When you perform an experiment that gives a series of observed values which you want to compare against theoretical values, the ​root-mean-square deviation (RMSD)​ or root-mean-square error (RMSE) lets you quantify this comparison. You calculate RMSD by finding the square root of the mean square error.

For a series of observations, you calculate mean square error by finding the difference between each experimental or observed value and the theoretical or predicted value, squaring each difference, adding them up, and dividing them by the number of observed values or predicted values there are.

This makes the RMSD formula:

for ​x_e​ expected values, ​x_o​ observed values, and ​n​ total number of values.

This method of finding a difference (or deviation), squaring each difference, summing them up and dividing by the number of data points (as you would when finding the average of a set of data), then taking the square root of the result is what gives the quantity its name, "root-mean-square deviation." You can use a step-by-step approach like this to calculate RMSD in Excel, which is great for large data sets.

​Standard deviation​ measures how much a set of data varies within itself. You can calculate it using

for each value ​x​ for ​n​ values with ​μ​ ("mu") average. Notice that this is the same formula for RMSD but, instead of expected and observed data values, you use the data value itself and the average of the set of data, respectively. Using this description, you can compare root mean square error vs standard deviation.

This means that, though it has a formula with similar structure to RMSD, standard deviation measures a specific hypothetical experimental scenario in which the expected values are all the average of the data set.

In this hypothetical scenario, the quantity inside the square root is called the ​variance​, how the data is distributed around the mean. Determining the variance lets you compare the data set to specific distributions that you would expect the data to take based on prior knowledge.

RMSD gives a specific, unified way of determining how errors of how predicted values differ from observed values for experiments. The lower the RMSD, the more accurate the experimental results are to theoretical predictions. They let you quantify how various sources of error affect the observed experimental results, such as air resistance affecting a pendulum's oscillation or surface tension between a fluid and its container preventing it from flowing.

You can further ensure that RMSD reflects the range of the set of data by dividing it by the difference between the maximum observed experimental value and the minimum to obtain the ​normalized root-mean-square deviation​ or error.

In the field of molecular docking, in which researchers compare the theoretical computer-generated structure of biomolecules to those from experimental results, RMSD can measure how closely experimental results reflect theoretical models. The more experimental results are able to reproduce what theoretical models predict, the lower the RMSD.

In addition to the example of molecular docking, meteorologists use RMSD to determine how closely mathematical models of climate predict atmospheric phenomena. Bioinformaticians, scientists who study biology through computer-based means, determine how distances between atomic positions of protein molecules vary from the average distance of those atoms in proteins using the RMSD as a measure of accuracy.

Economists use RMSD to figure out how closely economic models fit measured or observed results of economic activity. Psychologists use RMSD to compare observed behavior of psychological or psychology-based phenomena to computational models.

Neuroscientists use it to determine how artificial or biological-based systems can learn when compared to models of learning. Computer scientists studying imaging and vision compare the performance of how well a model can reconstruct images to the original images through different methods.