What important theorem in statistics implies that, for a large sample size, the possible sample proportions of that size have approximately a normal distribution?

The various sample proportions of large sample size have an essentially normal distribution.

In statistics, the central limit theorem is one of the most significant and extensively used theorems.

This theorem states that if the sample size is high enough, the sampling distribution of that proportion will be close to a normal distribution, regardless of whether the variable under discussion has a skewed or normal distribution. The theorem is valid even for smaller samples if the population is normal.

Values of a variable in a population can follow several probability distributions, such as left-skewed, right-skewed, normal, uniform, and so on. The central limit theorem applies to variables that are both independently and identically distributed. This indicates that the worth of one sample observation should not be influenced by the worth of another.

In addition, the sample size necessary for an essentially normal distribution is influenced by the fact that many shapes of the variable in the underlying population are not normal. This means that the central limit theorem demands a bigger sample size if the variable is substantially skewed in the population.