If you are doing a p-test, then it matters a lot.Adeimantus said:What is the preferred method of measuring how accurate the normal approximation to the binomial distribution is? I know that the rule of thumb is that the expected number of successes and failures should both be >5 for the approximation to be adequate. But what is a useful definition of "adequate"? The approximation is good for values near the mean, and terrible for values in the extreme tails. How much do those extreme values matter?
Thank you.
It seems to me that it would be better to use the binomial distribution for the p-test in that case.Adeimantus said:That is a good point. So in that case, how would you decide when the approximation is good enough?
If the decision about significance of the result is the same for the approximation and the actual distribution, then the approximation is good enough.Adeimantus said:Okay, that makes sense. In cases where you would want to use the approximation, how do you quantify how good the approximation is?
Good point. This implies that it is the accuracy of the cumulative distribution function at important and frequently used confidence values that really matters.tnich said:If the decision about significance of the result is the same for the approximation and the actual distribution, then the approximation is good enough.
tnich said:If the decision about significance of the result is the same for the approximation and the actual distribution, then the approximation is good enough.
Right. My point is, why would you want to use an approximation, when you can easily calculate or look up exact values for the binomial distribution? Exact values are unimpeachable.Adeimantus said:Okay, but it will never be exactly the same, right?
I think a normal distribution table would have been quite useful for computing approximate binomial probabilities by hand. Of course, DeMoivre would have realized that at extreme values where the approximation was not good, doing the computation using the actual binomial distribution would be fairly simple.Adeimantus said:Is that what Laplace and DeMoivre were using it for, or were they more interested in the values of the density/mass function?
Adeimantus said:I agree that since tables of binomial probabilities are readily available, it makes sense to use them. Especially since they are now accessible with the call of a function in R, for instance. But then I'm left wondering, why does anyone care about the limit theorem? Does it have a use? Perhaps it has more theoretical value than anything else.
StoneTemplePython said:As is often the case, it depends. It's worth remarking that the sum of two independent binomial distributions isn't necessarily binomial, but the sum of 2 independent Gaussians is Gaussian. If you are interesting in general for bounding the error of a Gaussian approximation, look into Stein's method. Other reasons: in math, people tend to like the exponential function a lot more than binomial coefficients, and the fact that the Gaussian is entirely characterized by its first two moments (and for a continuous distribution on ##(-\infty, \infty)## with a given finite variance, it has maximum entropy) and so on.
The Gaussian is one of the most important distributions in probability. In general the various central limit theorem proofs are not easy... it just so happens that the theorem for binomial -> normal, is an easy one, so it has pedagogic value.
The normal approximation to binomial distribution is a statistical method used to estimate the probability of a certain number of successes in a series of independent trials. It assumes that the data follows a normal distribution, which is a bell-shaped curve, and allows for easier calculations and predictions.
The normal approximation to binomial distribution is appropriate when the number of trials is large (typically more than 30) and the probability of success is not too extreme (between 0.1 and 0.9). This is because the normal distribution becomes a better fit with larger sample sizes.
The accuracy of the normal approximation to binomial distribution depends on the sample size and the probability of success. Generally, the larger the sample size and the closer the probability of success is to 0.5, the more accurate the approximation will be. It is important to note that the approximation is not exact and there will always be some level of error.
One limitation is that the normal approximation may not be accurate when the probability of success is very small or very large. In these cases, the binomial distribution should be used instead. Additionally, the normal approximation assumes that the trials are independent, which may not always be the case in real-world situations.
To determine if the normal approximation to binomial distribution is appropriate, you can check the sample size and the probability of success. If the sample size is large (typically more than 30) and the probability of success is not too extreme (between 0.1 and 0.9), then the approximation may be appropriate. It is also helpful to plot the data and see if it follows a normal distribution or if there are any outliers or non-normal patterns present.