Enter your observed and expected frequencies below to calculate the Chi-Square statistic, degrees of freedom, and p-value.
Category
Observed (O)
Expected (E)
Please enter valid numeric values for both Observed and Expected in at least two categories. Expected values must be greater than zero.
Chi-Square Statistic (χ²):
Degrees of Freedom (df):
P-Value:
Significance (α = 0.05):
What is the Chi-Square Test?
The Chi-Square (χ²) Goodness of Fit test is a statistical hypothesis test used to determine whether a variable's observed frequency distribution differs significantly from an expected theoretical distribution. It is widely used in genetics, marketing research, and social sciences to validate if observed data "fits" a specific model.
The Chi-Square Formula
The mathematical representation of the Chi-Square statistic is:
χ² = Σ [ (Oᵢ – Eᵢ)² / Eᵢ ]
Where:
Oᵢ = Observed frequency for category i
Eᵢ = Expected frequency for category i
Σ = The sum across all categories
How to Interpret the Result
To understand your results, look at the p-value:
p ≤ 0.05: Statistically significant. There is a significant difference between your observed data and the expected distribution. You reject the null hypothesis.
p > 0.05: Not statistically significant. The observed data fits the expected distribution well. You fail to reject the null hypothesis.
A Practical Example
Imagine you roll a six-sided die 60 times. Theoretically, you expect each number (1 through 6) to appear 10 times. However, you observe the following:
Number 1: 12 times
Number 2: 8 times
Number 3: 11 times
Number 4: 9 times
Number 5: 10 times
Number 6: 10 times
By entering these Observed values (12, 8, 11, 9, 10, 10) and Expected values (10 for each) into the calculator, you can determine if the die is biased or if the variations are just due to random chance.
function calculateChiSquare() {
var chiSq = 0;
var categoriesCount = 0;
var errorDiv = document.getElementById("errorMessage");
var resultDiv = document.getElementById("resultBox");
errorDiv.style.display = "none";
resultDiv.style.display = "none";
for (var i = 1; i <= 5; i++) {
var obsVal = document.getElementById("obs" + i).value;
var expVal = document.getElementById("exp" + i).value;
if (obsVal !== "" && expVal !== "") {
var o = parseFloat(obsVal);
var e = parseFloat(expVal);
if (isNaN(o) || isNaN(e) || e <= 0) {
errorDiv.innerText = "Error: Expected values must be positive numbers.";
errorDiv.style.display = "block";
return;
}
var term = Math.pow(o – e, 2) / e;
chiSq += term;
categoriesCount++;
}
}
if (categoriesCount < 2) {
errorDiv.innerText = "Please provide data for at least two categories.";
errorDiv.style.display = "block";
return;
}
var df = categoriesCount – 1;
var p = getChiSquareP(chiSq, df);
document.getElementById("chiValue").innerText = chiSq.toFixed(4);
document.getElementById("dfValue").innerText = df;
document.getElementById("pValue").innerText = p.toFixed(5);
var sigText = (p <= 0.05) ? "Significant (Reject Null Hypothesis)" : "Not Significant (Fail to Reject Null Hypothesis)";
document.getElementById("sigValue").innerText = sigText;
document.getElementById("sigValue").style.color = (p <= 0.05) ? "#e74c3c" : "#27ae60";
resultDiv.style.display = "block";
}
// Function to approximate P-value from Chi-Square and df
// Uses the Wilson-Hilferty transformation for an accurate approximation
function getChiSquareP(chi, df) {
if (chi < 0 || df 0) {
var a = 1 / (9 * df);
var z = (Math.pow(chi / df, 1/3) – (1 – a)) / Math.sqrt(a);
return 1 – normalCDF(z);
}
return 1.0;
}
function normalCDF(z) {
var t = 1 / (1 + 0.2316419 * Math.abs(z));
var d = 0.3989423 * Math.exp(-z * z / 2);
var p = d * t * (0.3193815 + t * (-0.3565638 + t * (1.781478 + t * (-1.821256 + t * 1.330274))));
if (z > 0) return 1 – p;
return p;
}