# Central Limit Theorem and Skewed Distribution

I'm looking for a simple answer to this question relating the central limit theorem and Gaussian and skewed distributions, if one exists. I used the binomial function to generate calculations of the probabilities of possible outcomes for 10 flips of an unfair coin (p=0.3, q=0.7) and obtained a skewed distribution. I have been thinking of this as a kind of sampling distribution of proportions. If the coin was flipped 10 trillion times, with 3 trillion heads and 7 trillion tails, and 10 flip samples are plotted into the sampling distribution, yielding my skewed curve.

Now I also "learned" that the central limit theorem says that the sampling distribution of any distribution is a Gaussian curve, but I acknowledge that that my studies of this are relatively superficial. Is a skewed curve still considered a Gaussian curve? Are there other important aspects about the central limit theorem that I am clearly unaware of? I'm not looking for a comprehensive explanation necessarily, but just some guidance about misconceptions that I may have.

Thanks.

• (1) The CLT does not apply to all distributions; it is required that the variance exist. (2) Even if the IID random variables being averaged are skewed, the CLT still applies and says that the average tends to (symmetrical) normal. Dec 4, 2020 at 21:26
• The central limit theorem relates to a limit distribution and not to the sample distribution of the sum of 10 coin flips. This question has been asked before here. Dec 4, 2020 at 21:49
• Of possible interest: stats.stackexchange.com/questions/473455/…
– Dave
Dec 4, 2020 at 22:03
• stats.stackexchange.com/questions/389590/… Dec 4, 2020 at 22:03
• The first portion of my post on the CLT aims at heading off many common misconceptions.
– whuber
Dec 4, 2020 at 22:08

Suppose $$X_i \stackrel{iid}{\sim} \mathsf{Binom}(n = 10, p=0.3),$$ a skewed distribution.

plot(x, PDF, type="h", lwd=3, col="blue",
main="PDF of BINOM(10, .3)")
abline(h=0, col="green2")
abline(v=0, col="green2")


Then the average $$\hat p = \bar X_{1000}$$ of $$m = 1000$$ of these $$X_i$$s is very nearly normal, as illustrated in the following simulation in R, based on 100,000 replications of this estimate $$\hat p$$ for $$p.$$

set.seed(2020)
p.est = replicate(10^5, mean(rbinom(1000, 10, .3)))
summary(p.est)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
2.814   2.969   3.000   3.000   3.031   3.195
sd(p.est)
[1] 0.04594069

hist(p.est, prob=T, col="skyblue2",
main="Simulated Sampling Dist'n")
curve(dnorm(x, mean(p.est), sd(p.est)), add=T,
col="orange", lwd=2)


According to a Shapiro-Wilk test on the first 5000 simulated values of $$\hat p,$$ they are consistent with a random sample from a normal distribution. [The S-W test in R is restricted to a maximum of 5000 observations.]

shapiro.test(p.est[0:5000])

Shapiro-Wilk normality test

data:  p.est[0:5000]
W = 0.99971, p-value = 0.727


Nevertheless, The distribution of $$\hat p$$s based on a thousand observations is discrete (even though the histogram doesn't reveal that). Among the 100,000 realizations of $$\hat p$$ from the simulation above, there are only 355 unique values.

length(unique(p.est))
[1] 355


This is to follow up from my initial question. I calculated frequency distributions for n = 10, 25, and 100 for an even more unfair coin (p=0.1, q=0.9) and found, by eye at least, that the data became much more symmetrical at higher numbers of coin flips. My takeaway is that the binomial function calculated frequency distribution is a sampling distribution of a Bernoulli type experiment with a global population of trillions of 0's and 1's in frequency ratio p and q, selected with individual sample size n. As n increases, the distribution appears to become increasingly symmetrical bell shaped.

I don't know however if the data becomes more normal per the Shapiro-Wilk normality test mentioned in BruceET's answer. I calculated the expected frequencies on a spreadsheet, and suspect that the data points will not meet SW criteria even though the curves look bell shaped. I think this because as a check of how the SW normality test works on statskingdom, I manually entered 1024 data points for the expected sampling distribution frequencies of flipping a fair coin 10 times, and webpage result was that the binomial distribution was not Gaussian. Perhaps it would have been different with a larger set of points from a binomial distribution.