Standard Error of the Mean vs. Standard Deviation: The Difference

Data Science, AI and ML
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The [standard deviation (SD)measures the amount of variability, or dispersion, from the individual data values to the mean, while the standard error of the mean (SEM) measures how far the sample mean (average) of the data is likely to be from the true population mean. The SEM is always smaller than the SD.

  • Standard deviation (SD) measures the dispersion of a dataset relative to its mean.
  • The standard error of the mean (SEM) measures how much discrepancy is likely in a sample’s mean compared with the population mean.
  • The SEM takes the SD and divides it by the square root of the sample size.

SEM vs. SD

Standard deviation and standard error are both used in all types of statistical studies, including those in finance, medicine, biology, engineering, and psychology. In these studies, the SD and the estimated SEM are used to present the characteristics of sample data and explain statistical analysis results.

However, some researchers occasionally confuse the SD and the SEM. Such researchers should remember that the calculations for SD and SEM include different statistical inferences, each of them with its own meaning. SD is the dispersion of individual data values. In other words, SD indicates how accurately the mean represents sample data.

However, the meaning of SEM includes statistical inference based on the sampling distribution. SEM is the SD of the theoretical distribution of the sample means (the sampling distribution).

Standard Error of the Mean

SEM is calculated by taking the standard deviation and dividing it by the square root of the sample size.

Standard error gives the accuracy of a sample mean by measuring the sample-to-sample variability of the sample means. The SEM describes how precise the mean of the sample is as an estimate of the true mean of the population. As the size of the sample data grows larger, the SEM decreases vs. the SD; hence, as the sample size increases, the sample mean estimates the true mean of the population with greater precision.

In contrast, increasing the sample size does not make the SD necessarily larger or smaller; it just becomes a more accurate estimate of the population SD.