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Linear Regression Equation (y = mx + b)
Enter data points to see the equation.
Understanding Linear Regression
Linear regression is a fundamental statistical method used to model the relationship between a dependent variable (y) and one or more independent variables (x). In its simplest form, called simple linear regression, we aim to find the best-fitting straight line through a set of data points. This line can then be used to predict the value of the dependent variable for new values of the independent variable.
The Linear Regression Equation
The equation of a straight line is typically represented as:
y = mx + b
Where:
y is the dependent variable (what we are trying to predict).
x is the independent variable (the predictor).
m is the slope of the line, representing the change in 'y' for a one-unit change in 'x'.
b is the y-intercept, representing the value of 'y' when 'x' is zero.
How the Calculator Works (Least Squares Method)
This calculator uses the method of least squares to find the values of 'm' and 'b' that best fit the given data points. The goal is to minimize the sum of the squared differences between the actual 'y' values and the 'y' values predicted by the line. The formulas are derived as follows:
Given n data points (x₁, y₁), (x₂, y₂), …, (x, y):
m = [ n(Σxy) – (Σx)(Σy) ] / [ n(Σx²) – (Σx)² ]
b = [ (Σy)(Σx²) – (Σx)(Σxy) ] / [ n(Σx²) – (Σx)² ]
or more simply, after calculating m:
b = ȳ – mx
Where:
n is the number of data points.
Σx is the sum of all x values.
Σy is the sum of all y values.
Σxy is the sum of the products of each corresponding x and y value.
Σx² is the sum of the squares of each x value.
x is the mean of the x values (Σx / n).
ȳ is the mean of the y values (Σy / n).
Use Cases for Linear Regression
Linear regression is a versatile tool with applications across numerous fields:
Finance: Assessing the relationship between risk and return, modeling asset prices.
Science: Studying the relationship between temperature and plant growth, analyzing experimental data.
Medicine: Investigating the correlation between drug dosage and patient response.
Social Sciences: Examining the relationship between education level and income, predicting population trends.
Machine Learning: As a foundational algorithm for more complex predictive models.
Interpreting the Results
The calculated equation y = mx + b allows you to:
Understand the strength and direction of the linear relationship between your variables. A positive 'm' indicates that as 'x' increases, 'y' tends to increase, while a negative 'm' suggests the opposite.
Predict the value of 'y' for a given value of 'x'. For example, if you have an equation y = 2x + 5, and you want to know the predicted 'y' when x = 10, you calculate y = 2(10) + 5 = 25.
function calculateLinearRegression() {
var xValues = [];
var yValues = [];
// Collect data points from input fields
for (var i = 1; i <= 5; i++) {
var xInput = document.getElementById('x' + i);
var yInput = document.getElementById('y' + i);
var x = parseFloat(xInput.value);
var y = parseFloat(yInput.value);
// Check if inputs are valid numbers and are not empty
if (!isNaN(x) && !isNaN(y) && xInput.value !== "" && yInput.value !== "") {
xValues.push(x);
yValues.push(y);
}
}
var n = xValues.length;
var resultDiv = document.getElementById('regressionResult');
// Need at least two data points to perform regression
if (n < 2) {
resultDiv.textContent = "Please enter at least two valid data points.";
return;
}
var sumX = 0;
var sumY = 0;
var sumXY = 0;
var sumX2 = 0;
for (var i = 0; i < n; i++) {
sumX += xValues[i];
sumY += yValues[i];
sumXY += xValues[i] * yValues[i];
sumX2 += xValues[i] * xValues[i];
}
var denominator = n * sumX2 – sumX * sumX;
// Avoid division by zero if all x values are the same
if (denominator === 0) {
resultDiv.textContent = "Cannot calculate regression: All X values are the same.";
return;
}
var m = (n * sumXY – sumX * sumY) / denominator;
var b = (sumY * sumX2 – sumX * sumXY) / denominator;
// Format the output string
var equation = "y = " + m.toFixed(4) + "x + " + b.toFixed(4);
resultDiv.textContent = equation;
}
// Update the count of data points in the article dynamically
// This is a small enhancement; ideally, the article would be updated by a CMS or templating engine
// For this single file, we'll just set a placeholder if we know max points is 5
document.getElementById('n_formula').textContent = "n";
document.getElementById('n_formula_dup').textContent = "n";