What is normal approximation? Show
 Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher) And how does it relate to statistics? Let’s find out, shall we… If certain conditions are met, then a continuous distribution can be used to approximate a discrete distribution? The Normal Distribution (continuous) is an excellent approximation for such discrete distributions as the Binomial and Poisson Distributions, and even the Hypergeometric Distribution. How? Thanks to the Central Limit Theorem and the Law of Large Numbers. Central Limit TheoremFirst, the Central Limit Theorem (CLT) states that for non-normal distribution, as the sample size increases, the distribution of the sample means becomes approximately Normal. So, as long as the sample size is large enough, the distribution looks normally distributed. Law Of Large NumbersSecondly, the Law of Large Numbers helps us to explain the long-run behavior. It states that if we observe more and more repetitions of any chance experiment, the proportion of times that a specific outcome occurs will approach a single value as noted by Lumen Learning. Coin Toss ExampleFor example, if we flip a coin repeatedly for more than 30 times, the probability of landing on heads becomes approximately 0.5. This is why we say you have a 50-50 shot of getting heads when you flip a coin because, over the long run, the chance or probability of getting heads occurs half the time. So, with these two essential theorems, we can say that with a large sample size of repeated trials, the closer a distribution will become normally distributed. Approximating Normal Density Curve But why do we care? Because for certain discrete distributions, namely the Binomial and Poisson distributions, summing large values can be tedious or not practical. Thankfully, the Normal Distribution allows us to approximate the probability of random variables that would otherwise be too difficult to calculate. Now, before we jump into the Normal Approximation, let’s quickly review and highlight the critical aspects of the Binomial and Poisson Distributions. Binomial DistributionA binomial random variable represents the number of successes in a fixed number of successive identical, independent trials. Each trial has the possibility of either two outcomes:
And the probability of the two outcomes remains constant for every attempt. And as the sample size grows large, the more symmetric, or bell shape, the binomial distribution becomes. Poisson DistributionNow the Poisson differs from the Binomial distribution as it is used for events that could occur a large number of times because it helps us find the probability of a certain number of events happening in a period of time or space. And once again, the Poisson distribution becomes more symmetric as the mean grows large. FormulasHow do we use the Normal Distribution to approximate non-normal, discrete distributions? By using the following formulas! Normal Approximation Formula Also, I should point out that because we are “approximating” a normal curve, we choose our x-value a little below or a little above our given value. For example, if we look at approximating the Binomial or Poisson distributions, we would say Hypergeometric Vs Binomial Vs Poisson Vs Normal Approximation Additionally, the Normal distribution can provide a practical approximation for the Hypergeometric probabilities too! Here’s a quick look at the conditions that must be met for these discrete distributions to be approximately normal. Conditions For Normal Approximation Example of BinomialOkay, so now that we know the conditions and how to standardize our discrete distributions, let’s look at a few examples. Suppose a manufacturing company specializing in semiconductor chips produces 50 defective chips out of 1,000. If 100 chips are sampled randomly, without replacement, approximate the probability that at least 1 of the chips is flawed in the sample. First, we notice that this is a binomial distribution, and we are told that. Binomial Challenging CDF – Example This would not be a very pleasant calculation to conduct. Thankfully, we are told to approximate, and that’s exactly what we’re going to do because our sample size is sufficiently large! So, using the Normal approximation, we get Normal Approximation To Binomial – Example Meaning, there is a probability of 0.9805 that at least one chip is defective in the sample. Example of PoissonNow let’s suppose the manufacturing company specializing in semiconductor chips follows a Poisson distribution with a mean production of 10,000 chips per day. Approximate the expected number of days in a year that the company produces more than 10,200 chips in a day. Poisson Approximation To Normal – Example This means that if the probability of producing 10,200 chips is 0.023, we would expect this to happen approximately 365(0.023) = 8.395 days per year. So, by the power of the Central Limit Theorem and the Law of Large Numbers, we can approximate non-normal distributions using the Standard Normal distribution where the mean becomes zero with a standard deviation of one! This video will look at countless examples of using the Normal distribution and use it as an approximation to the Binomial distribution and the Poisson distribution. Let’s jump on in! Normal Approximation – Lesson & Examples (Video)47 min
Get access to all the courses and over 450 HD videos with your subscription Monthly and Yearly Plans Available Get My Subscription Now Still wondering if CalcWorkshop is right for you? When should we use the normal distribution to approximate the binomial or Poisson distributions?The normal distribution can be used as an approximation to the binomial distribution, under certain circumstances, namely: If X ~ B(n, p) and if n is large and/or p is close to ½, then X is approximately N(np, npq)
When can you use a normal approximation?When n * p and n * q are greater than 5, you can use the normal approximation to the binomial to solve a problem.
When can Poisson approximation be used?When the value of n in a binomial distribution is large and the value of p is very small, the binomial distribution can be approximated by a Poisson distribution. If n > 20 and np < 5 OR nq < 5 then the Poisson is a good approximation.
When can we use a normal distribution to approximate data that is discrete?The short and strict answer to your question is no. By definition, all normally distributed random variables are continuous random variables. A random variable cannot be continuous and discrete at the same time, so the definition excludes the existence of normally distributed discrete random variables.
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