simulation by hand
A rather weird question on X validated this week was about devising a manual way to simulate (a few) normal variates. By manual I presume the author of the question means without resorting to a computer or any other business machine. Now, I do not know of any real phenomenon that is exactly and provably Normal. As analysed in a great philosophy of science paper by Aidan Lyon, the standard explanations for a real phenomenon to be Normal are almost invariably false, even those invoking the Central Limit Theorem. Hence I cannot think of a mechanical device that would directly return Normal generations from a Normal distribution with known parameters. However, since it is possible to simulate by hand Uniform U(0,1) variates [up to a given precision] using a chronometre or a wheel, calls to versions of the Box-Müller algorithm that do not rely on logarithmic or trigonometric functions are feasible, for instance by generating two Exponential variates, x and y, until 2y>(1-x)², x being the output. And generating Exponential variates is easy provided a radioactive material with known half-life is available, along with a Geiger counter. Or, if not, by calling von Neumann’s exponential generator. As detailed in Devroye’s simulation book.
After proposing this solution, I received a comment from the author of the question towards a simpler solution based, e.g., on the Central Limit Theorem. Presumably for simple iid random variables such as coin tosses or dice experiments. While I used the CLT for simulating Normal variables in my very early days [just after programming on punched cards!], I do not think this is a very good or efficient method, as the tails grow very slowly to normality. By comparison, using the same amount of coin tosses to create a sufficient number of binary digits of a Uniform variate produces a computer-precision exact Uniform variate, which can be exploited in Box-Müller-like algorithms to return exact Normal variates… Even by hand if necessary. [For some reason, this question attracted a lot of traffic and an encyclopaedic answer on X validated, despite being borderline to the point of being proposed for closure.]