How Do You Find the Correlation Coefficient with a Scatter Plot?

Finding the correlation coefficient from a scatter plot is a helpful way to see how two things are related. This coefficient, often called $r$, shows how strongly connected the two variables are.

Step 1: Know Your Variables

First, figure out the two things you are looking at. For example, let’s check the relationship between how many hours you study (X) and your exam scores (Y). On your scatter plot, put hours studied on the x-axis (horizontal line) and exam scores on the y-axis (vertical line).

Step 2: Make Your Scatter Plot

Next, you need to plot your data points on the graph. Let’s say you have these pairs of data:

• (1 hour, 50 points)
• (2 hours, 60 points)
• (3 hours, 70 points)
• (4 hours, 80 points)

You would mark these points on your scatter plot. As you connect the dots, you might notice a pattern: usually, it will go up, showing that as you study more hours, your exam scores go up too.

Step 3: Look Closely at the Plot

Before calculating $r$, take a good look at the scatter plot. If the points are close to a straight line, that means there is a strong connection. If they are spread out, that means there is a weak connection. Remember, $r$ can give you different values where:

• $r = 1$: Perfect positive correlation (strong connection)
• $r = -1$: Perfect negative correlation (strong negative connection)
• $r = 0$: No correlation (no connection)

Step 4: Calculate the Correlation Coefficient

Here’s the formula for calculating the correlation coefficient:

$$r = \frac{n(\sum{XY}) - (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} - (\sum{X})^2][n\sum{Y^2} - (\sum{Y})^2]}}$$

To break it down:

• $n$ is the number of pairs of scores,
• $X$ and $Y$ are your variables,
• $\sum$ means you will add up all the data points.

Using our example:

1. Calculate the sums:

   • $n = 4$ (four data pairs)
   • $\sum X = 10$, $\sum Y = 260$
   • $\sum XY = 1 \cdot 50 + 2 \cdot 60 + 3 \cdot 70 + 4 \cdot 80 = 380$
   • $\sum X^2 = 1^2 + 2^2 + 3^2 + 4^2 = 30$
   • $\sum Y^2 = 50^2 + 60^2 + 70^2 + 80^2 = 30500$

2. Plug those numbers into the formula for $r$.

After calculating, let’s say we got $r = 0.98$. This means there is a very strong positive correlation between hours studied and exam scores.

Conclusion

In short, to find the correlation coefficient using a scatter plot, you need to plot your data points, look at the relationships, and use the correlation formula. It’s an easy process that gives you useful information about how two variables are related. Just remember, a strong correlation does not mean one thing causes the other, so you might need to dig deeper to understand more!