Independent and identically distributed (i.i.d.) refers to a collection of random variables that share two key properties: independence and identical distribution. Independence means that the occurrence of one random variable does not affect the occurrence of another. Identical distribution indicates that all random variables have the same probability distribution.

For example, consider flipping a fair coin multiple times. Each flip is independent of the others, and each flip has the same probability distribution (0.5 for heads and 0.5 for tails). This scenario exemplifies i.i.d. variables because the outcome of one flip does not influence the outcome of another, and they all come from the same distribution ¹.

In statistical analysis and machine learning, the i.i.d. assumption is crucial as it simplifies the modeling process and underpins many statistical methods, including the central limit theorem. However, it’s important to note that real-world data may not always meet this assumption strictly, yet it can still be useful in analysis ².