continuity correction I have been told by some PhD's that the correction for continuity used in statistics is wrong and has been know to be wrong for at least the past few decades. Here I mean if we have a discrete random variable $X$ with p.m.f $\;p(x)$ and approximate it by a continuous random variable with p.d.f $\;f(x)$ then $$P(i\leq X \leq j)\approx\int_{i-\frac{1}{2}}^{j+\frac{1}{2}}f(x)dx$$ And this becomes precise in the limit. Can someone explain to me why it is wrong?
 A: The continuity correction works well for approximation of binomial distributions by normal distributions.  That is the purpose for which Abraham de Moivre introduced the normal distribution in the 18th century in his book The Doctrine of Chances (and in an earlier paper, which (I think?) differed from the book in that in included numerical bounds on the normalizing constant but did not state that the constant is $1/\sqrt{2\pi  {}}$).
Suppose you throw a die $n=14$ times.  Let $X$ be the number of times you get a $1$ or a $2$.  The probability of success on each trial is $p=1/3$.  The expected number of successes is $14/3=4.6666\ldots$.  What is the probability that $X\le 6$?  It is the same as the probability that $X<6$.  The continuity correction finds $\Pr(Y\le 6.5)$ where $Y$ is a normally distributed random variable with the same mean and the same variance.  The mean was noted above; the variance is $np(1-p)$ $=14(1/3)(2/3)$ $=3.11111\ldots$.  The exact probability is
$$
\Pr(X\le6)= \sum_{x=0}^6\binom{14}x \left(\frac13\right)^x\left(\frac23\right)^{14-x} \approx 0.8505378.
$$
The normal approximation is
$$
\Pr(Y\le6.5)=\Pr\left(\frac{Y-14/3}{\sqrt{28/9}}\le\frac{6.5-14/3}{\sqrt{28/9}}\right) = \Phi\left(\frac{6-14/3}{\sqrt{28/9}}\right) \approx 0.8506912
$$
The central limit theorem says it will work well for large $n$, and when $n$ is large, then it matters less whether one does a continuity correction or not.
How well it works for other discrete distributions I don't know, and I actually don't know a precise theorem on error bounds or rates of convergence or the like.
But here's something that may be surprising.  Try it for $n=1$ and $p=$ various values between $0$ and $1$.  You'll find that even for that extremely small sample size, the approximation always differs from the exact probability by less than $0.04$, with the largest errors near the two boundaries.
