how to calculate residual standard deviation
to calculate the residual standard deviation the difference between the predicted values and actual values formed around a fitted line must be calculated first this difference is known as the residual value or simply residuals or the distance between known data points and those data points predicted by the model to calculate the residual standard deviation plug the residuals into the residual standard deviation equation to solve the formula example of residual standard deviationstart by calculating residual values for example assuming you have a set of four observed values for an unnamed experiment the table below shows y values observed and recorded for given values of x if the linear equation or slope of the line predicted by the data in the model is given as yest 1 2 where yest predicted y value the residual for each observation can be found the residual is equal to y yest so for the first set the actual y value is 1 and the predicted yest value given by the equation is yest 1 1 2 3 the residual value is thus 1 3 2 a negative residual value for the second set of x and y data points the predicted 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 same so the residual value will be zero you would use the same process for arriving at the predicted values for y in the remaining two data sets once you ve calculated the residuals for all points using the table or a graph use the residual standard deviation formula expanding the table above you calculate the residual standard deviation observe 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 number of data points which is 4 in this case calculate the denominator of the equation as finally calculate the square root of the results the magnitude of a typical residual can give you a sense of generally how close your estimates are the smaller the residual standard deviation the closer is the fit of the estimate to the actual data in effect the smaller the residual standard deviation is compared to the sample standard deviation the more predictive or useful the model is the residual standard deviation can be calculated when a regression analysis has been performed as well as an analysis of variance anova when determining a limit of quantitation loq the use of a residual standard deviation is permissible instead of the standard deviation 