central limit theorem

A theorem about the shape of averages: for i.i.d.i.i.d., abbr.: Short for 'independent and identically distributed': every data point is produced the same way, and none influences any other — like repeated rolls of one fair die. random variables with mean μ and finite variancevariance, n.: A number that grows the more widely something's possible values scatter away from its average, and is zero if it never varies. Its square root, the standard deviation, uses the same units as the values themselves. σ², the mean of n of them, minus μ and multiplied by √n/σ, has a distribution that approaches the standard normal distributionnormal distribution, n.: The familiar bell-shaped curve: most values cluster near the average, and values become rapidly rarer the farther they lie from it. Measurement errors and adult heights roughly follow it. as n grows, no matter how the individual variables are distributed; versions exist for non-identical and weakly dependent variables. In practice a sample mean is roughly normal with mean μ and standard deviation σ/√n, so estimation error falls like 1/√n. The theorem justifies normal-based confidence intervalsconfidence interval, n.: A range of plausible values for something being estimated, such as 'accuracy 82 percent, give or take 2 points', reflecting the uncertainty that comes from measuring only a sample., error bars and significance testshypothesis testing, n.: A formal procedure for deciding whether data show a real effect or could easily be explained by chance — for instance, whether one AI model is genuinely better than another. — including the error bars that should accompany benchmarkbenchmark, n.: A fixed, shared test — a set of questions or tasks with an agreed way of scoring — used to compare AI systems with one another and to track progress over time. scores — and explains why Monte CarloMonte Carlo method, n.: Estimating an answer by running many random trials and averaging the results — like estimating the odds in a card game by dealing thousands of hands instead of calculating. error shrinks as 1/√N. It also motivates Gaussian models of noise and explains why sums over many random weights at initialization are roughly Gaussian, the basis of infinite-width limits such as network Gaussian processesGaussian process, n.: A way of predicting an unknown curve or surface that gives not just a best guess at each point but an honest error bar, narrow near measured points and wide far from them. and the neural tangent kernelneural tangent kernel, n.: A mathematical tool showing that extremely wide neural networks learn like a simpler, well-understood method, because their internal settings barely move during training. It explains some, but not all, of deep learning's behavior.. Convergence can be slow for skewed or heavy-tailed data, and the theorem fails without finite variance.

With 500 questions and 70 percent accuracy, the CLT gives a standard error of about 2 points.