Two categorical variables. Are they linked, or do they vary independently? Compare observed counts in each cell to the count you'd expect if rows and columns were unrelated: E = (row total × col total) / grand total.
χ² statistic—
df = (r−1)(c−1)—
p-value—
Decision (α=0.05)—
OBSERVED & EXPECTED COUNTS
Reading the cells: when association is 0, observed ≈ expected, all contributions near zero, χ² is small. As association grows, certain cells deviate more (big orange numbers) and χ² climbs.