Linear regression models a relationship between one or more forecasting variables to an outcome variable. Understanding this model allows us to predict an outcome based on its forecasting varibles.

For example, we could take a students hours per week spent studying and use it to predict how they will score on their final.

The equation is: \(y=mx+b\)

• \(x\) is the independent variable (Forecasting variables)

• \(y\) is the dependent variable (Outcome variable)

• \(m\) is the slope of the line

• \(b\) is the y-intercept

The formula to find the slope (m) is: \(r\frac{Sy}{Sx}\)

• \(r\) is the correlation coefficient

• \(Sy\) is the standard deviation of the outcome variable (Dependent variable)

• \(Sx\) is the standard deviation of the forecasting variable (independent variable)

The formula to find the y-intercept (b) is: \(\bar{y}-m * \bar{x}\)

• \(\bar{y}\) is the mean of the outcome variable (Dependent variable)

• \(\bar{x}\) is the mean of the forecasting variable (independent variable)

• \(m\)s is the slope

Lets do a simple linear regression using the iris dataset that’s prepackaged with r. We will predict Sepal Width using Sepal Length as the forecasting variable. The below code will create a scatter plot with an ordinary least squares trend line representing the linear regression equation that minimizes error.

ggplot(iris, aes(x = Sepal.Length, y = Sepal.Width)) + 
  geom_point() + 
  geom_smooth(method = lm) +
  ggtitle("Effect of Sepal Length on Sepal Width") +
  xlab("Sepal Width") +
  ylab("Sepal Width")

• Regularly used in finance and business to predict the price of securities or other capital holdings, as well as key metrics like sales and shrinkage.
• Cities use linear regression to predict things like fuel usage in the winter, or how growth will effect water usage and housing prices.
• Medication dosages are determined by using things like patient weight, height, and blood pressure levels as forecasting variables.