Standard Error

Statistics are derived from sample data, and because they are not taken from complete data, they inevitably vary from one sample to another. The standard error is a measure of the expected dispersion of sample estimates around the true population parameter. It is used to gauge the accuracy of a sample estimate: A larger standard error suggests less confidence in the sample statistic as an accurate measure of the population characteristic. Standard errors can be calculated for a range of survey statistics including means, percentages, totals, and differences in percentages. The discussion that follows focuses on sample means and proportions.

In a survey of families, let X represent family income, and let denote the mean family income, which is an example of a statistic resulting from the data. Although the survey may be designed to provide [Page 838]an estimate of the mean in the entire population of interest, it is highly unlikely that the estimate from the survey, , will be equal to μ, the mean income of all families in the targeted population.

Assuming that each family in the population has an equal chance of being selected for the survey, the true standard error of could be derived, hypotheti-cally, by conducting an infinite number of identical surveys of independent samples of the same size from the same population. The distribution of the values of comprises the sampling distribution of . The mean of this sampling distribution is the true value of the parameter, that is, μ, that the statistic is meant to estimate, and the standard deviation of the sampling distribution is the true standard error associated with the statistic. It can be denoted by σ .

In practice, however, the true population mean μ and the true standard error σ of are not derived from repeated surveys. In fact, they are rarely known. Instead, they are estimated from a single sample, and it is this estimate of the standard error that has become synonymous with the term standard error in survey research. With this understanding, the standard error of , denoted s, is defined in the following section.