Borrowing the bell curve
A binomial count with a large n is tedious to compute exactly and, by the central limit theorem, shaped increasingly like a normal curve with mean np and standard deviation √(np(1 − p)). The approximation reads areas off that curve instead of summing binomial terms. One adjustment makes it honest: the continuity correction. A discrete count k occupies the whole bar from k − 0.5 to k + 0.5 on the continuous axis, so P(X ≤ k) should be priced up to k + 0.5, not k.
One hundred fair coin flips, P(at most 45 heads). Exact binomial: 0.1841. Normal with the correction, using z = (45.5 − 50)/5 = −0.90: 0.1841, off by less than 0.0001. Without the correction, z = −1.00: 0.1587, off by 0.0254, a 14% relative error. The single point version shows the same gap: P(exactly 50) is 0.0796 exactly and 0.0797 approximated with the bar method. Half a unit of x is a lot of probability when the standard deviation is only 5.
When the approximation earns its keep
The working rule: trust it when np and n(1 − p) are both at least 10, so the distribution has room to spread on both sides of its mean. Near the edges (small p, counts near 0 or n) the binomial is skewed, the symmetric bell fits badly, and the Poisson approximation or an exact calculation is the better tool. The calculator above always shows the exact value next to both approximations, so the error is never a matter of faith: read it directly.
Applying the correction in the wrong direction. Going from a cumulative question to the curve, the boundary moves outward by 0.5 (at most k means up to k + 0.5; at least k means from k − 0.5). Moving it inward prices half the target bar for the other side.
Frequently asked questions
When can a binomial be approximated by a normal?
When np and n(1 - p) are both at least 10 (some texts say 5). The distribution then has a symmetric enough bell shape around its mean.
What does the continuity correction do?
It treats each integer count as a bar spanning plus and minus 0.5, matching how a histogram of discrete counts sits on a continuous axis. It typically cuts the approximation error by an order of magnitude.
Why not always compute the exact binomial?
On this site, do: the calculator shows it. The approximation survives for hand work, exam conditions, and understanding why binomial results look normal-shaped at all.