# Residual Standard Deviation Definition

## What Is Residual Standard Deviation?

Residual standard deviation is a statistical term used to describe the difference in standard deviations of observed values versus predicted values as shown by points in a regression analysis.

Regression analysis is a method used in statistics to show a relationship between two different variables, and to describe how well you can predict the behavior of one variable from the behavior of another.

Residual standard deviation is also referred to as the standard deviation of points around a fitted line or the standard error of estimate.

### Key Takeaways

• Residual standard deviation is the standard deviation of the residual values, or the difference between a set of observed and predicted values.
• The standard deviation of the residuals calculates how much the data points spread around the regression line.
• The result is used to measure the error of the regression line’s predictability.
• The smaller the residual standard deviation is compared to the sample standard deviation, the more predictive, or useful, the model is.

## Understanding Residual Standard Deviation

Residual standard deviation is a goodness-of-fit measure that can be used to analyze how well a set of data points fit with the actual model. In a business setting for example, after performing a regression analysis on multiple data points of costs over time, the residual standard deviation can provide a business owner with information on the difference between actual costs and projected costs, and an idea of how much-projected costs could vary from the mean of the historical cost data.

## Formula for Residual Standard Deviation

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Residual

=

(

Y

Y

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s

t

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S

r

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s

=

(

Y

Y

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n

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where:

S

r

e

s

=

Residual standard deviation

Y

=

Observed value

Y

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s

t

=

Estimated or projected value

n

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Data points in population

\begin{aligned} &\text{Residual}=\left(Y-Y_{est}\right)\\ &S_{res}=\sqrt{\frac{\sum \left(Y-Y_{est}\right)^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 points in population}\\ \end{aligned}

Residual=(YYest)Sres=n2(YYest)2where:Sres=Residual standard deviationY=Observed valueYest=Estimated or projected valuen=Data points in population﻿

## 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 Deviation

Start 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:

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