Glossary

12.7 Regression (Distance from School)

4 min read•june 25, 2024

helps us understand how distance from school affects student performance. We'll look at scatterplots, correlation coefficients, and to uncover patterns and make predictions.

We'll explore how to interpret scatterplots, calculate correlation coefficients, and use equations. These tools will help us figure out if there's a link between how far students live from school and how well they do academically.

Regression Analysis: Distance from School and Student Performance

Scatterplot interpretation for student performance

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  • Graphical representation of the relationship between two quantitative variables
    • (distance from school) plotted on the x-axis
    • (student performance) plotted on the y-axis
  • Each point represents an individual student's distance and academic performance
  • Overall pattern of points reveals the nature of the relationship
    • Positive : points trend upward from left to right (as distance increases, performance increases)
    • Negative linear relationship: points trend downward from left to right (as distance increases, performance decreases)
    • Non-linear relationship: curved pattern or no clear pattern
  • Strength of relationship visually assessed by proximity of points to an imaginary straight line
    • Points clustered tightly around a line indicate a strong relationship
    • Points scattered widely suggest a weak relationship
  • (points that deviate significantly from overall pattern) should be identified and considered when interpreting the relationship
  • can be visually assessed on the as the vertical distance between each data point and the regression line

Correlation coefficient of distance and achievement

  • Numerical measure of the strength and direction of the linear relationship between two variables
    • Ranges from -1 to +1
    • +1 indicates a perfect positive linear relationship
    • -1 indicates a perfect negative linear relationship
    • 0 indicates no linear relationship
  • Formula for the : [r](https://www.fiveableKeyTerm:R)=∑i=1n(xi−xˉ)(yi−yˉ)∑i=1n(xi−xˉ)2∑i=1n(yi−yˉ)2[r](https://www.fiveableKeyTerm:R) = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2} \sqrt{\sum_{i=1}^{n} (y_i - \bar{y})^2}}
    • xix_i and yiy_i are individual values of independent and dependent variables
    • xˉ\bar{x} and yˉ\bar{y} are means of independent and dependent variables
    • nn is the number of data points
  • Sign of correlation coefficient indicates direction of relationship
    • Positive value suggests positive linear relationship (as distance increases, performance increases)
    • Negative value suggests negative linear relationship (as distance increases, performance decreases)
  • Absolute value of correlation coefficient indicates strength of linear relationship
    • Values close to 0 suggest a weak linear relationship
    • Values close to 1 (positive or negative) suggest a strong linear relationship
  • Correlation does not imply causation; other factors may influence the relationship
  • The (r-squared) measures the proportion of variance in the dependent variable that is predictable from the independent variable

Linear regression for performance prediction

  • Models the relationship between independent variable (distance from school) and dependent variable (student performance)
    • Equation takes the form: y^=b0+b1x\hat{y} = b_0 + b_1x
      • y^\hat{y} is predicted value of dependent variable
      • b0b_0 is (value of y^\hat{y} when x=0x = 0)
      • b1b_1 is (change in y^\hat{y} for a one-unit change in xx)
      • xx is value of independent variable
  • Slope (b1b_1) and y-intercept (b0b_0) calculated using formulas:
    • b1=rsysxb_1 = r \frac{s_y}{s_x}
      • rr is correlation coefficient
      • sys_y is of dependent variable
      • sxs_x is standard deviation of independent variable
    • b0=yˉ−b1xˉb_0 = \bar{y} - b_1\bar{x}
      • yˉ\bar{y} is mean of dependent variable
      • xˉ\bar{x} is mean of independent variable
  • Usefulness of linear regression equation depends on strength of linear relationship
    • Strong linear relationship (high absolute value of rr) suggests equation can provide accurate predictions
    • Weak linear relationship (low absolute value of rr) indicates equation may not be reliable for predictions
  • Limitations of linear regression equation:
    • Only models linear relationships; non-linear relationships cannot be accurately represented
    • Sensitive to outliers, which can significantly influence slope and y-intercept
    • Assumes relationship between variables remains constant across entire range of data
    • Does not account for other factors that may affect student performance (socioeconomic status, school quality)
  • To evaluate usefulness of linear regression equation, consider strength of linear relationship, presence of outliers, and context of the problem
  • The is used to find the best-fitting line by minimizing the sum of squared residuals

