The TI-83 Plus graphing calculator is a staple in high school and college mathematics and science courses. One of its most powerful and frequently used features is its ability to perform statistical analysis, including linear regression. Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data.
This calculator emulates the core functionality of the TI-83 Plus's linear regression (LinReg(ax+b)) feature, allowing you to find the equation of the least-squares regression line (y = ax + b), the correlation coefficient (r), and the coefficient of determination (r²) for a given set of paired data points (x, y).
Understanding linear regression is crucial for predicting trends, analyzing relationships between variables, and making informed decisions in various fields, from economics to engineering. The TI-83 Plus simplifies this complex calculation, making it accessible to students and professionals alike.
How to Use This Calculator:
Enter X-Values: Input your independent variable data points, separated by commas (e.g., 1,2,3,4,5).
Enter Y-Values: Input your dependent variable data points, separated by commas (e.g., 2,4,5,4,5).
Click "Calculate Regression": The calculator will process your data and display the slope (a), y-intercept (b), correlation coefficient (r), coefficient of determination (r²), and the full regression equation.
Understanding the Results:
Slope (a): Represents the rate of change of Y with respect to X. For every one-unit increase in X, Y is predicted to change by 'a' units.
Y-Intercept (b): The predicted value of Y when X is 0.
Correlation Coefficient (r): A measure of the strength and direction of a linear relationship between two variables. It ranges from -1 to +1. A value close to +1 indicates a strong positive linear relationship, close to -1 indicates a strong negative linear relationship, and close to 0 indicates a weak or no linear relationship.
Coefficient of Determination (r²): Represents the proportion of the variance in the dependent variable (Y) that is predictable from the independent variable (X). For example, an r² of 0.75 means 75% of the variation in Y can be explained by the variation in X.
Regression Equation (y = ax + b): The mathematical equation of the line of best fit, which can be used to predict Y values for given X values.
Example Scenario:
Imagine a student is tracking their study hours (X) and their test scores (Y) for several exams. They want to see if there's a linear relationship and predict future scores.
X-Values (Study Hours):3, 5, 2, 8, 6
Y-Values (Test Scores):70, 85, 60, 95, 90
Inputting these values into the calculator would yield the regression equation, slope, intercept, and correlation coefficients, helping the student understand the impact of study hours on test performance, similar to how they would use a TI-83 Plus.
.calculator-container {
font-family: 'Arial', sans-serif;
background-color: #f9f9f9;
padding: 20px;
border-radius: 8px;
box-shadow: 0 2px 10px rgba(0, 0, 0, 0.1);
max-width: 700px;
margin: 20px auto;
border: 1px solid #ddd;
}
.calculator-container h2 {
color: #333;
text-align: center;
margin-bottom: 20px;
font-size: 24px;
}
.calculator-container h3 {
color: #555;
margin-top: 25px;
margin-bottom: 15px;
font-size: 18px;
}
.calculator-container p {
color: #666;
line-height: 1.6;
margin-bottom: 10px;
}
.calculator-form label {
display: block;
margin-bottom: 8px;
color: #333;
font-weight: bold;
}
.calculator-form input[type="text"] {
width: calc(100% – 22px);
padding: 10px;
margin-bottom: 15px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 16px;
}
.calculator-form button {
background-color: #007bff;
color: white;
padding: 12px 20px;
border: none;
border-radius: 4px;
cursor: pointer;
font-size: 16px;
width: 100%;
transition: background-color 0.3s ease;
}
.calculator-form button:hover {
background-color: #0056b3;
}
.calculator-result {
background-color: #e9ecef;
padding: 15px;
border-radius: 4px;
margin-top: 20px;
border: 1px solid #dee2e6;
min-height: 50px;
}
.calculator-result p {
margin: 5px 0;
color: #333;
}
.calculator-result p strong {
color: #000;
}
.calculator-container ol, .calculator-container ul {
color: #666;
margin-left: 20px;
margin-bottom: 10px;
}
.calculator-container ol li, .calculator-container ul li {
margin-bottom: 5px;
line-height: 1.5;
}
function calculateRegression() {
var xInput = document.getElementById("xValues").value;
var yInput = document.getElementById("yValues").value;
var resultOutput = document.getElementById("resultOutput");
var xValues = xInput.split(',').map(Number);
var yValues = yInput.split(',').map(Number);
// Input validation
if (xValues.some(isNaN) || yValues.some(isNaN)) {
resultOutput.innerHTML = "Error: Please enter valid numbers for X and Y values.";
return;
}
if (xValues.length !== yValues.length) {
resultOutput.innerHTML = "Error: The number of X values must match the number of Y values.";
return;
}
if (xValues.length < 2) {
resultOutput.innerHTML = "Error: At least two data points are required for linear regression.";
return;
}
var n = xValues.length;
var sum_x = 0;
var sum_y = 0;
var sum_xy = 0;
var sum_x2 = 0;
var sum_y2 = 0;
for (var i = 0; i < n; i++) {
sum_x += xValues[i];
sum_y += yValues[i];
sum_xy += (xValues[i] * yValues[i]);
sum_x2 += (xValues[i] * xValues[i]);
sum_y2 += (yValues[i] * yValues[i]);
}
var denominator_m = (n * sum_x2 – sum_x * sum_x);
if (denominator_m === 0) {
resultOutput.innerHTML = "Error: Cannot calculate slope. All X values are identical.";
return;
}
// Calculate slope (a)
var a = (n * sum_xy – sum_x * sum_y) / denominator_m;
// Calculate y-intercept (b)
var b = (sum_y – a * sum_x) / n;
// Calculate correlation coefficient (r)
var numerator_r = (n * sum_xy – sum_x * sum_y);
var denominator_r = Math.sqrt((n * sum_x2 – sum_x * sum_x) * (n * sum_y2 – sum_y * sum_y));
var r = 0;
if (denominator_r !== 0) {
r = numerator_r / denominator_r;
} else {
// Handle cases where all Y values are identical or all X values are identical (already handled for X)
// If all Y values are identical, r is 0 unless X values are also identical (which means denominator_r is 0)
// If all Y values are the same, the line is horizontal, r should be 0.
if (yValues.every(function(val) { return val === yValues[0]; })) {
r = 0;
} else {
resultOutput.innerHTML = "Error: Cannot calculate correlation coefficient due to data variance.";
return;
}
}
// Calculate coefficient of determination (r^2)
var rSquared = r * r;
resultOutput.innerHTML =
"Slope (a): " + a.toFixed(4) + "" +
"Y-Intercept (b): " + b.toFixed(4) + "" +
"Correlation Coefficient (r): " + r.toFixed(4) + "" +
"Coefficient of Determination (r²): " + rSquared.toFixed(4) + "" +
"Regression Equation: y = " + a.toFixed(4) + "x + " + b.toFixed(4) + "";
}