2.4.1 Correlation & Regression

Correlation

Linear correlation is a measure of how close a set of points lie to a straight line. Correlation is measured using a value called the Product Moment Correlation Coefficient, or $r$, for short.

How to Calculate the $r$ Value

Step 1

• Select the "Lin" option.

Step 2

• Choose $y = a + bx$.

Step 3

• Input data.

Step 4

• Use the "OPTN" button.

Step 5

• Select "Regression Calc".

Step 6

• The value r will be displayed.

---

In this case, $r = 1$. This means perfect linear positive correlation between $x$ and $y$.

Types of Correlation

There are different types of correlation that can be observed on the graph or through calculating the $r$ value.

Strong Positive Correlation

• The points lie close to a straight line with a positive slope.
• $r$ will be close to $1$.

Weak Positive Correlation

• The points show some positive correlation but are more spread out.
• $0 < r < 0.5$.

No Correlation

• The points do not show any trend or pattern.
• $r = 0$.

Weak Negative Correlation

• The points show some negative correlation but are more spread out.
• $-0.5 < r < 0$.

Strong Negative Correlation

• The points lie close to a straight line with a negative slope.
• $r$ is close to $-1$.

Perfect Negative Correlation

Regression Line

A regression line is the best line that passes through the mean point $(\bar{x}, \bar{y})$ and minimizes the distance between the line and the points.

• Red Line: The red line is a better line of best fit because each point lies closer to the red line than the blue.

---

---

Important Point

It is only appropriate to use the regression line when:

• The relationship exhibited by the data appears linear.
• The value of r is sufficient to suggest strong enough correlation. We only use the equation to make predictions close to or within the range of data we have. This is because we cannot be sure that the relationship will remain linear beyond our data.

---