Root mean square error

Root mean square error DEFAULT

Root Mean Square Error (RMSE) is a standard way to measure the error of a model in predicting quantitative data. Formally it is defined as follows:

Let’s try to explore why this measure of error makes sense from a mathematical perspective. Ignoring the division by n under the square root, the first thing we can notice is a resemblance to the formula for the Euclidean distance between two vectors in ℝⁿ:

This tells us heuristically that RMSE can be thought of as some kind of (normalized) distance between the vector of predicted values and the vector of observed values.

But why are we dividing by n under the square root here? If we keep n (the number of observations) fixed, all it does is rescale the Euclidean distance by a factor of √(1/n). It’s a bit tricky to see why this is the right thing to do, so let’s delve in a bit deeper.

Imagine that our observed values are determined by adding random “errors” to each of the predicted values, as follows:

These errors, thought of as random variables, might have Gaussian distribution with mean μ and standard deviation σ, but any other distribution with a square-integrable PDF (probability density function) would also work. We want to think of ŷᵢ as an underlying physical quantity, such as the exact distance from Mars to the Sun at a particular point in time. Our observed quantity yᵢ would then be the distance from Mars to the Sun as we measure it, with some errors coming from mis-calibration of our telescopes and measurement noise from atmospheric interference.

Picture of the Sun and Mars
Picture of the Sun and Mars

The mean μ of the distribution of our errors would correspond to a persistent bias coming from mis-calibration, while the standard deviation σ would correspond to the amount of measurement noise. Imagine now that we know the mean μ of the distribution for our errors exactly and would like to estimate the standard deviation σ. We can see through a bit of calculation that:

Here E[…] is the expectation, and Var(…) is the variance. We can replace the average of the expectations E[εᵢ²] on the third line with the E[ε²] on the fourth line where ε is a variable with the same distribution as each of the εᵢ, because the errors εᵢ are identically distributed, and thus their squares all have the same expectation.

Remember that we assumed we already knew μ exactly. That is, the persistent bias in our instruments is a known bias, rather than an unknown bias. So we might as well correct for this bias right off the bat by subtracting μ from all our raw observations. That is, we might as well suppose our errors are already distributed with mean μ = 0. Plugging this into the equation above and taking the square root of both sides then yields:

Notice the left hand side looks familiar! If we removed the expectation E[ … ] from inside the square root, it is exactly our formula for RMSE form before. The central limit theorem tells us that as n gets larger, the variance of the quantity Σᵢ (ŷᵢ — yᵢ)² / n = Σᵢ (εᵢ)² / n should converge to zero. In fact a sharper form of the central limit theorem tell us its variance should converge to 0 asymptotically like 1/n. This tells us that Σᵢ (ŷᵢ — yᵢ)² / n is a good estimator for E[Σᵢ (ŷᵢ — yᵢ)² / n] = σ². But then RMSE is a good estimator for the standard deviation σ of the distribution of our errors!

We should also now have an explanation for the division by n under the square root in RMSE: it allows us to estimate the standard deviation σ of the error for a typical single observation rather than some kind of “total error”. By dividing by n, we keep this measure of error consistent as we move from a small collection of observations to a larger collection (it just becomes more accurate as we increase the number of observations). To phrase it another way, RMSE is a good way to answer the question: “How far off should we expect our model to be on its next prediction?”

To sum up our discussion, RMSE is a good measure to use if we want to estimate the standard deviation σ of a typical observed value from our model’s prediction, assuming that our observed data can be decomposed as:

The random noise here could be anything that our model does not capture (e.g., unknown variables that might influence the observed values). If the noise is small, as estimated by RMSE, this generally means our model is good at predicting our observed data, and if RMSE is large, this generally means our model is failing to account for important features underlying our data.

RMSE in Data Science: Subtleties of Using RMSE

In data science, RMSE has a double purpose:

  • To serve as a heuristic for training models
  • To evaluate trained models for usefulness / accuracy

This raises an important question: What does it mean for RMSE to be “small”?

We should note first and foremost that “small” will depend on our choice of units, and on the specific application we are hoping for. 100 inches is a big error in a building design, but 100 nanometers is not. On the other hand, 100 nanometers is a small error in fabricating an ice cube tray, but perhaps a big error in fabricating an integrated circuit.

For training models, it doesn’t really matter what units we are using, since all we care about during training is having a heuristic to help us decrease the error with each iteration. We care only about relative size of the error from one step to the next, not the absolute size of the error.

