Linear regression is a fundamental statistical method used to model the relationship between variables. It helps us find the best straight line through a set of data points, allowing us to make predictions and identify patterns. The goal is to find the line that best represents the underlying relationship in the data.
The linear regression equation is y equals m x plus b. Here, y is the dependent variable we want to predict, x is the independent variable, m is the slope showing the rate of change, and b is the y-intercept where the line crosses the y-axis. Let's see how changing these parameters affects the line.
To find the best fit line, we use residuals - the vertical distances between data points and our line. The least squares method finds the line that minimizes the sum of squared residuals. Watch how different line positions result in different error values, and how we can find the optimal position.
Let's work through a practical example using study hours and test scores. Our data shows a positive correlation. The regression analysis gives us the equation: Score equals 8.5 times Hours plus 42. The slope of 8.5 means each additional study hour increases the test score by 8.5 points on average. The intercept of 42 represents the expected score with zero study hours.
Model evaluation uses R-squared to measure how well our model explains the data variance. Values range from 0 for poor fit to 1 for perfect fit. Linear regression assumes linearity, independence of observations, equal variance of residuals, and normally distributed errors. When these assumptions are violated, the model may not be reliable.