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This is an open investigation. It can be taken to various levels of complexity and note that the construction of non-trivial examples is, in itself, difficult.
I was looking at two seemingly random lists of 12 whole numbers
chosen from 1 to 5. The sample correlation was exactly zero.
Convince yourself that such a list of numbers is possible. Can you get a feel for the properties of such lists?
Explore the properties and frequencies of such lists of numbers, perhaps varying the two numbers involved.
For more investigations see our Stage 5 pages.