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What allows us to apply normal calculations to non normal distributions?

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I'm not sure I completely understand the questions, but I would say the following apply to non normal distributions; Margins of error, Randomness, Stratified sample Central, Limit Theorem and options.
User Pubudu Sachintha
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Answer:

Central limit theorem

Explanation:

The central limit theorem in Statistics states that whenever sample sizes become large, and random samples are drawn, the mean of all sample means will follow a normal distribution irrespective of the original distribution to which it belongs to.

In those cases, especially in hypothesis testing for random samples, inferences about population parameters we apply normal calculations to non normal distributions.

User ADB
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