Welcome back to the AI Bayeslab Statistics series. Today, we will discuss Yates' continuity correction and Finite Population Correction (FPC).

Are they identical?

No, Yates' continuity correction is not the same as the finite population correction factor\left(\frac{N-n}{N-1}\right). They serve entirely different purposes in statistics.

Let's illustrate more details for it:

1. What is the Yates' Continuity Correction

Purpose: Adjusts for the discontinuity when approximating a discrete distribution (e.g., binomial) with a continuous distribution (e.g., normal).

Applied to:

• Chi-square tests (e.g., r\chi^2 test for independence in 2×2 contingency tables).

• Z-tests for proportions (when testing or comparing two proportions).

  
  

Formula for Z-test (single proportion):

Z = \frac{|p' - p_0| - \frac{0.5}{n}}{\sqrt{\frac{p_0(1-p_0)}{n}}}

For two proportions:

Z = \frac{|p'_1 - p'_2| - \frac{0.5}{n_1} - \frac{0.5}{n_2}}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}

Effect: Reduces the absolute difference by \frac{0.5}{n} to account for discreteness.

Key Point:

• Used only for small samples or when expected counts are borderline (e.g., np < 5 ).

2. What is the finite population correction FPC?

Purpose: Adjusts the standard error when sampling without replacement from a finite population (where n > 5% of N ).

Formula:

\text{FPC} = \sqrt{\frac{N-n}{N-1}}

Applied to:

• Standard error of the mean:\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \cdot \text{FPC} .

• Standard error of proportions: \sqrt{\frac{p(1-p)}{n}} \cdot \text{FPC} .

Key Point:

• Only relevant for finite populations (e.g., surveying 100 out of 500 customers).

3. Why They’re Confused?

Both involve adjustments, but:

• Yates’ correction addresses bias resulting from the discrete-to-continuous approximation.

• FPC fixes bias due to non-independence in finite samples.

1) Example Comparison

2) Conclusion

Yates’ correction ( subtracting \frac{0.5}{n} ) is unrelated to the finite population correction (multiplying by\sqrt{\frac{N-n}{N-1}} ) .

• Use Yates’ for small-sample proportion tests

• FPC for finite populations.

Stay tuned, subscribe to Bayeslab, and let everyone master the wisdom of statistics at a low cost with the AI Agent Online tool.