Linear Regression Line Calculator

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Linear Regression Line Calculator

Example: 10, 20, 30, 40, 50
Example: 25, 45, 65, 85, 105
Enter a specific X value to predict its corresponding Y.

Results:

Enter your data and click 'Calculate Regression' to see the results.

function calculateLinearRegression() { var xValuesStr = document.getElementById('xValuesInput').value; var yValuesStr = document.getElementById('yValuesInput').value; var predictXStr = document.getElementById('predictXInput').value; var resultDiv = document.getElementById('regressionResult'); var xArray = xValuesStr.split(',').map(function(item) { return parseFloat(item.trim()); }).filter(function(item) { return !isNaN(item); }); var yArray = yValuesStr.split(',').map(function(item) { return parseFloat(item.trim()); }).filter(function(item) { return !isNaN(item); }); var predictX = parseFloat(predictXStr); if (xArray.length === 0 || yArray.length === 0) { resultDiv.innerHTML = '

Error:

Please enter at least two valid numbers for both X and Y values.'; return; } if (xArray.length !== yArray.length) { resultDiv.innerHTML = '

Error:

The number of X values must match the number of Y values.'; return; } if (xArray.length < 2) { resultDiv.innerHTML = '

Error:

At least two data points are required to calculate a regression line.'; return; } var n = xArray.length; var sumX = 0; var sumY = 0; var sumXY = 0; var sumX2 = 0; var 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 denominator = (n * sumX2 – sumX * sumX); if (denominator === 0) { var allXSame = true; for (var i = 1; i < n; i++) { if (xArray[i] !== xArray[0]) { allXSame = false; break; } } if (allXSame) { resultDiv.innerHTML = '

Error:

All X values are identical. A vertical line cannot be represented by y = mx + b.'; return; } // This case should ideally not be reached if allXSame is handled, but as a fallback for numerical instability resultDiv.innerHTML = '

Error:

Cannot calculate regression: denominator is zero. Check your data for identical X values.'; return; } var m = (n * sumXY – sumX * sumY) / denominator; var b = (sumY – m * sumX) / n; // Calculate R-squared var ssTotal = 0; var ssResidual = 0; var meanY = sumY / n; for (var i = 0; i < n; i++) { var predictedY_i = m * xArray[i] + b; ssTotal += Math.pow(yArray[i] – meanY, 2); ssResidual += Math.pow(yArray[i] – predictedY_i, 2); } var rSquared = (ssTotal === 0) ? 1 : (1 – (ssResidual / ssTotal)); // If all Y values are the same, R^2 is 1 var predictedY = 'N/A'; if (!isNaN(predictX)) { predictedY = (m * predictX + b).toFixed(4); } var resultsHtml = '

Linear Regression Results:

'; resultsHtml += 'Slope (m): ' + m.toFixed(4) + "; resultsHtml += 'Y-intercept (b): ' + b.toFixed(4) + "; resultsHtml += 'Regression Equation: Y = ' + m.toFixed(4) + 'X + ' + b.toFixed(4) + "; resultsHtml += 'Coefficient of Determination (R²): ' + rSquared.toFixed(4) + "; if (!isNaN(predictX)) { resultsHtml += 'Predicted Y for X = ' + predictX + ': ' + predictedY + "; } else { resultsHtml += 'Prediction Error: Please enter a valid number for "Predict Y for X =" to get a prediction.'; } resultDiv.innerHTML = resultsHtml; }

Understanding the Linear Regression Line Calculator

Linear regression is a fundamental statistical method used to model the relationship between two continuous variables. It aims to find the "best-fit" straight line (the regression line) that describes how an independent variable (X) relates to a dependent variable (Y). This line can then be used to predict the value of Y for a given X, or to understand the strength and direction of the relationship between the variables.

What is a Linear Regression Line?

A linear regression line is represented by the equation: Y = mX + b

  • Y: The dependent variable (the variable you are trying to predict or explain).
  • X: The independent variable (the variable used to predict Y).
  • m: The slope of the line. It represents the change in Y for every one-unit change in X. A positive slope indicates a positive relationship (as X increases, Y tends to increase), while a negative slope indicates a negative relationship (as X increases, Y tends to decrease).
  • b: The Y-intercept. This is the value of Y when X is 0. It represents the starting point of the line on the Y-axis.

How is the Line Calculated?

The "best-fit" line is typically determined using the method of Ordinary Least Squares (OLS). This method minimizes the sum of the squared differences between the observed Y values and the Y values predicted by the line. The formulas for calculating the slope (m) and Y-intercept (b) are derived from this principle:

Slope (m):
m = [ nΣ(XY) - ΣXΣY ] / [ nΣ(X²) - (ΣX)² ]

Y-intercept (b):
b = [ ΣY - mΣX ] / n

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 product of each X and Y pair.
  • ΣX² is the sum of the squares of each X value.

Understanding R-squared (Coefficient of Determination)

The Coefficient of Determination, denoted as R², is a crucial metric in linear regression. It tells you how well the regression line fits the observed data points. R² values range from 0 to 1:

  • An R² of 1 (or 100%) means that the model explains all the variability of the dependent variable around its mean. In other words, the regression line perfectly fits the data.
  • An R² of 0 means that the model explains none of the variability of the dependent variable around its mean. The regression line does not help predict Y.
  • An R² between 0 and 1 indicates the proportion of the variance in the dependent variable that is predictable from the independent variable(s). For example, an R² of 0.75 means that 75% of the variation in Y can be explained by X.

A higher R² generally indicates a better fit for the model, but it's important to consider the context and other statistical measures.

Example Use Case

Imagine a researcher wants to study the relationship between the number of hours a student spends studying (X) and their score on a final exam (Y). They collect data from several students:

  • X (Study Hours): 10, 20, 30, 40, 50
  • Y (Exam Score): 25, 45, 65, 85, 105

Using the calculator with these values, you would find:

  • Slope (m): Approximately 2.00
  • Y-intercept (b): Approximately 5.00
  • Regression Equation: Y = 2.00X + 5.00
  • R²: Approximately 1.00 (indicating a very strong, almost perfect linear relationship in this simplified example)

This means for every additional hour of study, the exam score is predicted to increase by 2 points. If a student studies 0 hours, their predicted score is 5. If the researcher wants to predict the score for a student who studies 60 hours, they would input X=60 into the calculator, yielding a predicted Y of 125.

This calculator provides a quick and easy way to determine the linear regression line, its equation, and the R-squared value for your own datasets, helping you understand and predict relationships between variables.

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