// — Helper functions for Gamma and Incomplete Gamma (required for Chi-Squared P-Value) —
// Lanczos Approximation for Gamma function
function gamma(z) {
var g = 7;
var p = [
0.99999999999980993, 676.52036812423232, -1259.1392167224028,
771.32342877765307, -176.61502916214059, 12.507343278686905,
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7
];
if (z < 0.5) {
return Math.PI / (Math.sin(Math.PI * z) * gamma(1 – z));
}
z -= 1;
var x = p[0];
for (var i = 1; i < g + 2; i++) {
x += p[i] / (z + i);
}
var t = z + g + 0.5;
return Math.sqrt(2 * Math.PI) * Math.pow(t, z + 0.5) * Math.exp(-t) * x;
}
// Lower Incomplete Gamma function
function gamma_inc_lower(a, x) {
if (x < 0) return 0;
if (x === 0) return 0;
if (a <= 0) return 0; // Or throw error for invalid 'a'
var sum = 0;
var term = 1 / a;
sum = term;
for (var i = 1; i < 100; i++) { // Series expansion
term *= x / (a + i);
sum += term;
if (Math.abs(term) < 1e-10 * Math.abs(sum)) break; // Convergence check
}
return Math.exp(-x + a * Math.log(x) – Math.log(gamma(a))) * sum;
}
// Upper Incomplete Gamma function
function gamma_inc_upper(a, x) {
if (x < 0) return 0;
if (a <= 0) return 0; // Or throw error for invalid 'a'
if (x < a + 1) { // Use lower incomplete gamma and subtract from complete gamma
return gamma(a) – gamma_inc_lower(a, x);
} else { // Use continued fraction for upper incomplete gamma
var C = 0;
var D = 1 / C; // Initialize D to avoid division by zero later
var H = D;
var i = 1;
// Initialize C and D for continued fraction
C = 1e-30; // Small number to avoid division by zero
D = 1 / C;
H = D;
while (i < 100) {
var An = 1;
var Bn = x + i – a;
var Cn = An * C + Bn * D;
var Dn = An * D + Bn * C;
if (Cn === 0) Cn = 1e-30; // Avoid division by zero
if (Dn === 0) Dn = 1e-30; // Avoid division by zero
C = 1 / Cn;
D = 1 / Dn;
var Del = C * D;
H *= Del;
if (Math.abs(Del – 1) < 1e-10) break; // Convergence check
i++;
}
return Math.exp(-x + a * Math.log(x) – Math.log(gamma(a))) * H;
}
}
// Regularized Upper Incomplete Gamma function (P-value for Chi-Squared)
function regularized_gamma_q(a, x) {
if (x < 0 || a <= 0) return 1.0; // Invalid inputs, return 1.0 (no evidence against null)
if (x === 0) return 1.0;
// The p-value for Chi-squared is Q(df/2, chi2/2)
return gamma_inc_upper(a, x) / gamma(a);
}
// — Main Calculator Function —
function calculateChiSquaredPValue() {
var chiSquaredStatistic = parseFloat(document.getElementById("chiSquaredStatistic").value);
var degreesOfFreedom = parseInt(document.getElementById("degreesOfFreedom").value);
var resultDiv = document.getElementById("chiSquaredResult");
if (isNaN(chiSquaredStatistic) || isNaN(degreesOfFreedom) || chiSquaredStatistic < 0 || degreesOfFreedom < 1) {
resultDiv.innerHTML = "Please enter valid positive numbers for Chi-Squared Statistic and Degrees of Freedom (df must be at least 1).";
return;
}
// Parameters for the regularized upper incomplete gamma function
// a = df / 2
// x = chiSquaredStatistic / 2
var a_param = degreesOfFreedom / 2;
var x_param = chiSquaredStatistic / 2;
var pValue = regularized_gamma_q(a_param, x_param);
resultDiv.innerHTML = `
Calculated P-Value: ${pValue.toFixed(6)}
This is the probability of observing a Chi-Squared statistic as extreme or more extreme than ${chiSquaredStatistic.toFixed(3)}, assuming the null hypothesis is true, with ${degreesOfFreedom} degrees of freedom.
