# Method of Semi-Averages

The secular trend can also be measured by the method of semi-averages. The steps are:

• Divide the time series data into two equal portions. If observations are odd then either omit the middle value or include the middle value in each half.
• Take the average of each part and place these average values against the midpoints of the two parts.
• Plot the semi-averages in the graph of the original values.
• Draw the required trend line through these two potted points and extend it to cover the whole period.
• It is simple to compute the slope and $y$-intercept of the line drawn from two points. The trend values can be found from the semi-average trend line or by estimated straight line as explained:

Let $y’_1$ and $y’_2$ be the semi-averages placed against the times $x_1$ and $x_2$. Let the estimated straight line $y’=a+bx$ is to pass through the points ($x_1$, $y’_1$) and ($x_2$, $y’_2$). The constant “$a$” and “$b$” can easily be determined. the equation of the line passing through the points ($x_1$, $y’_1$) and ($x_2$, $y’_2$) can be written as:

\begin{align*}
y’ – y’_1 &= \frac{y’_2-y’_1}{x_2-x_1}(x-x_1)\\
&= b(x-x_1)\\
\Rightarrow y’ &= (y’_1 – bx_1) + bx\\
&= a+bx, \quad \text{ where $a=y’_1-bx_1$}
\end{align*}

For even number of observations the slope of trend line can be found as:

\begin{align*}
b&=\frac{1}{n/2}\left(\frac{S_2}{n/2} – \frac{S_1}{n/2} \right)\\
&= \frac{1}{n/2} \left(\frac{S_2-S_1}{n/2}\right)\\
&= \frac{4(S_2-S_1)}{n^2},
\end{align*}

where $S_1$ is sum of $y$-values for the first half of the period, $S_2$ is sum of $y$-values of the second half of the period, and $n$ is the number of time units covered by the time-series.

Merits of Semi-Averages

• The method of semi-averages is simple, easy, and quick.
• It smooths out seasonal variations
• It gives a better approximation to the trend because it is based on a mathematical model.

Demeris of Semi-Averages

• It is a rough and objective method.
• The arithmetic mean used in Semi Average is greatly affected by very large or by very small values.
• The method of semi-averages is applicable when the trend is linear. This method is not appropriate if the trend is not linear.

Numerical Example 1:

The following table shows the property damaged by road accidents in Punjab for the year 1973 to 1979.

1. Obtain the semi-averages trend line
2. Find out the trend values.

Solution

Let $x=t-1973$

\begin{align*}
y’_1 &= 277, x_1 = 1, y’_2 = 625, x_2=5\\
b&=\frac{y’_2-y’_1}{x_2-x_1}=\frac{625-277}{5-1}=87\\
a&=y’_1 – bx_1 = 277-87(1)=190
\end{align*}

The semi-average trend line $y’=190+87x$ (with the origin at 1973)

Numerical Example 2:

The following table gives the number of books in thousands sold at a book store for the year 1973 to 1981

1. Find the equation of the semi-averages trend line
2. Compute the trend values
3. Estimate the number of books sold for the year 1982.

Solution

Let $x=t-1973$

\begin{align*}
y’_1 &= 35, x_1=1.5, y’_2=20, x_2=6.5\\
b &= \frac{y’_2 – y’_1}{x_2-x_1} = \frac{20-35}{6.5-1.5} =-3\\
a &= y’_1 – bx_1 = 35 – (-3)(1.5) = 39.5\\
y’&= 39.5 – 3x (\text{with origin at 1973})
\end{align*}

For the year 1982, the estimated number of books sold is: $y’=39.5-3(9)=12.5$. 