Yates Correction

Yates correction, also called Yates continuity correction, is an adjustment applied to the chi square test for independence when analyzing a 2x2 contingency table. The correction reduces the absolute difference between each observed and expected frequency by 0.5 before squaring, compensating for the fact that the discrete chi square statistic is being compared to a continuous chi square distribution.

The chi square test is one of the most frequently used statistical procedures in health sciences, social sciences, and epidemiology. When sample sizes are small or expected cell frequencies are low, the standard chi square test's p value is only an approximation, and it can overstate the significance of an association. Yates correction was designed to address this by making the test more conservative, but it usually overcorrects: its p values tend to be too large, so its actual Type I error rate falls below the nominal level and real associations are missed more often. Understanding when to apply the correction and when it may be overly conservative helps researchers choose the most appropriate analytical approach for their data.

The standard chi square statistic for a 2x2 table sums the squared differences between observed (O) and expected (E) cell frequencies divided by the expected frequencies. Yates correction modifies this by subtracting 0.5 from each absolute difference before squaring.

The corrected formula is: chi square = sum of ((|O - E| - 0.5)^2 / E) across all four cells.

This reduces the computed chi square value and produces a larger p value compared to the uncorrected test. Some textbooks advise applying the correction when any expected cell frequency falls below 10, others whenever a 2x2 table is analyzed, and some software applies it by default (R's chisq.test does so for every 2x2 table). Because the correction overcorrects, many statisticians now advise against it. In a large computational comparison, Campbell (2007) found that the N − 1 chi square test (the Pearson statistic multiplied by (N − 1)/N, where N is the total sample size) kept close to the nominal Type I error rate whenever every expected count was at least 1, and recommended Fisher's exact test only when an expected count fell below 1. A traditional and still common rule prefers Fisher's exact test over either chi square version whenever an expected frequency falls below 5.

Traditional textbook advice is to apply Yates correction when performing a chi square test on a 2x2 contingency table with small to moderate sample sizes, particularly when one or more expected cell frequencies fall between 5 and 10. The same tradition uses Fisher's exact test instead when any expected frequency is below 5. Because the correction overcorrects, many statisticians now skip it and use Fisher's exact test or the N − 1 chi square test for small tables. In large samples where all expected frequencies comfortably exceed 10, the uncorrected chi square is adequate and the correction is unnecessary. Whichever test is used, name it in the methods and choose it before seeing the results, because software often prints the corrected and uncorrected p values side by side.

Applying Yates correction to tables larger than 2x2 is inappropriate because the correction was designed specifically for the 2x2 case. Using the correction with large samples where expected frequencies are well above 10 makes the test unnecessarily conservative and reduces statistical power. Failing to recognize when Fisher's exact test is more appropriate than either version of the chi square test can lead to unreliable conclusions with very small samples. Another mistake is treating the corrected p value as automatically the more trustworthy one: the correction tends to overcorrect, and in the example below the corrected p value is larger even than Fisher's exact p value. Choosing between the corrected and uncorrected results after seeing which one is significant is a form of selective reporting.

An epidemiologist analyzes a 2x2 table examining the association between a dietary supplement and the occurrence of migraine episodes in 40 participants: 4 of the 16 who took the supplement (25%) and 14 of the 24 who did not (58%) had a migraine episode. The smallest expected cell frequency is 7.2 (16 × 18 ÷ 40, for supplement users with a migraine episode). The uncorrected chi square is 4.31 (p = 0.038), while the Yates corrected chi square is 3.07 (p = 0.080). For comparison, the N − 1 chi square gives p = 0.040 and Fisher's exact test gives p = 0.054 (two sided, as computed by R and SciPy), so the correction pushes the p value above even the exact test's. Because the methods fall on both sides of 0.05, the researcher reports the test named in the analysis plan alongside the proportions, describes the evidence as borderline rather than as a clear yes or no, and recommends a larger confirmatory study.