To understand the results of a scatter plot, especially when looking at regression and correlation, follow these simple steps:

1. Identify the Variables:

   • Start by naming your independent variable (this is usually $X$) and your dependent variable (this is usually $Y$).

2. Look for Patterns:

   • Check the scatter plot for any patterns:
     • Positive Correlation: If the points go up as you move from left to right, this shows a positive correlation.
     • Negative Correlation: If the points go down from left to right, this shows a negative correlation.
     • No Correlation: If the points are spread out randomly, there’s very little or no correlation.

3. Calculate the Correlation Coefficient ($r$):

   • This number helps you see how strong the relationship is between the two variables.
   • The value of $r$ can be between -1 and 1:
     • $r \approx 1$: This means a strong positive correlation.
     • $r \approx -1$: This means a strong negative correlation.
     • $r \approx 0$: This means there is little to no linear correlation.

4. Perform Linear Regression:

   • Use a method called least squares to find the best-fitting line.
   • This line can be represented by the equation $Y = a + bX$, where $b$ is the slope and $a$ is where the line crosses the Y-axis.

5. Check the Results:

   • Look at the residuals (the differences between the observed and predicted values) and find $R^2$ (which tells you how well your model works).
   • A higher $R^2$ value, which is closer to 1, shows that your model explains a big part of the change in $Y$.

Following these steps will help you make sense of what the scatter plot is telling you!