Regression Slope Calculator

About Regression Slope Calculator

The regression line minimizes the sum of squared residuals. R² measures how well the line fits the data (0 to 1).

How to use this calculator

1. Enter your paired data values as x, y points.
2. Check that each x value is matched with the correct y value.
3. Run the calculation to find the regression slope and intercept.
4. Use the results to write the line of best fit, usually in the form y = mx + b.

The formula explained

The slope formula computes the least-squares slope of the regression line, which shows the average change in y for each 1-unit increase in x. The intercept is the value of y when x = 0, found from the regression equation after the slope is known.

• m = the slope of the regression line
• n = the number of data pairs
• x = the x-values in the data set
• y = the y-values in the data set
• \(\sum xy\) = the sum of each x times its matching y
• \(\sum x\) = the sum of all x-values
• \(\sum y\) = the sum of all y-values
• \(\sum x^2\) = the sum of each x-value squared
• b = the y-intercept of the regression line

Step by step method

1. List the data as ordered pairs and find n, \(\sum x\), \(\sum y\), \(\sum xy\), and \(\sum x^2\).
2. Substitute those values into \(m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}\) to find the slope.
3. Use one data pair in \(y = mx + b\) and solve for b.
4. Write the regression line as \(y = mx + b\) and interpret the slope in context.

Worked example

Problem. Find the regression line for the data points (1, 2), (2, 3), (3, 5), and (4, 4).

1. First find the needed sums: \(n = 4\), \(\sum x = 1+2+3+4 = 10\), \(\sum y = 2+3+5+4 = 14\), \(\sum xy = 1\cdot2 + 2\cdot3 + 3\cdot5 + 4\cdot4 = 37\), and \(\sum x^2 = 1^2+2^2+3^2+4^2 = 30\).
2. Substitute into the slope formula: \(m = \frac{4(37) - (10)(14)}{4(30) - (10)^2} = \frac{148 - 140}{120 - 100} = \frac{8}{20} = 0.4\).
3. Use \(y = mx + b\) with point \((1,2)\): \(2 = 0.4(1) + b\), so \(b = 1.6\).

Tips and common mistakes

• Make sure each x-value is paired with the correct y-value, because switching pairs changes the result.
• A positive slope means y tends to increase as x increases, while a negative slope means y tends to decrease as x increases.

Frequently asked questions