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Method of Semi Averages (2020)

May 2, 2024Oct 4, 2020 by Muhammad Imdad Ullah

Post Views: 97

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 $y^{'} = a + b x$ 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:

$\begin{aligned} y^{'} - y_{1}^{'} & = \frac{y_{2}^{'} - y_{1}^{'}}{x_{2} - x_{1}} (x - x_{1}) \\ = b (x - x_{1}) \\ \Rightarrow y^{'} & = (y_{1}^{'} - b x_{1}) + b x \\ = a + b x, where a = y_{1}^{'} - b x_{1} \end{aligned}$

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

$\begin{aligned} b & = \frac{1}{n / 2} (\frac{S_{2}}{n / 2} - \frac{S_{1}}{n / 2}) \\ = \frac{1}{n / 2} (\frac{S_{2} - S_{1}}{n / 2}) \\ = \frac{4 (S_{2} - S_{1})}{n^{2}}, \end{aligned}$

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	$y^{'} = 190 + 87 (6) = 712$

$\begin{aligned} y_{1}^{'} & = 277, x_{1} = 1, y_{2}^{'} = 625, x_{2} = 5 \\ b & = \frac{y_{2}^{'} - y_{1}^{'}}{x_{2} - x_{1}} = \frac{625 - 277}{5 - 1} = 87 \\ a & = y_{1}^{'} - b x_{1} = 277 - 87 (1) = 190 \end{aligned}$

The semi-average trend line $y^{'} = 190 + 87 x$ (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	$y^{'} = 39.5 - 3 (8) = 15.5$

$\begin{aligned} y_{1}^{'} & = 35, x_{1} = 1.5, y_{2}^{'} = 20, x_{2} = 6.5 \\ b & = \frac{y_{2}^{'} - y_{1}^{'}}{x_{2} - x_{1}} = \frac{20 - 35}{6.5 - 1.5} = - 3 \\ a & = y_{1}^{'} - b x_{1} = 35 - (- 3) (1.5) = 39.5 \\ y^{'} & = 39.5 - 3 x (with origin at 1973) \end{aligned}$

For the year 1982, the estimated number of books sold is $y^{'} = 39.5 - 3 (9) = 12.5$ .

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The Method of Free Hand Curve (2020)

Mar 15, 2025Sep 28, 2020 by Muhammad Imdad Ullah

Post Views: 100

Introduction to Free Hand Curve

The Method of Free Hand Curve is a simple and non-mathematical technique used to fit a curve to a set of data points by visually estimating the trend. It provides a quick, approximate visualization of the relationship between variables.

It is used to

Plot Data Points: Scatter plot the data points on a graph.
Draw Curve: Using a smooth, freehand line, draw a curve that best represents the overall trend of the data points.
No Rigid Rules: The curve is drawn based on the analyst’s judgment, without strict mathematical formulas.

Exploring Data Patterns

This method is subjective and less precise than mathematical curve-fitting techniques but is useful for initial data exploration or when data patterns are irregular.

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.

Practical Example of Curve

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

Year	1972	1973	1974	1975	1976	1977	1978
No. of Accidents	2493	2638	2699	3038	3745	4079	4688

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:

Year	Value	Total	Mean	Trend value
1972	2493			2200
1973	2638			2550
1974	2699			2950
1975	3038	233380	3340	3340
1976	3745			3650
1977	4079			4050
1978	4688			4499

The usefulness of the Method of Free Hand Curve

The method of freehand curve is useful for:

Exploratory Data Analysis (EDA): As a preliminary step free free-hand curve method helps us to understand the basic characteristics of the data and identify potential relationships between variables.
Visual Communication: It also helps to present trends in the data in a clear and easily understandable way for non-statistical audiences.
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.

Real-Life Examples of Free-Hand Curve

Some real-life examples where the Method of Free Hand Curve can be applied:

Economics:
- Plotting and estimating trends in stock market prices over time.
- Visualizing the relationship between inflation and unemployment rates.
Meteorology:
- Drawing temperature trends over days or months based on scattered weather data.
- Estimating rainfall patterns over a year.
Education:
- Plotting student performance trends over semesters to identify improvements or declines.
- Visualizing the relationship between study hours and exam scores.
Healthcare:
- Tracking a patient’s blood pressure or glucose levels over time to observe trends.
- Estimating the progression of a disease based on irregular data points.
Business:
- Visualizing sales trends over months or years to identify seasonal patterns.
- Estimating the relationship between advertising expenditure and revenue growth.
Sports:
- Plotting an athlete’s performance metrics (e.g., running speed, scores) over time to observe trends.
- Visualizing the relationship between training hours and performance improvement.

