Residual Standard Deviation: Definition, Means, and Examples

What Is Residual Standard Deviation?

Residual standard deviation is a statistical time frame used to give an explanation for the difference in standard deviations of observed values versus predicted values as confirmed by the use of problems in a regression analysis.

Regression analysis is a method used in statistics to show a courting between two different variables, and to give an explanation for how well you can be expecting the behavior of one variable from the behavior of each and every different.

Residual standard deviation could also be referred to as the standard deviation of problems spherical a fitted line or the standard error of estimate.

Working out Residual Standard Deviation

Residual standard deviation is a goodness-of-fit measure that can be used to analyze how well a selection of wisdom problems fit with the actual taste. In a industry surroundings for example, after appearing a regression analysis on multiple wisdom problems of costs through the years, the residual standard deviation may give a industry owner with knowledge on the difference between precise costs and projected costs, and an idea of the way in which much-projected costs might simply vary from the indicate of the traditional value wisdom.

Means for Residual Standard Deviation


$\begin{array}{cc}& \text{Residual}=(Y-Y_{est})\\ & S_{res}=\sqrt{\frac{\sum {(Y-Y_{est})}^2}{n-2}}\\ & \text{where:}\\ & S_{res}=\text{Residual standard deviation}\\ & Y=\text{Observed value}\\ & Y_{est}=\text{Estimated or projected value}\\ & n=\text{Data problems in population}\end{array}$

get started{aligned} &text{Residual}=left(Y-Y_{est}correct) &S_{res}=sqrt{frac{sum left(Y-Y_{est}correct)^2}{n-2}} &textbf{where:} &S_{res}=text{Residual standard deviation} &Y=text{Observed value} &Y_{est}=text{Estimated or projected value} &n=text{Data problems in population} end{aligned} ​Residual=(Y−Yest​)Sres​=n−2∑(Y−Yest​)2​​where:Sres​=Residual standard deviationY=Observed valueYest​=Estimated or projected valuen=Data problems in population​

How you can Calculate Residual Standard Deviation

To calculate the residual standard deviation, the difference between the anticipated values and precise values formed spherical a fitted line should be calculated first. This difference is known as the residual value or, simply, residuals or the space between known wisdom problems and those wisdom problems predicted by the use of the way.

To calculate the residual standard deviation, plug the residuals into the residual standard deviation equation to get to the bottom of the components.

Example of Residual Standard Deviation 

Get began by the use of calculating residual values. For example, assuming you have got a selection of 4 observed values for an unnamed experiment, the table underneath displays y values observed and recorded for given values of x:

If the linear equation or slope of the street predicted by the use of the ideas inside the taste is given as y_est = 1x + 2 where y_est = predicted y value, the residual for each remark can be came upon.

The residual is equal to (y – y_est), so for the principle set, the actual y value is 1 and the anticipated y_est value given by the use of the equation is y_est = 1(1) + 2 = 3. The residual value is thus 1 – 3 = -2, a hostile residual value.

For the second set of x and y wisdom problems, the anticipated y value when x is 2 and y is 4 can be calculated as 1 (2) + 2 = 4.

In this case, the actual and predicted values are the equivalent, so the residual value might be 0. It’s possible you’ll use the equivalent process for arriving at the predicted values for y in the remainder two wisdom devices.

When you’ve calculated the residuals for all problems using the table or a graph, use the residual standard deviation components.

Expanding the table above, you calculate the residual standard deviation:

x	y	yest	Residual (y-yest)	Sum of each residual squared, or Σ(y-yest)2
1	1	3	-2	4
2	4	4	0	0
3	6	5	1	1
4	7	6	1	1

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

For the bottom portion or denominator of the residual standard deviation equation, n = the selection of wisdom problems, which is 4 in this case. Calculate the denominator of the equation as:

• (Collection of residuals – 2) = (4 – 2) = 2

Finally, calculate the sq. root of the consequences:

• Residual standard deviation: √(6/2) = √3 ≈ 1.732

The magnitude of an ordinary residual can give you some way of maximum steadily how close your estimates are. The smaller the residual standard deviation, the closer is the fit of the estimate to the actual wisdom. In have an effect on, the smaller the residual standard deviation is compared to the development standard deviation, the additional predictive, or useful, the way is.

The residual standard deviation can be calculated when a regression analysis has been performed, along with an analysis of variance (ANOVA). When understanding a limit of quantitation (LoQ), the use of a residual standard deviation is permissible as an alternative of the standard deviation.