.calc-container { background: #ffffff; padding: 25px; border-radius: 8px; box-shadow: 0 4px 6px rgba(0,0,0,0.1); margin-bottom: 30px; }
.calc-header { text-align: center; margin-bottom: 25px; }
.calc-header h2 { color: #2c3e50; margin-bottom: 10px; font-size: 24px; }
.input-group { margin-bottom: 20px; }
.input-group label { display: block; font-weight: 600; margin-bottom: 8px; color: #444; }
.input-group input { width: 100%; padding: 12px; border: 1px solid #ccc; border-radius: 5px; font-size: 16px; box-sizing: border-box; }
.calc-btn { background-color: #3498db; color: white; border: none; padding: 15px 20px; width: 100%; border-radius: 5px; font-size: 18px; font-weight: bold; cursor: pointer; transition: background 0.3s; }
.calc-btn:hover { background-color: #2980b9; }
#residual-result-area { margin-top: 25px; padding: 20px; border-radius: 5px; display: none; text-align: center; }
.result-positive { background-color: #e8f5e9; border: 1px solid #c8e6c9; color: #2e7d32; }
.result-negative { background-color: #ffebee; border: 1px solid #ffcdd2; color: #c62828; }
.result-zero { background-color: #e3f2fd; border: 1px solid #bbdefb; color: #1565c0; }
.result-value { font-size: 32px; font-weight: 800; display: block; margin: 10px 0; }
.formula-box { background: #eee; padding: 15px; border-radius: 5px; font-family: 'Courier New', Courier, monospace; margin: 20px 0; text-align: center; }
.seo-content h2 { color: #2c3e50; margin-top: 30px; border-bottom: 2px solid #3498db; padding-bottom: 5px; }
.seo-content h3 { color: #2980b9; margin-top: 20px; }
.example-table { width: 100%; border-collapse: collapse; margin: 20px 0; }
.example-table th, .example-table td { border: 1px solid #ddd; padding: 12px; text-align: left; }
.example-table th { background-color: #f2f2f2; }
function calculateResidual() {
var observed = document.getElementById('observedValue').value;
var predicted = document.getElementById('predictedValue').value;
var resultArea = document.getElementById('residual-result-area');
var output = document.getElementById('residual-output');
var interpretation = document.getElementById('interpretation-text');
if (observed === "" || predicted === "") {
alert("Please enter both observed and predicted values.");
return;
}
var y = parseFloat(observed);
var yHat = parseFloat(predicted);
if (isNaN(y) || isNaN(yHat)) {
alert("Please enter valid numeric values.");
return;
}
var residual = y – yHat;
// Round to 4 decimal places for precision
residual = Math.round(residual * 10000) / 10000;
output.innerHTML = residual;
resultArea.style.display = 'block';
// Remove previous classes
resultArea.className = ";
if (residual > 0) {
resultArea.classList.add('result-positive');
interpretation.innerHTML = "Positive Residual: The actual value is higher than the model's prediction (Underestimation).";
} else if (residual < 0) {
resultArea.classList.add('result-negative');
interpretation.innerHTML = "Negative Residual: The actual value is lower than the model's prediction (Overestimation).";
} else {
resultArea.classList.add('result-zero');
interpretation.innerHTML = "Zero Residual: The model's prediction perfectly matches the observed data.";
}
}
Residual Calculator
Determine the difference between observed and predicted values in regression analysis.
The Residual is:
What is a Residual in Statistics?
In statistics and regression analysis, a residual is the vertical distance between a data point and the regression line. It represents the "error" of a model, indicating how much the actual observed value deviates from the value predicted by the mathematical model.
Calculating residuals is a fundamental step in residual analysis, which helps data scientists and statisticians determine if a linear regression model is appropriate for a specific dataset or if there are hidden patterns that the model failed to capture.
Residual (e) = Observed Value (y) – Predicted Value (ŷ)
How to Calculate Residuals Step-by-Step
- Collect your observed data: This is the real-world measurement or data point you have recorded (the dependent variable).
- Generate a prediction: Use your regression equation (usually in the form y = mx + b) to calculate what the value should be based on your independent variables.
- Subtract: Take the observed value (y) and subtract the predicted value (ŷ).
- Analyze: A result of 0 means a perfect prediction. Any other number represents the deviation.
Interpreting Positive vs. Negative Residuals
- Positive Residual: When the observed value is greater than the predicted value, the residual is positive. This means the model underestimated the actual result.
- Negative Residual: When the observed value is less than the predicted value, the residual is negative. This means the model overestimated the actual result.
Residual Calculation Example
Suppose you are studying the relationship between study hours and exam scores. Your regression model predicts that a student who studies for 5 hours should receive a score of 82. However, the student actually receives a score of 88.
| Variable | Value |
|---|---|
| Observed Value (y) | 88 |
| Predicted Value (ŷ) | 82 |
| Residual (e) | +6 |
In this case, the residual is +6, indicating the model was 6 points off and underestimated the student's actual performance.
Importance of Residual Analysis
Calculating individual residuals is the first step toward more complex diagnostic checks, such as:
- Residual Plots: Plotting residuals on the y-axis against the independent variable on the x-axis to look for non-linear patterns.
- Homoscedasticity: Checking if the variance of the residuals is constant across all levels of the independent variables.
- Outlier Detection: Identifying data points with exceptionally large residuals that might skew the model's accuracy.
- Sum of Squared Residuals (SSR): Squaring all residuals and adding them together to measure the overall fit of the regression line.