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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
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    $\begingroup$ 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 ~$10^{-11}$? $\endgroup$
    – Glen_b
    Nov 29, 2016 at 1:58
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2 Answers 2

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I know of no reason to use the normal approximation to the binomial distribution in practice. There are a variety of exact algorithms that are more than good enough for general use, and these are what you get when you use the binomial RNGs from R, SciPy, etc. The only good reason I can think of to discuss the method in a statistics class is that you can use it to illustrate the central limit theorem.

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    $\begingroup$ I can perform Normal calculations quickly in my head (either from memory or with simple approximations to the integrals). I cannot do that for Binomial distributions. That means I have a better working knowledge of the Normal approximation than I do of the Binomial distributions. I leave it to individual readers to decide whether such a skill might have any value. $\endgroup$
    – whuber
    Nov 27, 2016 at 20:15
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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.

Regarding your question about calculating binomial probabilities on the computer, the computer can calculate these probabilities quickly and therefore you really don't need a normal approximation.

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    $\begingroup$ Can you help explain the advantages of using the approximation? Why would I want to use it? $\endgroup$
    – Hatshepsut
    Nov 26, 2016 at 0:07
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    $\begingroup$ @Hatshepsut: perhaps either you have a set of tables but no computer, or you are looking for asymptotic results $\endgroup$
    – Henry
    Nov 26, 2016 at 2:03

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