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I am doing a project focused on the Central Limit Theorem, and in one part of it I want to use real data to see how the histogram approximates to a normal distribution. I also want another part which features data for which it does not work (i.e the sample mean does not follow a normal distribution) but I have not been able to find any. Any suggestions? I wrote in the title the Cauchy distribution because it is a well-known example, but any other which may work is fine. I have already looked for data such as annual maximum one-day rainfalls or light luminosity, but it hasn't worked so far.

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    Please define the technical terms, "Central Limit Theorem" and "Cauchy distribution." You mention that the Cauchy distribution is a "well-known example." A well-known example of what? – csk Jul 25 at 15:40
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The Cauchy distribution has infinite variance, so you are not going to find simple, real world examples. However it is possible to mathematically construct examples.

One possibility:

The ratio of 2 normal distributions where the mean of the denominator random variable is 0 will follow a Cauchy (or at least Cauchy like (it has been a while since I was in a theory class)) distribution.

The location (x-coordinate) of the vertex (max or min) of a parabola is -b/(2a) where a is the coefficient on x-squared and b is the coefficient on x. So if you fit a parabola to data that is really linear, then the estimate of the vertex location should follow a Cauchy.

Here is some R code that does a simulation based on this:

n <- 25
n.sim <- 1e4
out <- replicate(n.sim, {
  x <- seq(0, 1, length.out=n)
  y <- 2 + 0.3*x + rnorm(n, 0, 0.1)
  fit <- lm(y ~ x + I(x^2))
  -coef(fit)[2]/(2*coef(fit)[3])
})

hist(out)
var(out)

Rerunning this code with a larger sample size (n) does not narrow the distribution like it does with finite variance distributions.

For showing when the CLT does not work, I would look at something very skewed instead of the Cauchy and just show that while in theory a big enough sample size will result in normality, "big enough" is very big. Consider a binomial based on a very rare event, e.g. the number (or proportion) of people in your sample that have disease X (or trait X) where the probability of each individual having X is 1 in 1,000. With a sample size of 5,000 (a "big" sample in most basic stats textbooks) the sampling distribution is still obviously skewed and not yet normal.

Here is a line of R code to show this with simulated data:

hist(rbinom(100000, 5000, 1/1000))

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