Proportion

The proportion is a statistic that is used to describe how much of a population has a particular characteristic or attribute and is usually expressed as a fraction or decimal. The defining characteristic of the proportion, as distinct from the ratio, is that every individual in the numerator of a proportion is also included in the denominator. Consider a population in which each member either has or does not have a specified attribute. The population proportion is the percentage (or rate) of the entire population that has the specified attribute. For examples, we might be interested in the proportion of U.S. adults who have health insurance or in comparing the proportions of prevalence of CF antibody to Para influenza I virus among boys and girls in the age group 5 to 9 years. In the first case, the population consists of all U.S. adults and the specified attribute is ‘has health insurance.’ For the second case, the population consists of all boys and girls in the age group of 5 to 9 years.

Frequently, the population under consideration is large, and determining the population proportion by taking a census is therefore usually impractical and often impossible; for instance, imagine trying to interview every U.S. adult for the purpose of ascertaining the proportion that have health insurance. Thus, in practice, we mostly rely on sampling and use the sample data to make inferences about the population proportion.

The sample proportion is the percentage of a sample from the population that has the specified attribute. The sample proportion p can be computed by the formula

where x denotes the number of members in the sample that have the specified attribute and n denotes the sample size. For example, a study is undertaken to [Page 848]compare the rates of prevalence of CF antibody to Para influenza I virus among boys and girls in the age group of 5 to 9 years. Among 113 boys tested, 34 are found to have the antibody; among 139 girls tested, 54 have the antibody. Let p1 denote the population proportion of boys who have the CF antibody and p2 the population of girls who have the CF antibody.

Then sample proportions are

We may use these sample proportions, in accordance with statistical theory, to make inferences about the difference of these two population proportions.