Sampling distribution and CLT visualizer

The Central Limit Theorem is a claim you can test. Pick an ugly population, draw samples, and watch what happens to the distribution of their means.

parent population (mu = 1.93, sigma = 1.82)

distribution of sample means (0 samples drawn)

indigo curve = the CLT's prediction N(mu, sigma/sqrt(n))

Draw one sample: 10 values will fall on the right-skewed population above, and their single mean drops into the empty plot below. The story is what thousands of those means pile up into.

Try to:○ shrink the predicted SE below 0.4○ make 2,000+ skewed samples come out normal○ pile up 5,000 means

What to try

  1. Shape. Choose the right-skewed population, set n = 2, and draw 2,000 samples: the histogram of means is still skewed. Now slide n to 30 and draw again: it snaps toward the indigo normal curve. That is the theorem.
  2. Spread. Watch the predicted standard error, sigma divided by the square root of n. Quadruple n (say 10 to 40) and the histogram gets half as wide, not a quarter. Root n is the whole story of why bigger samples help less than you hope.
  3. Center. No matter the shape or n, the histogram centers on the population mean. Sampling does not bias the mean; it only adds spread.

Compare the observed SD of the sample means with the predicted standard error in the readout: with a few thousand samples they agree to about two decimals. The reasoning lives in the CLT explained and sampling distributions; the formulas are on the formula sheet.

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