The secular trends 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 an 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 an even number of observations, the slope of the 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 the sum of $y$-values for the first half of the period, $S_2$ is the sum of $y$-values of the second half of the period, and $n$ is the number of time units covered by the time series.

The following merits and demerits of the Method of Semi Averages are as described:

**Merits of Method 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.

**Demerits of Method 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:** **Method of Semi Averages**

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

Year | 1973 | 1974 | 1975 | 1976 | 1977 | 1978 | 1979 |

Property Damage | 201 | 238 | 392 | 507 | 484 | 648 | 742 |

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

**Solution**

Let $x=t-1973$

Year | Property Damaged | Semi Total | Semi Average | Coded Year | Trend Values |
---|---|---|---|---|---|

1973 | 201 | 0 | $y’=190+87(0)=190$ | ||

1974 | 238 | 831 | 277 | 1 | $y’=190+87(1)=277$ |

1975 | 392 | 2 | $y’=190+87(2)=364$ | ||

1976 | 507 | 3 | $y’=190+87(3)=451$ | ||

1977 | 484 | 4 | $y’=190+87(4)=538$ | ||

1978 | 549 | 1875 | 625 | 5 | $y’=190+87(5)=625$ |

1979 | 742 | 6 | $y’=190+87(6)=712$ |

\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:** **Method of Semi Averages**

The following table gives the number of books in thousands sold at a bookstore for the years 1973 to 1981

Year | 1973 | 1974 | 1975 | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 |

No. of Books Sold | 42 | 38 | 35 | 25 | 32 | 24 | 20 | 19 | 17 |

- Find the equation of the semi-average trend line
- Compute the trend values
- Estimate the number of books sold for the year 1982.

**Solution**

Let $x=t-1973$

Year | No. of books (y) | Semi Total | Semi Average | Coded year | Trend Values |
---|---|---|---|---|---|

1973 | 42 | 0 | $y’=39.5 – 3(0)=39.5$ | ||

1974 | 38 | 140 | 35 | 1 | $y’=39.5 – 3(1)=36.5$ |

1975 | 35 | 2 | $y’=39.5 – 3(2)=33.5$ | ||

1976 | 25 | 3 | $y’=39.5 – 3(3)=30.5$ | ||

1977 | 32 | 4 | $y’=39.5 – 3(4)=27.5$ | ||

1978 | 24 | 5 | $y’=39.5 – 3(5)=24.5$ | ||

1979 | 20 | 80 | 20 | 6 | $y’=39.5 – 3(6)= 21.5$ |

1980 | 19 | 7 | $y’=39.5 – 3(7)=18.5$ | ||

1981 | 17 | 8 | $y’=39.5 – 3(8)=15.5$ |

\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 $y’=39.5-3(9)=12.5$.