Linear Regression Slope Calculator Formula
Understand the math behind the linear regression slope calculator. Each variable explained with a worked example.
Formulas Used
Slope (b1)
slope = slope_num / slope_denMean of x
x_mean = sum_x / nMean of y
y_mean = sum_y / nVariables
| Variable | Description | Default |
|---|---|---|
n | Number of Pairs | 5 |
sum_xy | Sum of (x*y) | 2350 |
sum_x | Sum of x | 75 |
sum_y | Sum of y | 150 |
sum_x2 | Sum of x-squared | 1175 |
slope_num | Derived value= n * sum_xy - sum_x * sum_y | calculated |
slope_den | Derived value= n * sum_x2 - pow(sum_x, 2) | calculated |
How It Works
How to Calculate the Regression Slope
Formula
b1 = [n*Sum(xy) - Sum(x)*Sum(y)] / [n*Sum(x^2) - (Sum(x))^2]
The slope of the least-squares regression line represents the predicted change in Y for a one-unit increase in X. A positive slope indicates a positive relationship; negative means Y decreases as X increases.
Worked Example
n = 5, Sum(xy) = 2350, Sum(x) = 75, Sum(y) = 150, Sum(x^2) = 1175.
- 01Numerator = 5*2350 - 75*150 = 11750 - 11250 = 500
- 02Denominator = 5*1175 - 75^2 = 5875 - 5625 = 250
- 03Slope b1 = 500 / 250 = 2.0
- 04For each unit increase in x, y increases by 2 on average
Frequently Asked Questions
What if the slope is zero?
A slope of zero means there is no linear relationship between X and Y. The best prediction for Y is simply the mean of Y, regardless of X.
Is the regression slope the same as correlation?
No. The slope depends on the scales of X and Y (it has units of Y/X). The correlation r is dimensionless and standardized between -1 and 1. They share the same sign but different magnitudes.
What does "least squares" mean?
The least-squares method finds the line that minimizes the sum of squared vertical distances (residuals) between observed Y values and predicted Y values. This gives the best linear unbiased estimate under certain assumptions.
Learn More
Guide
Regression Analysis Guide
Comprehensive guide to regression analysis. Learn how linear regression works, how to interpret slope and intercept, R-squared, residuals, and when to use regression.
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