# 10 4: The Least Squares Regression Line Statistics LibreTexts

The best-fit line minimizes the sum of the squares of these vertical distances. Note that the least-squares solution is unique in this case, since an orthogonal set is linearly independent, Fact 6.4.1 in Section 6.4. But for any specific observation, the actual value of Y can deviate from the predicted value.

## Setting up an example

Let us have a look at how the data points and the line of best fit obtained from the least squares method look when plotted on a graph. Imagine that you’ve plotted some data using a scatterplot, and that you fit a line for the mean of Y through the data. Let’s lock this line in place, and attach springs between the data points and the line. Use the least square method to determine the equation of line of best fit for the data. Suppose when we have to determine the equation of line of best fit for the given data, then we first use the following formula.

## Least-Squares Solutions

It’s noteworthy to note that more than half of students admitted the negative effects that excessive social media use had on both their personal and academic lives (64.6.1%) and students (77.1%). Of them, 69% claimed that social media prevented them from attending classes. Parallel to this, Able to Encourage (Alwagait et al., 2015) https://www.business-accounting.net/ examined how SM affected the academic achievement of 108 Saudi students. According to 60% of individuals, using social media excessively interfered with their capacity to function (Sarfraz et al., 2022). An additional survey included university students in Ghana who said WhatsApp negatively affected their academic performance.

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One of the main benefits of using this method is that it is easy to apply and understand. That’s because it only uses two variables (one that is shown along the x-axis and the other book profit calculator calculate profit from your book on amazon on the y-axis) while highlighting the best relationship between them. After having derived the force constant by least squares fitting, we predict the extension from Hooke’s law.

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The self-determination theory provided confirmation that this model is appropriate for fostering students’ feelings of competence, autonomy, and relatedness in order to increase their affective learning involvement. This, in turn, improves students’ satisfaction and achievement in higher education. Furthermore, a study carried out in Malaysia (Al-Rahmi et al., 2022a) used some elements from the previous research, including student satisfaction and collaborative learning, but not all of the elements. Furthermore, a study conducted in Saudi Arabia by Alturki and Aldraiweesh (2022) used a number of variables, such as task-technology fit, perceived utility, perceived ease of use, behavioral intention to use, and actual use of mobile M-learning. Social impact, performance expectations, effort expectations, behavior to utilize these devices, and actual use were some of the factors used in this study. The association between students’ academic achievement, level of satisfaction with their education, and affective engagement when using SM for learning is depicted in Fig.

- Gauss showed that the arithmetic mean is indeed the best estimate of the location parameter by changing both the probability density and the method of estimation.
- Before we jump into the formula and code, let’s define the data we’re going to use.
- Furthermore, by enhancing views of competence, independence, and relatedness, it can enhance affective engagement with learning in educational settings.
- But, this method doesn’t provide accurate results for unevenly distributed data or for data containing outliers.
- According to them, these platforms have little to no impact on academic achievement (Al-Rahmi et al., 2022c) and are improper for improving learning performance (Foroughi et al., 2022).

This helps us to fill in the missing points in a data table or forecast the data. Least square method is the process of finding a regression line or best-fitted line for any data set that is described by an equation. This method requires reducing the sum of the squares of the residual parts of the points from the curve or line and the trend of outcomes is found quantitatively. The method of curve fitting is seen while regression analysis and the fitting equations to derive the curve is the least square method. The least squares method is a form of mathematical regression analysis used to determine the line of best fit for a set of data, providing a visual demonstration of the relationship between the data points. Each point of data represents the relationship between a known independent variable and an unknown dependent variable.

On 1 January 1801, the Italian astronomer Giuseppe Piazzi discovered Ceres and was able to track its path for 40 days before it was lost in the glare of the Sun. Based on these data, astronomers desired to determine the location of Ceres after it emerged from behind the Sun without solving Kepler’s complicated nonlinear equations of planetary motion. The only predictions that successfully allowed Hungarian astronomer Franz Xaver von Zach to relocate Ceres were those performed by the 24-year-old Gauss using least-squares analysis. Polynomial least squares describes the variance in a prediction of the dependent variable as a function of the independent variable and the deviations from the fitted curve. This could mean offering workshops, seminars, and ongoing support to students in order to provide them with the knowledge and skills needed to create a motivating and supportive learning environment. As a result, the study emphasizes the need to promote effective learning engagement and demonstrates how crucial student emotional engagement is to achieving effective learning outcomes.

In order to clarify the meaning of the formulas we display the computations in tabular form. Next, find the difference between the actual value and the predicted value for each line. To do this, plug the $x$ values from the five points into each equation and solve. In particular, least squares seek to minimize the square of the difference between each data point and the predicted value. By the way, you might want to note that the only assumption relied on for the above calculations is that the relationship between the response \(y\) and the predictor \(x\) is linear. Updating the chart and cleaning the inputs of X and Y is very straightforward.

Approval was obtained from the ethics committee at King Saud University (KSU). The procedures used in this study adhere to the tenets of the Declaration of Helsinki. Therefore, in accordance with ethical standards, we hereby confirm that the research study mentioned above involved the collection of data from KSU, and prior ethical approval was duly obtained, dated March 28, 2024, and under KSU research project no.

PLS enables the simultaneous evaluation of multiple interdependent relationships. A structural model in PLS illustrates the connections between the theoretical ideas. Using the bootstrapping technique, 500 recommended random samples were generated with SmartPLS (Hair et al., 2019).

In 1718 the director of the Paris Observatory, Jacques Cassini, asserted on the basis of his own measurements that Earth has a prolate (lemon) shape. Let’s assume that an analyst wishes to test the relationship between a company’s stock returns, and the returns of the index for which the stock is a component. In this example, the analyst seeks to test the dependence of the stock returns on the index returns.

This method is called so as it aims at reducing the sum of squares of deviations as much as possible. It is quite obvious that the fitting of curves for a particular data set are not always unique. Thus, it is required to find a curve having a minimal deviation from all the measured data points. This is known as the best-fitting curve and is found by using the least-squares method.

One main limitation is the assumption that errors in the independent variable are negligible. This assumption can lead to estimation errors and affect hypothesis testing, especially when errors in the independent variables are significant. The Least Square Method minimizes the sum of the squared differences between observed values and the values predicted by the model. This minimization leads to the best estimate of the coefficients of the linear equation. We can conclude from the above graph that how the least squares method helps us to find a line that best fits the given data points and hence can be used to make further predictions about the value of the dependent variable where it is not known initially.

One way to promote student participation and teamwork in the classroom would be to integrate SM platforms. Secondly, it is imperative to attend to the academic achievement and contentment of pupils. The impact of SM on students’ academic achievement is a controversial subject. Therefore, generalizations on the impact of SM are untrue (Lepp et al., 2014).

However, some studies (Alturki & Aldraiweesh, 2022; Capriotti & Zeler, 2023) discovered that SM might be used as an instructional tool to promote communication, facilitate cooperative learning, and boost student engagement (Al-Rahmi et al., 2022c). Through this study, we hope to close a knowledge gap in the literature and provide further insight into the connections between students’ satisfaction with their academic progress and their use of SM. SDT is applied in a variety of settings, including the commercial world, the workplace, and educational institutions. According to Sun et al. (2019), it is regarded as one of the “most supported by evidence incentive theories” in use today. The goal of SDT, a macro-level theory concerning human incentive, is to make clear the relationships that exist between motivation, growth, and well-being. Because of these benefits, SM has drawn a sizable initial user base, particularly in the setting of higher education.

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