Understanding the Chi-Square Test for Independence
The Chi-Square (χ²) test is a fundamental statistical tool used to determine if there is a significant association between two categorical variables. In a 2×2 contingency table, we compare the frequencies observed in your data against the frequencies we would expect to see if the two variables were completely independent of each other.
The Chi-Square Formula
The formula for calculating the Chi-Square statistic is:
χ² = Σ [ (O – E)² / E ]
O: Observed frequency (the numbers you entered)
E: Expected frequency (calculated based on row and column totals)
Σ: Sum of all cells in the table
Example Calculation
Imagine you are testing if a new medication (Treatment A vs. Placebo) affects recovery. Your observed data might look like this:
Recovered
Not Recovered
Treatment
45 (Cell A)
5 (Cell B)
Placebo
30 (Cell C)
20 (Cell D)
The calculator first finds the expected values for each cell. For example, the expected value for Cell A is (Row Total * Column Total) / Grand Total. Once the χ² value is calculated, it is compared against a distribution table to find the P-value.
Interpreting the Results
In most scientific research, a P-value of 0.05 or less is considered statistically significant.
P < 0.05: You reject the null hypothesis. There is a significant relationship between the variables.
P > 0.05: You fail to reject the null hypothesis. The variables are likely independent, and any observed difference is probably due to chance.
function calculateChiSquare() {
var a = parseFloat(document.getElementById('cell_a').value);
var b = parseFloat(document.getElementById('cell_b').value);
var c = parseFloat(document.getElementById('cell_c').value);
var d = parseFloat(document.getElementById('cell_d').value);
// Validation
if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(d) || a < 0 || b < 0 || c < 0 || d < 0) {
alert("Please enter valid non-negative numbers for all cells.");
return;
}
// Totals
var row1_total = a + b;
var row2_total = c + d;
var col1_total = a + c;
var col2_total = b + d;
var grand_total = a + b + c + d;
if (grand_total === 0) {
alert("Total sum cannot be zero.");
return;
}
// Expected Values
var exp_a = (row1_total * col1_total) / grand_total;
var exp_b = (row1_total * col2_total) / grand_total;
var exp_c = (row2_total * col1_total) / grand_total;
var exp_d = (row2_total * col2_total) / grand_total;
// Chi-Square Components
var chi_sq = 0;
chi_sq += Math.pow(a – exp_a, 2) / exp_a;
chi_sq += Math.pow(b – exp_b, 2) / exp_b;
chi_sq += Math.pow(c – exp_c, 2) / exp_c;
chi_sq += Math.pow(d – exp_d, 2) / exp_d;
// P-Value approximation for df=1 using the Normal Distribution
// For df=1, Chi-Square = Z^2, so we can find P from Z-score
var z = Math.sqrt(chi_sq);
var p_val = getPFromZ(z);
// Update Display
document.getElementById('results-area').style.display = 'block';
document.getElementById('chi_stat').innerText = chi_sq.toFixed(4);
document.getElementById('p_val').innerText = p_val.toFixed(6);
var interpretationDiv = document.getElementById('interpretation');
var sigRes = document.getElementById('sig_res');
if (p_val < 0.05) {
sigRes.innerText = "Significant (p 0.05)";
sigRes.style.color = "#d9534f";
interpretationDiv.innerHTML = "Result is not statistically significant. We fail to reject the null hypothesis; the variables appear to be independent.";
}
}
// Helper function to calculate P-value from Z-score (Standard Normal Cumulative Distribution)
function getPFromZ(z) {
// Standard normal error function approximation
var t = 1 / (1 + 0.2316419 * Math.abs(z));
var d = 0.3989423 * Math.exp(-z * z / 2);
var prob = d * t * (0.3193815 + t * (-0.3565638 + t * (1.781478 + t * (-1.821256 + t * 1.330274))));
if (z > 0) prob = 1 – prob;
// This gives area under one tail; for chi-sq df=1, it's equivalent to 2*(1-norm) or 1-erf
// Because we are looking at the upper tail of the chi-square distribution:
return 2 * (1 – prob);
}
// Initial calculation if needed
// calculateChiSquare();