Distribution Proportions

What is a sampling distribution?

The probability distribution of a given statistic based on a random sample of size n is called as a sampling distribution.

What is a proportion?

A special ratio in which the denominator includes the numerator is defined as a proportion.

More common usage of the term is in the context of a portion or part in its relation with the whole.

Sampling distribution of Proportion:

A distribution of values of p got by repeated samples of size N (taken from the population and the proportion (p) were determined for each sample) is called as a sampling distribution of proportion.

The sampling distribution of a proportion is always equal to the binomial distribution. The mean and the standard deviation of the binomial distribution are as follows:

μ = π

σ_p= √π(1 - π)/N

This is also the mean and standard deviation of a proportion. The shape of the sampling distribution depends upon both N and π. With large values of N and values of π in the neighbourhood of 0.5, the sampling distribution is very nearer to a normal distribution.

Correction for Continuity:

As the normal distribution is a continuous distribution, the probability that a sample value would exactly equal to any specific value is zero. But, this is not true when the normal distribution is used to approximate the sampling distribution of a proportion. To improve the approximation, a correction called the correction for continuity can be used.

The fundamental idea is to estimate the probability of, say, 5 successes out of 10 when p is 0.4, one should compute the area between 4.5 and 5.5.

Summary:

For small sample sizes n, the correction can make a much bigger difference than it would end up in the above example.