> [!tldr] Continuity Correction > A **continuity correction** is a technique applied when approximating discrete distributions (e.g. binomial) with continuous ones (e.g. Gaussians). > > This is because the mass of discrete distributions at a point $x$ corresponds to a an area $(x \pm \Delta x / 2)$ when integrating the continuous distribution, where $\Delta x$ is the distance between atoms in the discrete distribution. > ![[ContinuityCorrectionExample.png#invert|center|w60]] > In this example, the binomial mass at $x=12$ corresponds to the Gaussian's mass over $(11.5, 12.5)$, and $\Delta x=1$. - Besides estimating point masses, discrete CDF can also be estimated as $\mathbb{P}[X \le k] \approx F(k+\Delta x / 2),$where $F$ is the CDF of the continuous approximation. In the case of Gaussian approximations, this is simply $\Phi(k+\Delta x / 2)$. - While omitting the correction usually still gives asymptotically correct results, the correction improves the accuracy of finite-sample approximations. > [!examples] Example: Gaussian approximation of Binomials > Suppose $X \sim \mathrm{Binom}(n, p)$ for some large $n$. Then $\mu_{X}=np$, $\mathrm{Var}(X)=np(1-p)$, so $X \overset{D}{\approx}N(np, np(1-p))$by the CLT. Therefore the naive approximation of the cdf of $X$ gives $\mathbb{P}[X \le k]\approx \Phi\left( \frac{k-np}{\sqrt{ np(1-p) }} \right),$while the continuity correction gives $\mathbb{P}[X \le k] \approx \Phi\left( \frac{k+0.5-np}{\sqrt{ np(1-p) }} \right).$ > > Its pmf estimate does not rely on the continuity correction, which is simply $\mathbb{P}[X=k] \approx \Delta x \cdot\phi(k;np, np(1-p)),$where $\phi$ is the Gaussian density.