The standard error of the mean is a statistic that calculates how accurately a sample distribution represents a population using standard deviation.

Consider a group of feral cats in a neighborhood. Randomly sample five cats and note the mean weight and standard deviation. Repeating this a few more times with different cats from the same neighborhood, one can see each random sampling yields a slightly different mean and standard deviation.

The standard deviation of all these sample means is the standard error of the mean, expressed as sigma x bar. It represents the variation between the mean weight of the cats among different random samples.

The standard error of the sample means is calculated using this formula. Here, n represents the sample size and sigma is the population standard deviation.

Although the terms standard deviation and standard error are related, the standard deviation measures the variation within a sample. In contrast, the standard error of the mean measures the variation between the means of two or more samples from the same population.

The sampling variability of a statistic is defined as how much the statistic varies from one sample to another. The sampling variability of a statistic is typically measured by measuring its standard error.

The standard error of the mean is an example of a standard error. It is a unique standard deviation known as the standard deviation of the sampling distribution of the mean. The standard error of the mean is a statistic that calculates how correctly a sample distribution represents a population using standard deviation. The standard deviation of all the sample means is denoted as , which is also called the standard error of the mean.

This text is adapted from Openstax, Introductory Statistics, Section 2.7 Measures of the Spread of the Data

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4.4 : Standard Error of the Mean