# Why use the normal approximation to the binomial?

In school, I was taught about the normal approximation to the binomial, and it was suggested that I could use it effectively under some conditions, because it can be 'easier to calculate'.

I understand how this could be more convenient if I were using paper tables. Are there still advantages to using the normal approximation when all my computations are done using computers? Is it easier to do algebraic manipulations or calculus using the approximation? What are some examples of the advantages?

I don't know what the right benchmark test would be, but perhaps this gives an idea:

> benchmark(rbinom(1, 1, .5), replications=1000000)
test replications elapsed relative user.self sys.self user.child
1 rbinom(1, 1, 0.5)      1000000   3.593        1     3.476    0.156          0
sys.child
1         0
> benchmark(rnorm(1), replications=1000000)
test replications elapsed relative user.self sys.self user.child
1 rnorm(1)      1000000   3.724        1     3.564      0.2          0
sys.child
1         0

• One advantage of using the normal is it often gives enough information to quickly tell whether it's even worth calculating the answer more precisely. For example one can (say) compute a sample size in one's head using a normal approximation that's often within a couple of the binomial calculation; in many cases that's sufficient to figure out what needs to be known (e.g. will the current budget cover the sample size we need?). Or if you're say 7 standard errors from the hypothesized mean, can it matter that the binomial p-value is ~$10^{-12}$ rather than say ~$10^{-11}$? Commented Nov 29, 2016 at 1:58
• Commented Apr 27, 2017 at 17:26

The central limit theorem provides the reason why the normal can approximate the binomial in sufficiently large sample sizes. Sufficiently large depends on the success parameter $$p$$. When $$p=0.5$$, the binomial is symmetric and so the sample size does not need to be as much as if $$p=0.95$$, in which case the binomial could be highly skewed. Also, you get a better approximation when the continuity correction is applied.