Göçken et al. define the root mean square percentage error (RMSPE) as \begin{equation} \text{RMSPE} = \sqrt{\frac{100\%}{n} \cdot \sum_{i=1}^n \Delta X^2_{\text{rel},i}} \end{equation} with \begin{equation} \Delta X_{\text{rel},i}=\frac{X_i}{T_i}-1, \end{equation} where $T_i$ is the desired value and $X_i$ is the actual value.

However, Göçken et al. and Webber et al. define the root mean square relative error (RMSRE) as:

\begin{equation} \text{RMSRE} = \sqrt{\frac{1}{n}\cdot\sum_{i=1}^{n}\Delta X^2_{\text{rel},i}} \end{equation}

If we express the actual error $\Delta X_{\text{rel},i}$ as a percentage and name it $\Delta X_{\%,i}$, then we have: \begin{equation} \Delta X_{\%,i}=\left(\frac{X_i}{T_i}-1\right)\cdot 100\%=\Delta X_{\text{rel},i} \cdot 100\% \end{equation}

From my understanding, RMSPE should be the same as RMSRE, where $\Delta X_{\text{rel},i}$ is substituted by $\Delta X_{\text{%},i}$. However, this would yield \begin{equation} \text{RMSPE} = \sqrt{\frac{1}{n} \cdot \sum_{i=1}^n \Delta X^2_{\text{rel},i}} \cdot 100\%, \end{equation} which differs from the original definition of Göçken et al. by a factor of 10. Are my considerations correct and if so, are there alternative sources for the RMSPE?

  • $\begingroup$ $\sqrt{100\%}=100\%=1$. There is no factor of $10$ difference, though putting it inside the square root is misleading. $\endgroup$
    – robjohn
    Jun 8, 2022 at 8:24
  • $\begingroup$ That... makes sense. And it just took 3 years for someone to point it out. :-) Thanks a lot! $\endgroup$
    – Nos
    Jun 8, 2022 at 8:55

1 Answer 1


There are several alternative sources (Swanson et al., Fomby, Shcherbakov et al.), which agree that the RMSPE is defined as: \begin{equation} \text{RMSPE} = \sqrt{\frac{1}{n} \cdot \sum_{i=1}^n \Delta X^2_{\text{rel},i}} \cdot 100\% \end{equation}


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