How to Find the Correlation Coefficient of a Scatter Plot
A scatter plot is a graphical representation of the relationship between two variables, where each data point is plotted as a coordinate on a Cartesian plane. When analyzing such plots, one of the most important statistical measures to determine is the correlation coefficient, which quantifies the strength and direction of the linear relationship between the variables. The most commonly used correlation coefficient is Pearson’s r, which ranges from -1 to 1. This article will guide you through the step-by-step process of calculating the correlation coefficient from a scatter plot, including interpretation and common pitfalls Most people skip this — try not to..
Steps to Find the Correlation Coefficient
1. Understand the Scatter Plot
Before calculating the correlation coefficient, examine the scatter plot to identify patterns. A positive correlation appears as points trending upward from left to right, while a negative correlation trends downward. A correlation coefficient near 0 suggests no linear relationship It's one of those things that adds up..
2. Calculate the Means
Compute the mean (average) of the x-values and y-values in the dataset. The formulas are:
- Mean of x (( \bar{x} )) = ( \frac{\sum x}{n} )
- Mean of y (( \bar{y} )) = ( \frac{\sum y}{n} )
3. Determine Deviations from the Mean
For each data point, subtract the mean of x and the mean of y to find the deviations:
- Deviation for x: ( x_i - \bar{x} )
- Deviation for y: ( y_i - \bar{y} )
4. Compute Products and Squares
Multiply the deviations for each pair of values (( (x_i - \bar{x})(y_i - \bar{y}) )) and sum these products. Similarly, square the deviations for x and y:
- Sum of products: ( \sum (x_i - \bar{x})(y_i - \bar{y}) )
- Sum of squared deviations for x: ( \sum (x_i - \bar{x})^2 )
- Sum of squared deviations for y: ( \sum (y_i - \bar{y})^2 )
5. Calculate Covariance and Standard Deviations
The covariance between x and y is given by:
[ \text{Cov}(x, y) = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{n} ]
The standard deviations of x and y are:
[ \sigma_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} ]
[ \sigma_y = \sqrt{\frac{\sum (y_i - \bar{y})^2}{n}} ]
6. Apply the Pearson Correlation Formula
Finally, use the formula for Pearson’s r:
[ r = \frac{\text{Cov}(x, y)}{\sigma_x \cdot \sigma_y} ]
Alternatively, express it using sums directly:
[ r = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} ]
Example Calculation
Let’s walk through an example with the following dataset:
| x | y |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
| 5 | 10 |
