Standard Error of the Mean

The standard error of the mean refers to the standard deviation of the sampling distribution of the sample mean. This distribution represents all possible sample means that could be computed from samples selected according to a specified sample size and sampling design. The standard error of the mean quantifies how much variation is expected to be present in the sample means that would be computed from each and every possible sample, of a given size, taken from the population. The standard error of the mean is measured in the same units as the original data and is often denoted by SE() or simply as SE. Larger SE values imply more variation in sample means across possible samples of the same size; smaller SE values imply that the sample mean is more precise, or varies less from one sample to another.

The SE is typically estimated by dividing the estimate of the population standard deviation by the square root of the sample size: . Generally, increases in sample size imply decreases in the SE. Additionally, the SE is usually much smaller than the sample standard deviation with the degree of the difference being inversely proportional to the square root of the sample size.

In the context of a simple random sample of size n, selected without replacement from a finite population of size N with a population standard deviation σ, the standard error of the mean is given by

where the last part of the formula represents the “finite population correction.” If the population is much larger compared to the actual size of the sample, then the two SE formulas will be approximately equal. If σ is unknown, it can be estimated using information from the latest version of the survey (i.e. estimate from previous cycle) or from the sample (i.e. sample standard deviation).

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Uses of the Standard Error of the Mean in Survey Research

The design effect for the mean for a given survey sampling design is the square of the quotient of the standard error of the mean, based on the particular design, to the standard error of the mean based on simple random sampling without replacement.

Confidence intervals for the population mean are also computed using the sample mean estimate along with an estimate of the standard error of the mean. Typically formulas for a (1 − α) × 100% confidence interval for the population mean are presented in the form

where the “critical value” is computed according to some statistical reference distribution such as the standard normal or t-distribution.

The coefficient of variation is simply the quotient of the standard error of the mean to the sample mean. Because the standard error of the mean is influenced by the units of the data, the coefficient of variation allows researchers to compare variability in the sample means across different samples using the same variables that are perhaps measured on different scales, such as income ranging in the thousands of dollars compared to income measured in millions of dollars.

For example, suppose that there is an interest in estimating the mean number of days in the past year that teenagers living in a rural community consumed at least one alcoholic beverage, and data from a probability sample, such as that provided by the National Survey on Drug Use and Health, are used to make this estimate. Assuming that a simple random sample of 100 teenagers from the rural community of 1,000 teenagers produces a sample mean of 25.75 and a sample standard deviation of 30.0 days, then the estimated . Using the finite population version, the estimated SE

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