2025-06-08

What is Linear Regression?

Linear regression is a method used to model the relationship between a dependent variable and an independent variable.

The formula for a simple linear regression line is:

\[ y = mx + b \]

When Do We Use It?

  • Predicting outcomes (e.g. exam scores, home prices, blood pressure)
  • Exploring trends and correlations
  • Making future projections

The Math Behind It

The slope of the line is calculated using:

\[ m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \]

The intercept is calculated with:

\[ b = \frac{\sum y - m \sum x}{n} \]

Creating a Model in R

data(mtcars)
model <- lm(mpg ~ wt, data = mtcars)
summary(model)
## 
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5432 -2.3647 -0.1252  1.4096  6.8727 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
## wt           -5.3445     0.5591  -9.559 1.29e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.046 on 30 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7446 
## F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10
install.packages("ggplot2")
## Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.4'
## (as 'lib' is unspecified)

Visualizing the Model with ggplot2

Interactive MPG vs Weight Plot (Plotly)

Actual vs Predicted MPG