Understanding Linear Regression
Linear regression is a fundamental statistical method used to model the relationship between two continuous variables: a dependent variable (Y) and an independent variable (X). The goal is to find the best-fitting straight line (the regression line) that describes how Y changes as X changes. This line can then be used for prediction or to understand the strength and direction of the relationship between the variables.
How Linear Regression Works
The equation of a straight line is typically represented as Y = a + bX, where:
Y is the dependent variable (the one you're trying to predict).X is the independent variable (the one used for prediction).a is the Y-intercept, which is the value of Y when X is 0.b is the slope of the line, indicating how much Y changes for every one-unit change in X.
The "best-fit" line is determined using the method of least squares, which minimizes the sum of the squared differences between the observed Y values and the Y values predicted by the line. This method provides specific formulas for calculating the slope (b) and the Y-intercept (a).
Formulas Used in This Calculator
Given a set of n paired data points (Xi, Yi):
Slope (b):
b = (n * Σ(XiYi) - ΣXi * ΣYi) / (n * Σ(Xi2) - (ΣXi)2)
Y-intercept (a):
a = (ΣYi - b * ΣXi) / n
Where:
n is the number of data points.ΣXi is the sum of all X values.ΣYi is the sum of all Y values.Σ(XiYi) is the sum of the products of each X and Y pair.Σ(Xi2) is the sum of the squares of each X value.
Example Calculation
Let's use the default values provided in the calculator:
X Values: 1, 2, 3, 4, 5
Y Values: 2, 4, 5, 4, 5
First, we calculate the necessary sums:
n = 5ΣX = 1 + 2 + 3 + 4 + 5 = 15ΣY = 2 + 4 + 5 + 4 + 5 = 20ΣXY = (1*2) + (2*4) + (3*5) + (4*4) + (5*5) = 2 + 8 + 15 + 16 + 25 = 66ΣX2 = (12) + (22) + (32) + (42) + (52) = 1 + 4 + 9 + 16 + 25 = 55
Now, calculate the slope (b):
b = (5 * 66 - 15 * 20) / (5 * 55 - 152)
b = (330 - 300) / (275 - 225)
b = 30 / 50 = 0.6
Next, calculate the Y-intercept (a):
a = (20 - 0.6 * 15) / 5
a = (20 - 9) / 5
a = 11 / 5 = 2.2
Therefore, the linear regression equation for this data set is Y = 2.2 + 0.6X.
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function calculateLinearRegression() {
var xValuesInput = document.getElementById("xValues").value;
var yValuesInput = document.getElementById("yValues").value;
var resultDiv = document.getElementById("regressionResult");
var xArray = xValuesInput.split(',').map(function(item) {
return parseFloat(item.trim());
});
var yArray = yValuesInput.split(',').map(function(item) {
return parseFloat(item.trim());
});
// Filter out NaN values and ensure arrays are clean
xArray = xArray.filter(function(value) { return !isNaN(value); });
yArray = yArray.filter(function(value) { return !isNaN(value); });
if (xArray.length === 0 || yArray.length === 0) {
resultDiv.innerHTML = "Please enter valid numeric X and Y values.";
return;
}
if (xArray.length !== yArray.length) {
resultDiv.innerHTML = "The number of X values must match the number of Y values.";
return;
}
var n = xArray.length;
var sumX = 0;
var sumY = 0;
var sumXY = 0;
var sumX2 = 0;
for (var i = 0; i < n; i++) {
sumX += xArray[i];
sumY += yArray[i];
sumXY += xArray[i] * yArray[i];
sumX2 += xArray[i] * xArray[i];
}
var denominator = (n * sumX2 – sumX * sumX);
if (denominator === 0) {
resultDiv.innerHTML = "Cannot calculate linear regression: All X values are identical. This would result in a vertical line, which cannot be represented by y = ax + b.";
return;
}
var slopeB = (n * sumXY – sumX * sumY) / denominator;
var interceptA = (sumY – slopeB * sumX) / n;
// Round to a reasonable number of decimal places
slopeB = parseFloat(slopeB.toFixed(4));
interceptA = parseFloat(interceptA.toFixed(4));
var regressionEquation = "Y = " + interceptA + " + " + slopeB + "X";
resultDiv.innerHTML =
"