Method of Semi-Averages

This method is also simple and relatively objective than the free hand method. The data is divided in two equal halves and the arithmetic mean of the two sets of values of Y is plotted against the center of the relative time span. If the numbers of observations are even the division into halves will be straight forward, however, if the number of observations are odd, then the middle most i.e.,\left( {\frac{{n + 1}}{2}} \right)th, item is dropped. The two points so obtained are joined through a straight line which shows the trend. The trend values of Y i.e., \widehat Y can then be read from the graph corresponding to each time period.

The arithmetic mean, since greatly affected by extreme values, is subjected to misleading values hence the trend obtained by plotting by means might be distorted. However, if extreme values are not apparent, this method may be fruitfully employed. To understand the estimation of trend, using the above noted two methods, consider the following worked example.

Example:

Measure the trend by the method of semi-average by using the following table given below. Also write the equation of the trend line with origin at 1984-85.

Years
Value in Million
1984 – 85
18.6
1985 – 86
22.6
1986 – 87
38.1
1987 – 88
40.9
1988 – 89
41.4
1989 – 90
40.1
1990 – 91
46.6
1991 – 92
60.7
1992 – 93
57.2
1993 – 94
53.4

           
Solution:

Years
Values
Semi-Totals
Semi-Average
Trend Values
1984 – 85
18.6
28.664 – 3.656  = 25.008
1985 – 86
22.6
32.32 – 3.656 = 28.664
1986 – 87
38.1
161.6
32.32
32.32
1987 – 88
40.9
32.32 + 3.656 = 35.976
1988 – 89
41.4
35.976 + 3.656 = 39.632
1989 – 90
40.1
39.632 + 3.656 = 43.288
1990 – 91
46.6
43.288 + 3.656 = 46.944
1991 – 92
60.7
253.0
5.60
50.60
1992 – 93
57.2
50.60 + 3.656 = 54.256
1993 – 94
53.4
54.256 + 3.656 = 57.912

Trend for 1991 – 92 = 50.60
Trend for 1986 – 87 = 32.32
Increase in trend in 5 years = 18.28
Increase in trend in 1 year = 3.656

Trend for one year is 3.656. It is called slope of the trend line and is denoted by b. Thus, b = 3.656. The trend for 1987 – 88 is calculated by adding 3.656 to 32.32 and similar calculations are done for the subsequent years. Trend for 1985 – 86 is less than the trend for 1986 – 87. Thus trend for 1985 – 86 is 32.32 – 3.656 = 28.664. Trend for the year 1984 – 85 = 25.008. This is called the intercept because 1984 – 85 is the origin. Intercept is the value of Y when X = 0.

Intercept is denoted by a. The equation of trend line is \widehat Y = a + bX = 25.008 + 3.656X (1984 – 85 = 0) where \widehat Y shows the trend values. This equation can be used to calculate the trend values of the time series. It can also be used for forecasting the future values of the variable.


semi-average