future moment of time”.