Linear Regression and Correlation

When analyzing data on a scatter plot, we often want to know if there is a relationship between the two variables. Linear regression helps us model this relationship using a straight line, while correlation tells us how strong that relationship is.

The Line of Best Fit

A line of best fit (or trend line) is a straight line drawn through the center of the data points on a scatter plot. It best represents the general trend of the data.

• The line should follow the general direction of the data cluster.
• It should have roughly an equal number of points above and below it.

Once you have a line of best fit, you can use its equation to make predictions.

Example: Suppose you draw a line of best fit for a scatter plot, and you determine its equation is $y = 2.5x + 4$. If you need to predict the value of $y$ when $x = 10$, you simply substitute $10$ for $x$: $y = 2.5(10) + 4 = 25 + 4 = 29$

The Correlation Coefficient ($r$)

The correlation coefficient, denoted by the letter $r$, measures both the strength and the direction of a linear relationship between two variables.

The value of $r$ always falls in the range of $-1$ to $1$: $-1 \le r \le 1$

Direction

• Positive Correlation ($r > 0$): As $x$ increases, $y$ increases. The line of best fit slopes upward.
• Negative Correlation ($r < 0$): As $x$ increases, $y$ decreases. The line of best fit slopes downward.

Strength

• Strong Relationship: The closer $r$ is to $1$ or $-1$, the tighter the data points cluster around the line of best fit.
• Weak Relationship: The closer $r$ is to $0$, the more scattered the points are.
• No Relationship: If $r = 0$, there is no linear correlation at all.

Example: Interpreting $r$

Problem: Interpret a correlation coefficient of $r = -0.87$.

Solution:

1. Look at the sign: The negative sign means there is a negative correlation (as one variable goes up, the other goes down).
2. Look at the number: The absolute value $0.87$ is very close to $1$.

Therefore, $r = -0.87$ indicates a strong, negative linear relationship between the two variables.