## The Method of Free Hand Curve (2020)

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

• Take the time periods along the $x$-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.

#### Merits (Free Hand Curve)

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

#### Demerits (Free Hand Curve)

• 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 from 1972 to 1978.

• 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

Solution:

The method of free hand curve is useful for:

1. Exploratory Data Analysis (EDA): As a preliminary step free hand curve method helps us to understand the basic characteristics of the data and identify potential relationships between variables.
2. Visual Communication: It also helps to present trends in the data in a clear and easily understandable way for non-statistical audiences.
3. Limited Data: When you have a relatively small dataset, a free hand curve might be sufficient to get a basic idea of the central tendency.

By understanding the method of free hand curves and its limitations, one can use it as a valuable tool for initial data exploration and visualization alongside other statistical techniques for a more robust analysis.

MCQs Intermediate Mathematics Part-I Quadratic Equations

## The Secular Trend Example and Uses (2020)

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.

### The Secular Trend is used

• For 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:

The secular trend may be used 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 to isolate non-trend fluctuations.

Note that

• These trends can be positive or negative. For example, the advancement of technology offers new opportunities but also raises concerns about job displacement and privacy.
• These trends can be interrelated. For instance, urbanization might be fueled by technological advancements that allow people to work remotely.
• Identifying secular trends can be challenging, as they unfold over a long period. However, by analyzing historical data, monitoring current developments, and considering expert opinions, one can gain valuable insights into the long-term direction of change.

By understanding and utilizing secular trends, individuals, businesses, and policymakers can make informed decisions, prepare for future challenges, and capitalize on emerging opportunities in a constantly evolving world.

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