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Time Series Analysis

Method of Semi Averages

This method is very simple and relatively objective as a freehand method. In this method, we classify the time series data into two equal parts and then calculate averages for each half. If the data is for even number of years, it is easily divided into two. If the data is for an odd number of years, then the year at the middle of the time series is left and the two halves are constituted with the period on each side of the mid-year. Let us discuss the Method of Semi Averages in detail.

Method of Semi Averages

In this method, we can find the solution of a secular trend. For this, we have to show our time series on graph paper. For example, we can take sales on X-axis and data of production on Y-axis. Now make the original graph by plotting points on graph paper with time and value pairs. After plotting original data we can calculate the trend line. For calculating the trend line, we will calculate semi-average.

We divide the data into two equal parts with respect to time. And then we plot the arithmetic mean of the sets of values of Y against the center of the relative time span. If the number of observations is even then the division into halves will be done easily.

But, for an odd number of observations, we will drop the middle most item, i.e. \(\frac{n+1}{2}\) th term. We need to join these two points together through a straight line which shows the trend. The trend values can then be read from the graph corresponding to each time period.

Since extreme values greatly affect the arithmetic mean, and it is subjected to misleading values. Due to this, these trends may give distorted plots.  But, if extreme values are not apparent, we may easily use and employ this method. To understand the estimation of trends, using the above mentioned two methods, consider the following working example.

Method of Semi Averages


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  1. This method is simple to understand as compare to other methods for measuring the secular trends.
  2. Everyone who applies this method will get the same result.


  1. The method assumes a straight line relationship between the plotted points without considering the fact whether that relationship exists or not.
  2. If we add more data to the original data then we have to do the complete process again for the new data to get the trend values and the trend line also changes.

Explanation of the Method

Here are two cases of calculating semi-average of data:

When data is even: In this case, the time series will be into two parts and then we calculate the average of each part. Suppose if we have 10 years data then we divide it into 5 -5 years and then we will calculate the first five-year average and the next five-year average after this we have to plot this on the graph paper. This will show the trend line as shown in the picture.

When data is odd: In this case, we just leave the middle data and we will follow the above-said procedure for the rest.

Solved Example on Method of Semi Averages

Year production Semi averages
1971 40 \(\frac{40 + 45 +40 + 42}{4}\) = 41.75


1972 45
1973 40
1974 42
1975 46 \(\frac{46 + 52 + 56 + 61}{4}\) = 53.75


1976 52
1977 56
1978 61

 Thus we get two points 41.75 and 53.75 which we shall plot corresponding to their middle years i.e. 1972.5 and 1976.5. By joining these points we will obtain the required trend line.

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Time series consist of two mathematical modals.
What are they?


additive and multiplicative models


1.Decreases in the employment in sugar factory during off-season.
2.Fall in death rate due to advances in science.
what type of component

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