Additional Regression Considerations

  • : When independent variables are highly correlated, it can affect the stability and interpretation of regression coefficients
  • : When the variability of residuals is not constant across all levels of the independent variable, it can affect the reliability of statistical tests
  • These issues should be assessed and addressed to ensure the validity of regression analysis results

Key Terms to Review (20)

"OR" Event: An 'OR' event in probability occurs when at least one of multiple events happens. The probability of an 'OR' event is calculated by adding the probabilities of individual events and subtracting the probability of their intersection.
Coefficient of determination: The coefficient of determination, denoted as $R^2$, measures the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It ranges from 0 to 1, where a higher value indicates a better fit of the model.
Coefficient of Determination: The coefficient of determination, denoted as $R^2$, is a statistical measure that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s) in a regression model. It is a key concept in understanding the strength and predictive power of a regression analysis.
Correlation Coefficient: The correlation coefficient is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, with -1 indicating a perfect negative correlation, 0 indicating no correlation, and 1 indicating a perfect positive correlation.
Dependent Variable: The dependent variable is the outcome or response variable in a study or experiment. It is the variable that is measured or observed to determine the effect of the independent variable. The dependent variable depends on or is influenced by the independent variable.
Heteroscedasticity: Heteroscedasticity refers to the condition where the variability of a variable is unequal across the range of values of a second variable that predicts it. This concept is particularly relevant in the context of regression analysis, where it can impact the validity of statistical inferences.
Independent Variable: The independent variable is a variable that is manipulated or changed by the researcher in an experiment to observe its effect on the dependent variable. It is the variable that the researcher has control over and intentionally varies to measure its impact on the outcome.
Least Squares Method: The least squares method is a statistical technique used to determine the best-fitting line or curve that minimizes the sum of the squared differences between the observed data points and the predicted values from the model. It is a fundamental concept in regression analysis and is widely applied across various fields, including 12.1 Linear Equations, 12.7 Regression (Distance from School), and 12.9 Regression (Fuel Efficiency).
Linear regression: Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. It aims to predict the value of the dependent variable based on the values of the independent variables.
Linear Regression: Linear regression is a statistical technique used to model the relationship between a dependent variable and one or more independent variables. It aims to find the best-fitting straight line that describes the linear association between the variables.
Linear Relationship: A linear relationship is a mathematical relationship between two variables where the change in one variable is proportional to the change in the other variable. This type of relationship is often depicted visually through a scatter plot and can be further analyzed using regression techniques.
Multicollinearity: Multicollinearity is a statistical phenomenon in which two or more predictor variables in a multiple regression model are highly correlated with each other, making it difficult to determine the individual effects of the variables on the dependent variable.
Outliers: Outliers are data points that significantly differ from the rest of the data in a dataset. They can skew the results and lead to misleading interpretations, affecting measures of central tendency, variability, and visual representations.
R: R is a programming language and software environment for statistical computing and graphics. It is widely used in various fields, including statistics, data analysis, and scientific research, due to its powerful capabilities in handling and analyzing data.
Regression Analysis: Regression analysis is a statistical technique used to model the relationship between a dependent variable and one or more independent variables. It allows researchers to estimate the average change in the dependent variable associated with a one-unit change in the independent variable, while controlling for other factors.
Residuals: Residuals, in the context of statistical analysis, refer to the differences between the observed values and the predicted values from a regression model. They represent the unexplained or unaccounted-for portion of the variability in the dependent variable, providing insights into the quality and fit of the regression model.
Scatterplot: A scatterplot is a type of data visualization that displays the relationship between two variables by plotting individual data points on a coordinate plane. It allows for the visual exploration of the strength and direction of the association between the variables.
Slope: Slope is a measure of the steepness or incline of a line, typically represented as the ratio of the change in the vertical direction (rise) to the change in the horizontal direction (run) between two points on the line. It serves as a key component in understanding linear relationships and is vital for forming predictions based on data trends.
Standard Deviation: Standard deviation is a statistic that measures the dispersion or spread of a set of values around the mean. It helps quantify how much individual data points differ from the average, indicating the extent to which values deviate from the central tendency in a dataset.
Y-intercept: The y-intercept is the point at which a linear equation or regression line intersects the y-axis, representing the value of the dependent variable when the independent variable is zero. It is a crucial parameter in understanding the relationship between two variables and making predictions.
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