Least Squares Regression Calculator

Find the best-fit line y = mx + b using the least squares method. Calculate slope, intercept, R-squared, and correlation coefficient.

Enter Data Points (x, y)

1. x: y:
2. x: y:
3. x: y:
4. x: y:
5. x: y:

Result

Best-Fit Line
y = 0.7x + 1.5

Step-by-Step Solution

y = mx + b

What is Least Squares Regression?

Least squares regression is a statistical method used to find the best-fitting straight line through a set of data points. The "best fit" is determined by minimizing the sum of the squared differences (residuals) between observed values and the values predicted by the line.

The resulting equation y = mx + b gives a linear model that can be used to predict values and understand the relationship between two variables.

Key Components

Slope (m)

The rate of change of y with respect to x. A positive slope indicates an upward trend; negative indicates downward.

m = (n*sum(xy) - sum(x)*sum(y)) / (n*sum(x^2) - (sum(x))^2)

Y-Intercept (b)

The value of y when x = 0. It is the point where the line crosses the y-axis.

b = (sum(y) - m*sum(x)) / n

R-Squared (R²)

The coefficient of determination. Ranges from 0 to 1, indicating how well the line fits the data.

R^2 = 1 - SS_res / SS_tot

Correlation Coefficient (r)

Measures the strength and direction of the linear relationship. Ranges from -1 to +1.

r = n*sum(xy) - sum(x)*sum(y) / sqrt(...)

How to Calculate Least Squares Regression

  1. Collect all data points (x, y) pairs.
  2. Calculate the sums: sum(x), sum(y), sum(xy), sum(x²), sum(y²).
  3. Use the slope formula: m = [n * sum(xy) - sum(x) * sum(y)] / [n * sum(x²) - (sum(x))²].
  4. Calculate the y-intercept: b = [sum(y) - m * sum(x)] / n.
  5. Write the equation: y = mx + b.
  6. Calculate R² to assess the quality of the fit.

Interpreting R-Squared

  • R² = 1: Perfect fit. The line passes through every data point.
  • R² > 0.9: Excellent fit. Very strong linear relationship.
  • R² > 0.7: Good fit. Strong linear relationship.
  • R² > 0.5: Moderate fit. Some linear relationship exists.
  • R² < 0.3: Weak fit. Little to no linear relationship.

Applications of Least Squares Regression

  • Science: Analyzing experimental data and finding trends.
  • Economics: Predicting market trends, supply and demand.
  • Engineering: Calibration curves and quality control.
  • Medicine: Dose-response relationships.
  • Finance: Stock price prediction and risk analysis.