`;
}
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background-color: #f9f9f9;
border: 1px solid #ddd;
padding: 20px;
border-radius: 8px;
max-width: 600px;
margin: 20px auto;
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
}
.calculator-container h2 {
color: #333;
text-align: center;
margin-bottom: 20px;
}
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margin-bottom: 15px;
}
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display: block;
margin-bottom: 5px;
color: #555;
font-weight: bold;
}
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width: calc(100% – 22px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
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display: block;
width: 100%;
padding: 12px 20px;
background-color: #007bff;
color: white;
border: none;
border-radius: 4px;
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transition: background-color 0.3s ease;
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background-color: #0056b3;
}
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margin-top: 20px;
padding: 15px;
background-color: #e9f7ef;
border: 1px solid #d4edda;
border-radius: 4px;
color: #155724;
font-size: 1.1em;
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margin: 0 0 8px 0;
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margin-bottom: 0;
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font-size: 0.9em;
color: #386d4a;
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Understanding the Chi-Squared (χ²) Distribution and P-Value
The Chi-Squared (χ²) distribution is a fundamental concept in statistics, particularly in hypothesis testing. It's a continuous probability distribution that arises in various statistical tests, most notably in tests of independence, goodness-of-fit, and homogeneity.
What is the Chi-Squared Distribution?
The Chi-Squared distribution describes the distribution of the sum of the squares of independent standard normal random variables. Its shape is determined by a single parameter: the degrees of freedom (df). As the degrees of freedom increase, the Chi-Squared distribution approaches a normal distribution.
Key Applications:
Goodness-of-Fit Test: Used to determine if observed sample data fits an expected distribution. For example, testing if the observed frequencies of categories in a sample match the frequencies expected under a theoretical model.
Test of Independence: Used to determine if there is a significant association between two categorical variables in a contingency table. For example, testing if gender is independent of political affiliation.
Test of Homogeneity: Similar to the test of independence, but used when comparing the distribution of a single categorical variable across different populations.
Degrees of Freedom (df)
Degrees of freedom refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. In the context of Chi-Squared tests:
For a goodness-of-fit test, df = (number of categories – 1).
For a test of independence in a contingency table, df = (number of rows – 1) * (number of columns – 1).
The degrees of freedom directly influence the shape of the Chi-Squared distribution, which in turn affects the critical values and p-values.
What is a P-Value?
In hypothesis testing, the p-value is the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. For the Chi-Squared distribution:
A small p-value (typically less than a chosen significance level like 0.05) suggests that the observed Chi-Squared statistic is unlikely to have occurred by chance if the null hypothesis were true. This leads to rejecting the null hypothesis.
A large p-value suggests that the observed Chi-Squared statistic is consistent with the null hypothesis, and we fail to reject the null hypothesis.
Our calculator helps you find this p-value directly from your calculated Chi-Squared statistic and degrees of freedom.
How to Use This Calculator:
Enter your Chi-Squared Statistic (χ²): This is the value you would have calculated from your observed and expected frequencies in your statistical test.
Enter your Degrees of Freedom (df): This is determined by the structure of your data (e.g., number of categories or rows/columns in a table).
Click "Calculate P-Value": The calculator will then provide the corresponding p-value.
Example Scenario:
Imagine you are conducting a goodness-of-fit test to see if a die is fair. You roll the die 60 times and observe the frequencies of each face. You then calculate an expected frequency for each face (10 rolls per face if fair). After comparing observed vs. expected, you calculate a Chi-Squared statistic.
Let's say your calculated Chi-Squared Statistic (χ²) is 7.5.
Since there are 6 categories (faces of the die), your Degrees of Freedom (df) would be 6 – 1 = 5.
Using the calculator with χ² = 7.5 and df = 5, you would find a p-value of approximately 0.185. Since 0.185 is greater than the common significance level of 0.05, you would fail to reject the null hypothesis. This suggests there isn't enough evidence to conclude that the die is unfair based on your observations.