# Why is a sum of skewed left distribution normal distributed according to the central limit theorem

The central limit theorem says that a sum of i.i.d. random variables is normal distributed. Now I have a large number of random variables (100000 responses to an item of a questionaire) but the distribution is skewed left. So with so many respondants I can even ask 1Million the distribution will not change.

So, maybe I do not understand it right, but should not (according to the central limit theorem) this distribution be normal?

$$S_n = X_1 + ... + X_n$$ and

$$Z_n = \frac{S_n - \mu n}{\sigma \sqrt(n)}$$

Does that mean that every distribution that is changable to a normal distribution via linear transformation is normal distributued? (this should be almost everything).

Does that mean that every distribution [...] is normal distributued?

Not not the distribution is normal, but the sum or mean of a lot of values drawn from that distribution tends to be drawn form a normal distribution.

A distribution by definition does not change whether you draw one sample, a million samples or $10^{10}$ samples. But the sum or the mean of the samples do not stem from the same distribution aus the samples.

Maybe a little example in R can help clear this up. A beta distribution with shape parameters 1 and 50 is far from a normal distribution. You can plot it using R and the command:

curve(expr = dbeta(x, 1, 50))


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Now let'S take 10000 samples from that distribution, each size of 100, and compute their sums. Now lets draw a histogram of these sums:

hist(replicate(10000, sum(rbeta(100, 1, 50))), breaks = 50)


you can easily see, how this is nearing normal distribution. Not the beta distribution becomes normal, but the distribution of the sums.

The skewness decreases for the sum $$Z_n = \frac{S_n - \mu n}{\sigma \sqrt(n)}$$ when $$n$$ is increasing.

You have for first three central moments that

$$\begin{array}{} \mu_1(Z_n) &=& 0 \\ \mu_2(Z_n) &=& 1 \\ \mu_3(Z_n) &=& \frac{\mu_3(X)}{\sigma\sqrt{n}} \end{array}$$

this you could derive by using the properties of cumulants for which the first three relate to the central moments. You could derive similar relations for higher order moments (but more complicated) and end up with the central limit theorem which you could see as the vanishing of the cumulants with order above 2.