# Understanding Residual Standard Deviation: Key Concepts, Formula & Examples

![Residual Standard Deviation](https://www.investopedia.com/thmb/CQ-m1R-U8JYlMI-X2qu1fDYLWK0=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/ResidualStandardDeviation-6a2af23eade24ef08485171ab9502033.png)

*— Julie Bang / Investopedia*

## What Is Residual Standard Deviation?

Residual standard deviation lets you measure how far data points fall from a regression line. This tells you how much the actual values differ from the predicted values, indicating the accuracy of a model. A smaller residual [standard deviation](https://www.investopedia.com/terms/s/standarddeviation.asp) means the model’s predictions are closer to reality, while a larger one suggests it's less accurate. In this article, we explore the formula, examples, and explain how to calculate and interpret residual standard deviation.

### Key Takeaways

*   Residual standard deviation measures the discrepancy between observed and predicted values in regression analysis.  
    
*   It helps evaluate the accuracy of a regression model's predictive capacity.  
    
*   A smaller residual standard deviation compared to the sample standard deviation indicates a more effective and predictive model.  
    
*   Calculating residual standard deviation involves determining the difference, or residuals, between observed and predicted values.  
    
*   Residual standard deviation is useful in businesses for assessing discrepancies between actual and anticipated costs.

## Why Residual Standard Deviation Matters in Regression Analysis 

Residual standard deviation measures [goodness-of-fit](https://www.investopedia.com/terms/g/goodness-of-fit.asp), assessing how well data points match the model. In business, after a regression analysis on cost data over time, the residual standard deviation shows the difference between actual and projected costs and indicates cost variability from the historical mean.

## Calculating the Residual Standard Deviation Step-by-Step

1.  Calculate the residuals by subtracting predicted values from observed values.
2.  Compute the sum of squared residuals.
3.  Divide the sum by (n-2) where n equals the number of data points.
4.  Take the square root of the final result to determine the residual standard deviation.  
    

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

## Applying the Residual Standard Deviation Formula

To calculate residual standard deviation, first find the difference between predicted and actual values around a fitted line. This difference is called the residual value or residuals, the distance between actual data points and model predictions.

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

## Residual Standard Deviation: A Practical Example

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:

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](https://www.investopedia.com/terms/r/residual-sum-of-squares.asp) = 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:

*   (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 indicates 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](https://www.investopedia.com/terms/s/sample.asp) standard deviation, the more predictive, or useful, the model is.

The residual standard deviation can be calculated when a [regression](https://www.investopedia.com/terms/r/regression.asp) analysis has been performed, as well as an [analysis of variance (ANOVA)](https://www.investopedia.com/terms/a/anova.asp). When determining a limit of quantitation (LoQ), the use of a residual standard deviation is permissible instead of the standard deviation.

## What Type of Measure Is 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. Goodness-of-fit is a statistical test that determines how well sample data fits a distribution from a population with a [normal distribution](https://www.investopedia.com/terms/n/normaldistribution.asp).

## How Can a Residual Standard Deviation Be Used in Business?

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. It can also give an idea of how much projected costs could vary from the mean of the historical cost data.

## How Do I Calculate Residual Standard Deviation?

First, you have to calculate the difference between the predicted values and actual values formed around a fitted line. This difference is known as the residual value, or residuals. To calculate the residual standard deviation, plug the residuals into the residual standard deviation equation to solve the formula.

## The Bottom Line

Residual standard deviation is used in statistics to show how accurate a regression model is by measuring how far actual values are from predicted values. A smaller measure means the data points fit more closely to the regression line, making the model more reliable for predictions. It is key in checking how well one variable can predict another, which is useful for businesses. That's because it helps companies compare projected to actual costs so they can make better data-driven decisions.