The standard deviation (SD) measures the amount of variability, or dispersion, for a subject set of data from the mean, while the standard error of the mean (SEM) measures how far the sample mean of the data is likely to be from the true population mean. The SEM is always smaller than the SD.

Standard deviation and standard error are often used in clinical experimental studies. In these studies, the standard deviation (SD) and the estimated standard error of the mean (SEM) are used to present the characteristics of sample data and to explain statistical analysis results. However, some researchers occasionally confuse the SD and SEM in medical literature. 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 data in a normal distribution. 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).

Calculating Standard Error of the Mean

standard deviation σ=∑i=1n(xi−x¯)2n−1variance=σ2standard error (σx¯)=σnwhere:x¯=the sample’s meann=the sample size\begin{aligned} &\text{standard deviation } \sigma = \sqrt{ \frac{ \sum_{i=1}^n{\left(x_i - \bar{x}\right)^2} }{n-1} } \\ &\text{variance} = {\sigma ^2 } \\ &\text{standard error }\left( \sigma_{\bar x} \right) = \frac{{\sigma }}{\sqrt{n}} \\ &\textbf{where:}\\ &\bar{x}=\text{the sample's mean}\\ &n=\text{the sample size}\\ \end{aligned}​standard deviation σ=n−1∑i=1n​(xi​−x¯)2​​variance=σ2standard error (σx¯​)=n​σ​where:x¯=the sample’s meann=the sample size​

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

The formula for the SD requires a few steps:

  1. First, take the square of the difference between each data point and the sample mean, finding the sum of those values.
  2. Then, divide that sum by the sample size minus one, which is the variance.
  3. Finally, take the square root of the variance to get the SD.

Standard error functions as a way to validate the accuracy of a sample or the accuracy of multiple samples by analyzing deviation within the means. The SEM describes how precise the mean of the sample is versus the true mean of the population. As the size of the sample data grows larger, the SEM decreases versus the SD. As the sample size increases, the true mean of the population is known with greater specificity. In contrast, increasing the sample size also provides a more specific measure of the SD. However, the SD may be more or less depending on the dispersion of the additional data added to the sample.

The standard error is considered part of descriptive statistics. It represents the standard deviation of the mean within a dataset. This serves as a measure of variation for random variables, providing a measurement for the spread. The smaller the spread, the more accurate the dataset.

However, the standard deviation is a measure of volatility and can be used as a risk measure for an investment. Assets with higher prices have a higher SD than assets with lower prices. The SD can be used to measure the importance of a price move in an asset. Assuming a normal distribution, around 68% of daily price changes are within one SD of the mean, with around 95% of daily price changes within two SDs of the mean.