McNemar's Test Calculator

Test whether the marginal proportions of paired nominal data changed. Enter the two discordant counts (b and c) from your 2×2 table to get the chi-square statistic and p-value.

Quick Facts

Method
Chi-square on discordant pairs: χ² = (b−c)²/(b+c), df = 1
Concordant cells a and d are ignored; only the pairs that switched matter.

Your Results

Calculated
Chi-square statistic
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χ² with df = 1
p-value
-
Two-sided
Discordant total (b + c)
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Informative pairs
Significance at α = 0.05
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Interpretation

Ready

Enter your discordant counts b and c, then run the test.

What McNemar's test measures

McNemar's test is a statistical test for paired nominal data — the same subjects measured twice on a yes/no (binary) outcome. It answers one question: did the proportion of "yes" responses change between the two measurements? Typical uses are before-and-after studies, two diagnostic tests run on the same patients, or matched case-control pairs. Because the two observations come from the same subject, an ordinary chi-square test of independence is invalid; McNemar's test is the correct paired alternative.

The 2×2 table

Arrange the paired results in a 2×2 table. The rows are the first measurement, the columns are the second:

  • a — positive on both measurements (concordant)
  • b — positive then negative (discordant)
  • c — negative then positive (discordant)
  • d — negative on both measurements (concordant)

The two concordant cells, a and d, contain subjects who did not change, so they carry no information about a shift in the marginal proportion. McNemar's test discards them and works only with the two discordant cells, b and c. That is why this calculator asks only for b and c.

The formula

The uncorrected McNemar statistic is:

χ² = (b − c)² / (b + c)

Under the null hypothesis that b and c are equally likely, this statistic follows a chi-square distribution with 1 degree of freedom. The p-value is the upper-tail area of that distribution beyond the observed χ². A large gap between b and c produces a large statistic and a small p-value, signalling a real change in the marginal proportion.

For small discordant totals (a common threshold is b + c < 25), the chi-square approximation is inaccurate. Edwards' continuity correction, χ² = (|b − c| − 1)² / (b + c), pulls the statistic toward a more conservative value. For very small samples, the exact binomial test — treating b as a draw from Binomial(b + c, 0.5) — is the gold standard.

Worked example

Suppose 52 patients switched results between two diagnostic tests: b = 12 went from positive to negative and c = 40 went from negative to positive. Then χ² = (12 − 40)² / (12 + 40) = (−28)² / 52 = 784 / 52 ≈ 15.08. With df = 1 the two-sided p-value is about 0.0001, so the two tests disagree far more than chance would allow.

Frequently Asked Questions

Why does the test ignore the a and d cells?
Subjects in cells a and d gave the same answer on both measurements, so they provide no evidence about whether the overall proportion changed. Only the discordant pairs — b and c — record a switch, and McNemar's test compares those two counts against each other.
When should I use the continuity correction or an exact test?
Use the uncorrected statistic when the discordant total b + c is reasonably large (roughly 25 or more). When b + c is small, the chi-square approximation overstates significance; switch to Edwards' continuity correction, (|b−c|−1)²/(b+c), or use the exact binomial test with n = b + c and p = 0.5, which is always valid.
What is the difference between McNemar's test and a chi-square test of independence?
A chi-square test of independence compares two independent groups. McNemar's test is for paired data — the same subjects measured twice, or matched pairs. Applying an independence test to paired data ignores the pairing and gives the wrong answer, usually understating significance.

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