Chi Distribution Calculator

Chi-Squared Distribution P-Value Calculator

Use this calculator to determine the p-value for a given Chi-Squared statistic and degrees of freedom.

Result:

// Lanczos approximation for gamma function function gamma(z) { var p = [ 0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 ]; var g = 7; if (z < 0.5) { return Math.PI / (Math.sin(Math.PI * z) * gamma(1 – z)); } z -= 1; var a = p[0]; var t = z + g + 0.5; for (var i = 1; i < p.length; i++) { a += p[i] / (z + i); } return Math.sqrt(2 * Math.PI) * Math.pow(t, z + 0.5) * Math.exp(-t) * a; } // Log Gamma function (for efficiency) function logGamma(z) { return Math.log(gamma(z)); } // Regularized lower incomplete gamma function P(s, x) function gammap(s, x) { if (x < 0 || s <= 0) { return 0; } var gln = logGamma(s); if (x < s + 1) { return gser(s, x, gln); } else { return 1 – gcf(s, x, gln); } } // Regularized upper incomplete gamma function Q(s, x) function gammac(s, x) { if (x < 0 || s <= 0) { return 0; } var gln = logGamma(s); if (x < s + 1) { return 1 – gser(s, x, gln); } else { return gcf(s, x, gln); } } // Series representation for P(s, x) function gser(s, x, gln) { var itmax = 100; var eps = 3.0e-7; var sum = 1.0; var del = 1.0; var ap = s; for (var n = 0; n < itmax; n++) { ap++; del *= x / ap; sum += del; if (Math.abs(del) < Math.abs(sum) * eps) { return sum * Math.exp(-x + s * Math.log(x) – gln); } } return 0; // Should not happen with reasonable inputs } // Continued fraction representation for Q(s, x) function gcf(s, x, gln) { var itmax = 100; var eps = 3.0e-7; var b0 = x + 1 – s; var b1 = 1 / b0; var b2 = 1; var b3 = 1; var c = 1 / b1; var d = 0; var h = c; for (var i = 1; i <= itmax; i++) { var an = i * (s – i); var bn = x + (2 * i) – s + 1; d = bn + an * d; if (Math.abs(d) < eps) d = eps; c = bn + an / c; if (Math.abs(c) < eps) c = eps; d = 1 / d; var del = d * c; h *= del; if (Math.abs(del – 1.0) < eps) { return h * Math.exp(-x + s * Math.log(x) – gln); } } return 0; // Should not happen with reasonable inputs } function calculateChiSquaredPValue() { var chiSquaredValueInput = document.getElementById("chiSquaredValue").value; var degreesOfFreedomInput = document.getElementById("degreesOfFreedom").value; var resultDiv = document.getElementById("pValueResult"); var chi2 = parseFloat(chiSquaredValueInput); var df = parseInt(degreesOfFreedomInput); if (isNaN(chi2) || isNaN(df) || chi2 < 0 || df <= 0 || !Number.isInteger(df)) { resultDiv.innerHTML = "Please enter valid positive numbers for Chi-Squared Statistic (non-negative) and Degrees of Freedom (positive integer)."; return; } var s = df / 2; var x = chi2 / 2; var pValue = gammac(s, x); resultDiv.innerHTML = "The P-value is: " + pValue.toFixed(6) + ""; if (pValue < 0.05) { resultDiv.innerHTML += "(Typically considered statistically significant at the 0.05 level)"; } else { resultDiv.innerHTML += "(Typically not considered statistically significant at the 0.05 level)"; } }

Understanding the Chi-Squared Distribution

The Chi-Squared (χ²) distribution is a continuous probability distribution that arises in statistics. It is primarily used in hypothesis testing, particularly for tests involving categorical data or the variance of a normally distributed population. The shape of the Chi-Squared distribution is determined by a single parameter: its degrees of freedom (df).

Key Concepts:

• Chi-Squared Statistic (χ²): This is the calculated test statistic from your data. It quantifies the difference between observed and expected frequencies (for goodness-of-fit or independence tests) or relates to the sample variance (for variance tests). A larger χ² value generally indicates a greater discrepancy from the null hypothesis.
• Degrees of Freedom (df): This parameter dictates the specific shape of the Chi-Squared distribution. It is related to the number of independent pieces of information used to calculate the statistic. For example, in a goodness-of-fit test, df = (number of categories – 1). In a test of independence for a contingency table, df = (rows – 1) * (columns – 1).
• P-value: The p-value is the probability of observing a chi-squared statistic as extreme as, or more extreme than, the one calculated from your sample data, assuming the null hypothesis is true. In the context of the Chi-Squared distribution, it represents the area under the curve to the right of your calculated χ² statistic.

Interpreting the P-value:

The p-value helps you decide whether to reject or fail to reject the null hypothesis. A common significance level (alpha, α) is 0.05.

• If P-value < α (e.g., < 0.05), you typically reject the null hypothesis. This suggests that the observed data is unlikely to have occurred by chance if the null hypothesis were true, indicating a statistically significant result.
• If P-value ≥ α (e.g., ≥ 0.05), you typically fail to reject the null hypothesis. This suggests that the observed data is reasonably likely to have occurred by chance if the null hypothesis were true, meaning there isn't enough evidence to conclude a statistically significant effect.

Common Applications:

• Chi-Squared Goodness-of-Fit Test: Used to determine if a sample distribution matches a hypothesized population distribution.
• Chi-Squared Test of Independence: Used to determine if there is a significant association between two categorical variables in a contingency table.
• Chi-Squared Test for Variance: Used to test hypotheses about the population variance of a normally distributed variable.

Example Calculation:

Imagine you are conducting a goodness-of-fit test to see if the observed frequencies of customer preferences for three product colors (Red, Green, Blue) match an expected distribution (e.g., equal preference). After collecting data and performing the necessary calculations, you arrive at a Chi-Squared statistic (χ²) of 7.81. Since there are 3 categories, the degrees of freedom (df) would be (3 – 1) = 2.

Using the calculator above with:

• Chi-Squared Statistic (χ²): 7.81
• Degrees of Freedom (df): 2

The calculated P-value would be approximately 0.02016. Since this p-value (0.02016) is less than the common significance level of 0.05, you would reject the null hypothesis, concluding that the observed customer preferences are significantly different from the expected equal distribution.