Summary

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. The freehand curve provides a quick, intuitive way to understand trends or relationships without relying on complex mathematical models.

MCQs Intermediate Mathematics Part-I Quadratic Equations

Heteroscedasticity Residual Plot (2020)

Apr 7, 2024Sep 27, 2020 by Muhammad Imdad Ullah

Post Views: 72

The post is about Heteroscedasticity Residual Plot.

Heteroscedasticity and Heteroscedasticity Residual Plot

One of the assumptions of the classical linear regression model is that there is no heteroscedasticity (error terms have constant error terms) meaning that ordinary least square (OLS) estimators are (BLUE, best linear unbiased estimator) and their variances are the lowest of all other unbiased estimators (Gauss Markov Theorem).

If the assumption of constant variance does not hold then this means that the Gauss Markov Theorem does not apply. For heteroscedastic data, regression analysis provides an unbiased estimate of the relationship between the predictors and the outcome variables.

As we have discussed heteroscedasticity occurs when the error variance has non-constant variance. In this case, we can think of the disturbance for each observation as being drawn from a different distribution with a different variance. Stated equivalently, the variance of the observed value of the dependent variable around the regression line is non-constant.

We can think of each observed value of the dependent variable as being drawn from a different conditional probability distribution with a different conditional variance. A general linear regression model with the assumption of heteroscedasticity can be expressed as follows

$\begin{aligned} y_{i} & = β_{0} + β_{1} X_{i 1} + β_{2} X_{i 2} + \dots + β_{p} X_{i} p + ε_{i} \\ V a r (ε_{i}) & = E (ε_{i}^{2}) \\ = σ_{i}^{2}; \dots i = 1, 2, \dots, n \end{aligned}$

Note that we have a $i$ subscript attached to sigma squared. This indicates that the disturbance for each of the $n$ units is drawn from a probability distribution that has a different variance.

If the error term has non-constant variance, but all other assumptions of the classical linear regression model are satisfied, then the consequences of using the OLS estimator to obtain estimates of the population parameters are:

The OLS estimator is still unbiased
The OLS estimator is inefficient; that is, it is not BLUE
The estimated variances and covariances of the OLS estimates are biased and inconsistent
Hypothesis tests are not valid

Detection of Heteroscedasticity Residual Plot

The residual for the $i$ th observation, $\hat{ε_{i}}$ , is an unbiased estimate of the unknown and unobservable error for that observation, $\hat{ε_{i}}$ . Thus the squared residuals, $\hat{ε_{i}^{2}}$ , can be used as an estimate of the unknown and unobservable error variance, $σ_{i}^{2} = E (\hat{ε_{i}})$ .

One can calculate the squared residuals and then plot them against an explanatory variable that you believe might be related to the error variance. If you believe that the error variance may be related to more than one of the explanatory variables, you can plot the squared residuals against each one of these variables. Alternatively, you could plot the squared residuals against the fitted value of the dependent variable obtained from the OLS estimates. Most statistical programs (software) have a command to do these residual plots. It must be emphasized that this is not a formal test for heteroscedasticity. It would only suggest whether heteroscedasticity may exist.

Below there are residual plots showing the three typical patterns. The first plot shows a random pattern that indicates a good fit for a linear model. The other two plot patterns of residual plots are non-random (U-shaped and inverted U), suggesting a better fit for a non-linear model, than a linear regression model.

Heteroscedasticity Regression Residual Plot 3 — Heteroscedasticity Residual Plot 1

Heteroscedasticity Residual Plot 1 — Heteroscedasticity Residual Residual Plot 2

Heteroscedasticity Residual Plot 2 — Heteroscedasticity Residual Plot 3

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Merits of Method of Semi Averages

Demerits of Method of Semi Averages

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Introduction to Free Hand Curve

Table of Contents

Exploring Data Patterns

Merits (Free Hand Curve)

Demerits (Free Hand Curve)

Practical Example of Curve

The usefulness of the Method of Free Hand Curve

Real-Life Examples of Free-Hand Curve

Summary

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Heteroscedasticity and Heteroscedasticity Residual Plot

Detection of Heteroscedasticity Residual Plot

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