# Why a sample of skewed normal distribution is not normal?

I was under the impression that if I randomly sample from a skewed normal distribution, the distribution of my sample would be normal based on central limit theorem, but the graph clearly shows that it's not the case.

Can someone help me understand where I'm wrong in my assumptions?

import random
import numpy as np
from scipy.stats import skewnorm
import matplotlib.pyplot as plt

skewed = skewnorm(4)
sample = skewed.rvs(100000)

sampled = []
[sampled.append(random.sample(set(sample), 1)) for _ in range(100)]

fig, ax = plt.subplots(1, 1)

ax.hist(sampled)
plt.show() • Please see this question of mine from last year: stats.stackexchange.com/q/473455/247274. You’re making the same mistake about the CLT that most everyone makes for a while.
– Dave
Feb 15, 2021 at 22:40
• Instead of your complicated and incorrect way involving list comprehension to obtain sampled (incorrect because you may e.g. obtain as a "sample" only the first element 100 times), you may simply write sampled = random.sample(sample, 100) Feb 16, 2021 at 0:32
• Maybe the terminology here is confusing? I can understand that some people would expect a skew normal distribution to be normal, just as a long beard is also a beard, a heteroskedastic regression is still a regression, ... Mar 5, 2022 at 15:20

I was under the impression that if I randomly sample from a skewed normal distribution, the distribution of my sample would be normal based on central limit theorem

You are incorrect in your understanding of the central limit theorem (it is a pretty common misconception, as Dave pointed out). The CLT states that under certain conditions the limiting distribution of the sample mean is normal, not that data sampled from a non-normal population will have a normal distribution.

You can see this in action if you run a different simulation, where you simulate the sample means:

import random
import numpy as np
from scipy.stats import skewnorm, norm
import seaborn as sns
import matplotlib.pyplot as plt

skewed = skewnorm(4)

simulated_means = []

for i in range(10000):
data = skewed.rvs(100)
simulated_means.append(np.mean(data))

sns.distplot(simulated_means, fit=norm)
plt.show() In this particular case, we see that the sample distribution of the mean is more or less normal when n=100; the normal fit is the black line. This will not always be true, since the CLT is an asymptotic result, but simulations like this help us understand how the sampling distribution from a particular population with a particular sample size might look.

Consider this:

If you take as a sample the whole population (i.e. the very very large “sample”), then by some miracle your skewed population suddenly will be changed to a normal one?

• I don't think the first analogy is useful: by "normal" the OP is referring to the Normal (Gaussian) distribution, not "normal" in the sense of what one would normally expect. You could alter the answer to talk about heights (which do follow a roughly Gaussian distribution) and it would be more useful.
– JDL
Feb 16, 2021 at 9:13
• @JDL, analogy is just an analogy, isn't it? Feb 16, 2021 at 9:41
• it isn't actually analogous though (your second argument is fine, it's just the first one I find unhelpful)
– JDL
Feb 16, 2021 at 9:46