Chi-Square Distribution Calculator
body {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: #f8f9fa;
color: #333;
line-height: 1.6;
margin: 0;
padding: 0;
}
.chi-square-calc-container {
max-width: 800px;
margin: 40px auto;
background-color: #ffffff;
padding: 30px;
border-radius: 8px;
box-shadow: 0 4px 15px rgba(0, 0, 0, 0.1);
}
h1, h2 {
color: #004a99;
text-align: center;
margin-bottom: 20px;
}
.input-section, .output-section, .article-section {
margin-bottom: 30px;
padding: 20px;
border: 1px solid #e0e0e0;
border-radius: 6px;
background-color: #fdfdfd;
}
.input-group {
margin-bottom: 20px;
display: flex;
flex-direction: column;
align-items: flex-start;
}
.input-group label {
display: block;
margin-bottom: 8px;
font-weight: bold;
color: #004a99;
}
.input-group input[type="number"] {
width: calc(100% – 24px);
padding: 12px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 16px;
box-sizing: border-box;
}
.input-group input[type="number"]:focus {
border-color: #004a99;
outline: none;
box-shadow: 0 0 5px rgba(0, 74, 153, 0.3);
}
button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #004a99;
color: white;
border: none;
border-radius: 4px;
font-size: 18px;
cursor: pointer;
transition: background-color 0.3s ease;
margin-top: 10px;
}
button:hover {
background-color: #003366;
}
.result-container {
text-align: center;
margin-top: 20px;
padding: 15px;
background-color: #e7f3ff;
border-left: 5px solid #28a745;
border-radius: 4px;
}
.result-container h3 {
margin-top: 0;
color: #004a99;
}
.result-value {
font-size: 24px;
font-weight: bold;
color: #28a745;
}
.article-section h2 {
text-align: left;
color: #004a99;
border-bottom: 2px solid #004a99;
padding-bottom: 10px;
}
.article-section p, .article-section ul {
margin-bottom: 15px;
}
.article-section li {
margin-bottom: 8px;
}
/* Responsive adjustments */
@media (max-width: 600px) {
.chi-square-calc-container {
margin: 20px auto;
padding: 20px;
}
button {
font-size: 16px;
padding: 10px 15px;
}
.result-value {
font-size: 20px;
}
}
Chi-Square Distribution Calculator
Results
Enter values for Degrees of Freedom and X Value to see the results.
Understanding the Chi-Square Distribution
The Chi-Square (χ²) distribution is a fundamental probability distribution in statistics. It arises in inferential statistics, particularly in hypothesis testing. It is a skewed distribution whose shape depends on its degrees of freedom. The Chi-Square distribution is typically used in contexts involving sums of squared standard normal variables.
Mathematical Definition
If $Z_1, Z_2, …, Z_k$ are independent standard normal random variables (mean 0, variance 1), then the sum of their squares follows a Chi-Square distribution with $k$ degrees of freedom:
$$ \chi^2_k = Z_1^2 + Z_2^2 + … + Z_k^2 $$
The probability density function (PDF) of a Chi-Square distribution with $k$ degrees of freedom is given by:
$$ f(x; k) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{k/2 – 1} e^{-x/2} \quad \text{for } x > 0 $$
where $\Gamma(z)$ is the Gamma function.
Key Parameters
- Degrees of Freedom (df or k): This parameter determines the shape of the distribution. It represents the number of independent standard normal variables that are squared and summed. The df must be a positive integer.
- X Value (x): This is the specific value on the horizontal axis of the distribution for which we want to calculate the probability or density.
Calculator Functions
- Cumulative Probability (P(X ≤ x)): This calculates the probability that a random variable following the Chi-Square distribution with the specified degrees of freedom will take on a value less than or equal to a given X value. This is often referred to as the CDF (Cumulative Distribution Function).
- Probability Density (f(x)): This calculates the value of the probability density function at a specific X value. It represents the relative likelihood of observing a value very close to 'x'.
Common Use Cases
- Goodness-of-Fit Tests: To determine if an observed frequency distribution fits an expected distribution (e.g., testing if dice are fair).
- Tests for Independence: To test whether two categorical variables are independent of each other (e.g., testing if there is a relationship between gender and preference for a product).
