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Standard Deviation of Residuals Calculator

Standard Deviation of Residuals Calculator

Calculate Standard Deviation of Residuals

Compute the standard deviation of residuals (RMSE, RSD, RSE) from observed and predicted values.

Enter Your Data

Paste two columns, separated by a space or comma. Each row is a data point.

Must be an integer ≥1. $k$ is the number of coefficients in your model (including the intercept).

How to Use This Calculator

This **Standard Deviation of Residuals Calculator** helps you analyze the error of your regression model.

  1. **Enter Data:** In the text area, paste two columns of numbers: the observed values ($y_i$) and the predicted values ($\hat{y}_i$). You can separate the numbers with a space or a comma.
  2. **Set Parameters:** Input the **Number of Model Parameters ($k$)**. For a simple linear regression ($y = a + bx$), this value is 2 (one for the slope and one for the intercept).
  3. **Choose Your Metric:** Select the desired metric: **RMSE** (Root Mean Squared Error), **RSD** (Residual Standard Deviation), or **RSE** (Residual Standard Error).
  4. **Calculate:** Click the **"Calculate Residuals"** button to see the result, key statistics, and diagnostic plots.

Formula Breakdown

The calculator first finds the residuals ($e_i$) by subtracting the predicted value from the observed value:

$e_i = y_i - \hat{y}_i$

Then, it computes the Residual Sum of Squares (RSS), which is the sum of the squared residuals:

$\text{RSS} = \sum e_i^2$

The final result depends on the metric you choose, using different denominators:

  • **RMSE:** $\sqrt{\dfrac{\text{RSS}}{n}}$ (divides by the number of data points, $n$)
  • **RSD:** $\sqrt{\dfrac{\text{RSS}}{n-1}}$ (divides by $n-1$, similar to sample standard deviation)
  • **RSE:** $\sqrt{\dfrac{\text{RSS}}{n-k}}$ (divides by degrees of freedom, $n-k$, where $k$ is the number of model parameters)

The **RSE** is generally the most common and statistically robust measure for a regression model, as it accounts for the number of parameters estimated.

Frequently Asked Questions (FAQs)

What is the difference between RMSE and residual standard error?

RMSE (Root Mean Squared Error) is a common metric for a model's predictive performance and divides by $n$. Residual standard error (RSE) is a statistical measure that divides by the degrees of freedom ($n-k$), making it an unbiased estimator of the standard deviation of the error term in the population.

How does sample size change SEM?

The Standard Error of the Mean (SEM) decreases as the sample size ($n$) increases, specifically in proportion to the square root of $n$. This means larger samples provide a more precise estimate of the population mean.

When should I use pooled SD vs separate SDs?

You should use pooled standard deviation when you have two or more groups and can reasonably assume they have the same population variance. If this assumption is not met, it's safer to use separate standard deviations, which is a more conservative approach.

Keywords: standard deviation of residuals calculator, RMSE calculator, residual standard error calculator, regression, data analysis

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