Residual Standard Deviation

Residual standard deviation is a statistical term used to describe the difference in the standard deviations of observed values from expected values.

Residual standard deviation is also called the standard deviation of points around an adjusted line or standard error of estimate.

How to calculate the residual standard deviation

To calculate the residual standard deviation, you must first calculate the difference between the predicted values and the actual values formed around an adjusted line. This difference is known as the residual value or, simply, residuals or distance between the known data points and those data points predicted by the model.

To calculate the residual standard deviation, enter the residuals into the residual standard deviation equation to solve the formula.

Example of residual standard deviation

Start by calculating the residual values. For example, assuming you have a set of four observed values for an unnamed experiment, the table below shows the observed and recorded y-values for given values of x:

If the linear equation or slope of the line predicted by the data in the model is given as y est = 1x + 2 where y est = predicted y value, you can find the residual for each observation.

The residual is equal to (y - y est ), so for the first set, the actual y value is 1 and the predicted y est value given by the equation is y est = 1 (1) + 2 = 3. The residual value is then 1 - 3 = -2, a negative residual value.

For the second set of data points xey, the expected y value when x is 2 ey is 4 can be calculated as 1 (2) + 2 = 4.

In this case, the actual and expected values are the same, so the residual value will be zero. You will use the same process to obtain the predicted values for y in the remaining two data sets.

After calculating the residuals for all points using the table or a graph, use the residual standard deviation formula.

Note that the sum of the residuals squared = 6, which is the numerator of the residual standard deviation equation.

For the lower part or denominator of the residual standard deviation equation, n = the number of data points, which in this case is 4. Calculate the denominator of the equation as:

(Number of residuals - 2) = (4-2) = 2

Finally, calculate the square root of the results:

Residual standard deviation:  √ (6/2) = √3 ≈ 1,732

The size of a typical residual can give you a general idea of how close your estimates are. The smaller the residual standard deviation, the closer the fit of the estimate to the actual data. In fact, the smaller the residual standard deviation from the sample standard deviation, the more predictive or useful the model.

The residual standard deviation can be calculated when a regression analysis as well as an analysis of variance (ANOVA) has been performed. When determining a limit of quantification (LoQ), residual standard deviation may be used instead of standard deviation.