The Central Limit Theorem: the mean of n independent draws from a distribution with finite variance is approximately normal for large n, whatever the population's shape. That is what justifies normal-based confidence intervals and t-tests on non-normal data, provided the sample is large enough for the data's skew.
Card 1 of 259. Answer: The Central Limit Theorem: the mean of n independent draws from a distribution with finite variance is approximately normal for large n, whatever the population's shape. That is what justifies normal-based confidence intervals and t-tests on non-normal data, provided the sample is large enough for the data's skew.
About · Quizzes · History of AI · To suggest new questions or report an error send me a dm.
The central limit theorem. The mean of n independent draws from any distribution with finite variance σ² is approximately normal for large n, centered on the true mean μ with standard deviation σ/√n, whatever the shape of the data. That is why the normal curve appears throughout statistics, even for non-normal data.
Example: a fair die is flat on 1 to 6, with mean 3.5 and standard deviation about 1.71. Average 30 rolls, repeat many times, and the averages form a bell centered on 3.5 with standard deviation 1.71/√30 ≈ 0.31. Abraham de Moivre found the coin-tossing case in 1733, Laplace generalized it, and George Pólya named it in 1920.
Follow-ups test the conditions. It describes the sampling distribution of the mean, so it never makes a skewed dataset normal. The n needed depends on skew: 30 is a rule of thumb, and heavily skewed data may need hundreds. Without finite variance it fails: the mean of n Cauchy draws is distributed like a single draw.
