Probability Sampling

Probability sampling is the term used to describe sampling where the probability of a population unit being selected in the sample is known. In nonprobability sampling, this probability is not known. When probability sampling is used to survey a population, population parameters can be inferred from the sample. Examples of probability sampling are simple random sampling, stratified sampling, and cluster sampling. In all of these examples, the probability that an individual unit is selected in the sample can be calculated.

For simple random sampling (without replacement), the probability that a unit appears in the sample is the same for every unit and is n/N. Consider a population of size five, say, five children. A simple random sample of size three is selected. This could be done by writing the children's names on five separate balls and putting the balls in an urn. The balls are mixed well and three balls are drawn out. The balls are all equal in size, shape, and weight, and so each ball will have an equal chance of being selected. The number of possible samples that could be selected is

The probability that an individual unit i is selected in the sample is

where

πi = number of samples that include the unit,

i = number of possible samples.

The numerator in this equation uses N – 1 and n – 1. If the sample includes unit i, then there are N – 1 units left in the population from which to choose the remaining sample of size n – 1. In the sample of children, after choosing the first ball out of five, there are only four balls left to choose the second ball and three balls left to choose the third ball in the sample. With probability sampling, sampling theory then can be used to estimate the sample mean, and the variance of the sample mean to infer the population mean.

Nonprobability sampling is called haphazard sampling, convenience sampling, and judgment sampling, among other terms. Examples of nonprobability sampling would be to stand on a street corner and interview pedestrians who walk by when the population of interest is all citizens who live in the city, running survey lines in a forest adjacent to roads when the population of interest is all the forest, and surveying only those businesses that appear to be representative. The samples could, in fact, give informative and reliable results. Pedestrians who walk may be representative of all the citizens, the forest adjacent to the road may be representative of the entire forest, and the judgment of what are representative businesses may be correct. However, without using probability sampling, sampling theory cannot be used, and there is no way of knowing if the resultant samples are informative and reliable. It is not possible to estimate how accurate the samples are or how precise they are.