Category: Analysing the Secular Trend

The Method of Free Hand Curve

The secular trend is measured by the method of the free-hand curve in the following steps:

  • Take the time periods along $x$-axis by taking appropriate scaling
  • Plot the points for observed values of the $Y$ variable as the dependent variable against the given time periods
  • Join these plotted points by line segments to get a historigram
  • Draw a free hand smooth curve (or a straight line) through the histogram

In this method we draw the given times series data on graph paper, then we draw a free hand trend line through the plotted graph according to the trend of the graph. Then we read trend values from this free-hand trend line.

It is generally preferred to use a curve instead of a straight line to show the secular trend.


  • The free-hand curve method is a simple, easy, and quick method for measuring secular trends.
  • A well-fitted trend line (or curve) gives a close approximation to the trend based on a mathematical model.


  • It is a rough and crude method.
  • It is greatly affected by personal bias as different persons may fit different trends to the same data.
    The estimates are not reliable due to personal bias.

Question: The following time series shows the number of road accidents in Punjab for the year 1972 to 1978.

No. of Accidents2493263826993038374540794688
  • Obtain the historigram showing the number of road accidents and a free hand trend line by drawing a straight line
  • Find the trend values for this time series


Method of Free Hand Curve
YearValueTotalMeanTrend Value
19722493  2200
19732638  2550
19742699  2950
19763745  3650
19774079  4050
19784688  4499

The Secular Trend

For the estimation of the secular trend of a time series, the most commonly used method is to fit a straight line $\hat{y} = a+bx$, an exponential curve $\hat{y}=ab^x$, and a second-degree parabola $\hat{y}=a +bx+ cx^2$, etc, where $y$ is the value of a time series variable, $x$ representing the time and all others are constants (the intercept $a$, and the slope $b$). The method of least squares is a widely used method to determine the values of the constants appearing in such an equation.

It is used

  • For the purpose of prediction (or projection) into the future
  • The detrending process (removal of trend) in a time series for studying other non-trend fluctuations.
  • It is used for historical description

The secular trend can be represented either by a straight line or by some type of smooth curve. It is measured by the following methods:

Least Squares Method (secular trend)

The secular trend may be used whether in determining how a time series has grown in the past or in making a forecast. The trend line is used to adjust a series to eliminate the effect of the secular trend in order to isolate non-trend fluctuations.

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