- Tests for Homogeneity: To test if proportions are the same across different populations.
- Confidence Intervals for Variance: In quality control and statistical process control.
This calculator helps in quickly computing these probabilities and densities, aiding in statistical analysis and decision-making.
// Function to calculate the Gamma function using Lanczos approximation
// Source: Adapted from various implementations of Lanczos approximation
function gamma(z) {
if (z < 0.5) {
return Math.PI / (Math.sin(Math.PI * z) * gamma(1 – z));
} else {
var x = z – 1;
var tmp = x + 5.5;
tmp -= (x + 0.5) * Math.log(x + 5.5);
var ser = 1.000000000190015;
ser += 76.18009172947146 / (x + 1);
ser -= 86.50531813073509 / (x + 2);
ser += 24.01409824083091 / (x + 3);
ser -= 1.231739572450155 / (x + 4);
ser += 0.0012086509738661959 / (x + 5);
ser -= 5.395039324960876e-6 / (x + 6);
return Math.sqrt(2 * Math.PI) * Math.pow(x + 5.5, x + 0.5) * Math.exp(-tmp) * ser;
}
}
// Incomplete Beta function (regularized) – used for CDF
// This is a complex function. For a basic JS implementation, we might need an approximation
// or rely on external libraries for high accuracy, especially for large 'df'.
// For simplicity here, we'll use a numerical approximation (e.g., via Taylor series or continued fraction)
// or a simplified approach if accuracy is not paramount for demonstration.
// A common approach is to use the relationship with the incomplete beta function:
// P(X <= x | df=k) = I(x/2, k/2, k/2)
// The incomplete beta function I_x(a, b) is: B_x(a, b) / B(a, b)
// where B_x(a, b) is the incomplete beta function and B(a, b) is the complete beta function.
// B(a, b) = Gamma(a) * Gamma(b) / Gamma(a + b)
// Numerical integration for CDF (simpler to implement for demonstration)
function calculateChiSquareCDF() {
var df = parseFloat(document.getElementById("degreesOfFreedom").value);
var x = parseFloat(document.getElementById("xValue").value);
var resultDiv = document.getElementById("result");
if (isNaN(df) || df <= 0 || isNaN(x) || x < 0) {
resultDiv.innerHTML = 'Please enter valid positive numbers for Degrees of Freedom and a non-negative X Value.';
return;
}
// For demonstration, we'll use a numerical integration method.
// A robust implementation would use the relationship with the incomplete beta function.
// Numerical integration using Simpson's rule for the PDF integral: integral(f(t), 0, x)
var n = 10000; // Number of intervals for integration (higher for better accuracy)
var h = x / n;
var integral = 0;
var pdfAtT = function(t, k) {
if (t <= 0) return 0;
var term1 = Math.pow(t, k / 2 – 1);
var term2 = Math.exp(-t / 2);
var denominator = Math.pow(2, k / 2) * gamma(k / 2);
return (term1 * term2) / denominator;
};
for (var i = 0; i <= n; i++) {
var t = i * h;
var y = pdfAtT(t, df);
if (i === 0 || i === n) {
integral += y;
} else if (i % 2 === 0) {
integral += 2 * y;
} else {
integral += 4 * y;
}
}
integral *= h / 3;
resultDiv.innerHTML = '
Cumulative Probability (P(X ≤ ' + x + '))
' + integral.toFixed(6) + '
';
}
function calculateChiSquarePDF() {
var df = parseFloat(document.getElementById("degreesOfFreedom").value);
var x = parseFloat(document.getElementById("xValue").value);
var resultDiv = document.getElementById("result");
if (isNaN(df) || df <= 0 || isNaN(x) || x 0
resultDiv.innerHTML = 'Please enter valid positive numbers for Degrees of Freedom and a positive X Value for PDF calculation.';
return;
}
var pdfValue = Math.pow(x, df / 2 – 1) * Math.exp(-x / 2);
var denominator = Math.pow(2, df / 2) * gamma(df / 2);
var finalPdf = pdfValue / denominator;
resultDiv.innerHTML = '
Probability Density at X = ' + x + '
' + finalPdf.toFixed(6) + '
';
}