Calculate the line of best fit (y = mx + b) for a given set of data points.
Slope (m): N/A
Y-Intercept (b): N/A
Correlation Coefficient (r): N/A
Equation: y = N/A x + N/A
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, simple linear regression, we aim to find a straight line that best represents the trend in a scatter plot of data points. This line is often referred to as the "line of best fit".
The Math Behind the Line of Best Fit
The goal of linear regression is to find the equation of a line, typically represented as y = mx + b, where:
y is the dependent variable (the value we want to predict).
x is the independent variable (the predictor).
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 zero.
The method used to find the "best" line is typically Ordinary Least Squares (OLS). OLS minimizes the sum of the squared differences between the observed y values and the y values predicted by the regression line. The formulas for calculating the slope (m) and the y-intercept (b) are:
Calculating the Slope (m):
m = Σ[(xi - x̄)(yi - ȳ)] / Σ[(xi - x̄)²]
Where:
xi and yi are the individual data points.
x̄ (x-bar) is the mean (average) of the x values.
ȳ (y-bar) is the mean (average) of the y values.
Σ denotes summation.
An alternative, computationally simpler formula for the slope is:
m = [n(Σxy) - (Σx)(Σy)] / [n(Σx²) - (Σx)²]
Where n is the number of data points.
Calculating the Y-Intercept (b):
Once the slope m is calculated, the y-intercept b is found using the means:
b = ȳ - m * x̄
Correlation Coefficient (r):
The correlation coefficient (r) measures the strength and direction of the linear relationship between x and y. It ranges from -1 to +1:
+1 indicates a perfect positive linear correlation.
-1 indicates a perfect negative linear correlation.
Linear regression is widely applicable across various fields:
Economics: Predicting trends in stock prices, GDP, or inflation.
Finance: Forecasting sales, analyzing investment performance, and risk assessment.
Science: Modeling relationships between physical quantities, such as temperature and pressure.
Social Sciences: Understanding relationships between variables like education level and income.
Machine Learning: As a foundational algorithm for more complex predictive models.
By calculating the line of best fit, we gain valuable insights into data patterns and can make informed predictions about future outcomes.
function calculateLinearRegression() {
var xValuesStr = document.getElementById("xValues").value;
var yValuesStr = document.getElementById("yValues").value;
var xArray = xValuesStr.split(',').map(function(x) { return parseFloat(x.trim()); });
var yArray = yValuesStr.split(',').map(function(y) { return parseFloat(y.trim()); });
if (isNaN(xArray[0]) || isNaN(yArray[0])) {
alert("Please enter valid numbers for X and Y values.");
return;
}
if (xArray.length !== yArray.length) {
alert("The number of X values must match the number of Y values.");
return;
}
var n = xArray.length;
var sumX = 0, sumY = 0, sumXY = 0, sumX2 = 0, sumY2 = 0;
for (var i = 0; i < n; i++) {
sumX += xArray[i];
sumY += yArray[i];
sumXY += xArray[i] * yArray[i];
sumX2 += xArray[i] * xArray[i];
sumY2 += yArray[i] * yArray[i];
}
var meanX = sumX / n;
var meanY = sumY / n;
// Calculate slope (m) using the formula: m = [n(Σxy) – (Σx)(Σy)] / [n(Σx²) – (Σx)²]
var denominatorM = n * sumX2 – sumX * sumX;
if (denominatorM === 0) {
alert("Cannot calculate slope: denominator is zero (all X values might be the same).");
return;
}
var m = (n * sumXY – sumX * sumY) / denominatorM;
// Calculate y-intercept (b) using the formula: b = ȳ – m * x̄
var b = meanY – m * meanX;
// Calculate correlation coefficient (r)
var numeratorR = n * sumXY – sumX * sumY;
var denominatorR = Math.sqrt((n * sumX2 – sumX * sumX) * (n * sumY2 – sumY * sumY));
var r = 0; // Default to 0 if denominator is 0
if (denominatorR !== 0) {
r = numeratorR / denominatorR;
} else if (numeratorR === 0) {
r = 1; // Case where all points are identical
}
document.getElementById("slopeResult").textContent = "Slope (m): " + m.toFixed(4);
document.getElementById("interceptResult").textContent = "Y-Intercept (b): " + b.toFixed(4);
document.getElementById("correlationResult").textContent = "Correlation Coefficient (r): " + r.toFixed(4);
document.getElementById("equationResult").textContent = "Equation: y = " + m.toFixed(4) + " x + " + b.toFixed(4);
}