Enter your paired data points (x, y) to find the best-fit line.
Results
Slope (m): –
Y-intercept (b): –
Regression Equation: y = -x + –
R-squared Value: –
Understanding Least Squares Regression
Linear regression is a fundamental statistical technique used to model the relationship between a dependent variable (y) and one or more independent variables (x). The Least Squares Regression Line is the line that best fits a set of data points by minimizing the sum of the squares of the vertical distances (residuals) between the observed data points and the line itself.
The equation of a straight line is typically represented as y = mx + b, where:
y is the dependent variable.
x is the independent variable.
m is the slope of the line, indicating how much y changes for a one-unit increase in x.
b is the y-intercept, the value of y when x is 0.
The Math Behind It
For a simple linear regression with paired data points (x₁, y₁), (x₂, y₂), …, (xn, yn), the formulas for calculating the slope (m) and y-intercept (b) using the least squares method are derived as follows:
First, we need some preliminary sums:
Σx: Sum of all x values
Σy: Sum of all y values
Σxy: Sum of the product of each corresponding x and y pair
Σx²: Sum of the squares of all x values
n: The number of data points
The formulas are:
Slope (m): m = (n * Σxy - Σx * Σy) / (n * Σx² - (Σx)²)
Y-intercept (b): b = (Σy - m * Σx) / n
Alternatively, b = ȳ - m * x̄, where ȳ is the mean of y values and x̄ is the mean of x values.
The R-squared value (Coefficient of Determination) measures how well the regression line approximates the real data points. An R-squared value of 1 indicates that the regression line perfectly fits the data.
R² = [ (n * Σxy - Σx * Σy)² ] / [ (n * Σx² - (Σx)²) * (n * Σy² - (Σy)²) ]
Use Cases
Least squares regression is widely used in various fields:
Economics: Forecasting demand, analyzing the relationship between price and sales.
Science: Modeling experimental data, understanding relationships between variables (e.g., temperature vs. reaction rate).
Social Sciences: Analyzing trends in population growth, understanding factors influencing social behavior.
Engineering: Optimizing processes, predicting material performance.
By understanding the linear relationship and its strength, professionals can make more informed decisions and predictions.
function calculateRegressionLine() {
var xValuesInput = document.getElementById("xValues").value;
var yValuesInput = document.getElementById("yValues").value;
var errorMsg = document.getElementById("errorMsg");
errorMsg.style.display = 'none'; // Hide previous error messages
// Clear previous results
document.getElementById("slopeValue").textContent = "-";
document.getElementById("interceptValue").textContent = "-";
document.getElementById("equationValue").textContent = "y = -x + -";
document.getElementById("rSquaredValue").textContent = "-";
// Parse input values
var xValues = xValuesInput.split(',').map(function(val) { return parseFloat(val.trim()); });
var yValues = yValuesInput.split(',').map(function(val) { return parseFloat(val.trim()); });
// Validate input data
if (xValues.length !== yValues.length) {
errorMsg.textContent = "Error: The number of X values must match the number of Y values.";
errorMsg.style.display = 'block';
return;
}
var n = xValues.length;
if (n < 2) {
errorMsg.textContent = "Error: At least two data points are required.";
errorMsg.style.display = 'block';
return;
}
var sumX = 0;
var sumY = 0;
var sumXY = 0;
var sumX2 = 0;
var sumY2 = 0;
for (var i = 0; i 0 && denominatorR2_part2 > 0) {
rSquared = numeratorR2 / (denominatorR2_part1 * denominatorR2_part2);
} else if (numeratorR2 === 0 && denominatorR2_part1 * denominatorR2_part2 === 0) {
// Perfect fit for a horizontal line with all y-values the same
rSquared = 1;
}
// Display results
document.getElementById("slopeValue").textContent = m.toFixed(4);
document.getElementById("interceptValue").textContent = b.toFixed(4);
document.getElementById("equationValue").textContent = "y = " + m.toFixed(4) + "x + " + b.toFixed(4);
document.getElementById("rSquaredValue").textContent = rSquared.toFixed(4);
}