Yates’ Continuity Correction | Binomial Tests and 2×2 Tables

Yates' continuity correction is a correction that reduces the difference by 0.5 when approximating the discrete frequencies in a 2×2 table with a continuous chi-square distribution. The purpose is to prevent the uncorrected Pearson test from becoming too significant in small samples.

Where does 0.5 come from?#

discrete events

P(X\geq8)

continuous approximation

P(X\geq8)\approx P(Y\geq7.5)

moving boundaries

8-7.5=0.5

The interval centered on the integer 8 is from 7.5 to 8.5, so approximating 8 or more with a continuous distribution moves the boundary to 7.5. This half unit is the continuity correction.

From Pearson statistics to Yates correction#

Uncorrected Pearson

\chi_P^2=\sum\frac{(O-E)^2}{E}

Yates correction

\chi_Y^2=\sum\frac{\left(|O-E|-0.5\right)^2}{E}

Simplified formula for 2×2 table#

\chi_Y^2=\frac{N\left(|AD-BC|-N/2\right)^2}{(A+B)(C+D)(A+C)(B+D)}

method	simplified molecule
Uncorrected Pearson	(AD−BC)²
Yates correction	(\|AD−BC\|−N/2)²

\chi_Y^2\leq\chi_P^2\quad\Longrightarrow\quad p_Y\geq p_P

How to Interpret “Expected Count Less Than 5”#

The target of judgment is not the observed count O, but the expected count E below H₀. However, it is not a mechanical rule that says, ``If even one cell has E<5, it will always be Yates.'' Consider the expected frequencies for the entire table, the sample design, the purpose of the inference, and the available exact methods.

Strengths and Limitations#

Strong Points	limit
Reduce errors in approximating discrete distributions with continuous distributions	Often overly conservative and reduces power
Easy to calculate and widely used	It is not an exact test itself.
Easier to reduce false positives than uncorrected Pearson	If the sample is sufficient, the need for correction is small.