But in evaluating trained models in data science for usefulness / accuracy , we do care about units, because we aren’t just trying to see if we’re doing better than last time: we want to know if our model can actually help us solve a practical problem. The subtlety here is that evaluating whether RMSE is sufficiently small or not will depend on how accurate we need our model to be for our given application. There is never going to be a mathematical formula for this, because it depends on things like human intentions (“What are you intending to do with this model?”), risk aversion (“How much harm would be caused be if this model made a bad prediction?”), etc.

Besides units, there is another consideration too: “small” also needs to be measured relative to the type of model being used, the number of data points, and the history of training the model went through before you evaluated it for accuracy. At first this may sound counter-intuitive, but not when you remember the problem of over-fitting.

There is a risk of over-fitting whenever the number of parameters in your model is large relative to the number of data points you have. For example, if we are trying to predict one real quantity y as a function of another real quantity x, and our observations are (xᵢ, yᵢ) with x₁ < x₂ < x₃ … , a general interpolation theorem tells us there is some polynomial f(x) of degree at most n+1 with f(xᵢ) = yᵢ for i = 1, … , n. This means if we chose our model to be a degree n+1 polynomial, by tweaking the parameters of our model (the coefficients of the polynomial), we would be able to bring RMSE all the way down to 0. This is true regardless of what our y values are. In this case RMSE isn’t really telling us anything about the accuracy of our underlying model: we were guaranteed to be able to tweak parameters to get RMSE = 0 as measured measured on our existing data points regardless of whether there is any relationship between the two real quantities at all.

But it’s not only when the number of parameters exceeds the number of data points that we might run into problems. Even if we don’t have an absurdly excessive amount of parameters, it may be that general mathematical principles together with mild background assumptions on our data guarantee us with a high probability that by tweaking the parameters in our model, we can bring the RMSE below a certain threshold. If we are in such a situation, then RMSE being below this threshold may not say anything meaningful about our model’s predictive power.

If we wanted to think like a statistician, the question we would be asking is not “Is the RMSE of our trained model small?” but rather, “What is the probability the RMSE of our trained model on such-and-such set of observations would be this small by random chance?”

These kinds of questions get a bit complicated (you actually have to do statistics), but hopefully y’all get the picture of why there is no predetermined threshold for “small enough RMSE”, as easy as that would make our lives.


RMSE - Root mean square Error

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John D'Errico
Yes, it is different. The Root Mean Squared Error is exactly what it says.
(y - yhat).^2 % Squared Error
mean((y - yhat).^2) % Mean Squared Error
RMSE = sqrt(mean((y - yhat).^2)); % Root Mean Squared Error
What you have written is different, in that you have divided by dates, effectively normalizing the result. Also, there is no mean, only a sum. The difference is that a mean divides by the number of elements. It is an average.
Thus, you have written what could be described as a "normalized sum of the squared errors", but it is NOT an RMSE. Perhaps a Normalized SSE.

More Answers (6)

Sadiq Akbar
If I have 100 vectors of error and each error vector has got four elements, then how can we we find its MSE, RMSE and any other performance metric? e.g. If I have my desired vector as u=[0.5 1 0.6981 0.7854] and I have estimated vectors like as: Est1=[0.499 0.99 0.689 0.779], Est2=[0.500 1.002 0.699 0.77], Est3=[0.489 0.989 0.698 0.787],---Est100=[---],
Then Error1=u-Est1; Error2=u-Est2 and so on up to Error100=u-Est100. Now how can we find the MSE, RMSE and tell me others as well that are used to indicate the perofrmance of the algorithm. please tell me in the form of easy code.

Kardelen Darilmaz
x = hwydata(:,14); %Population of states
y = hwydata(:,4); %Accidents per state
xlabel('Population of state')
ylabel('Fatal traffic accidents per state')
title('Linear Regression Relation Between Accidents & Population')
X = [ones(length(x),1) x];
legend('Data','Slope','Slope & Intercept','Location','best');
Rsq1 = 1 - sum((y - yCalc1).^2)/sum((y - mean(y)).^2)
Rsq2 = 1 - sum((y - yCalc2).^2)/sum((y - mean(y)).^2)
I also want to add MSE and RMSE calculations to this code. Can you help me?*
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What is Root Mean Square Error (RMSE)?

Root mean square error or root mean square deviation is one of the most commonly used measures for evaluating the quality of predictions. It shows how far predictions fall from measured true values using Euclidean distance.

To compute RMSE, calculate the residual (difference between prediction and truth) for each data point, compute the norm of residual for each data point, compute the mean of residuals and take the square root of that mean. RMSE is commonly used in supervised learning applications, as RMSE uses and needs true measurements at each predicted data point.

Root mean square error can be expressed as

RMSE formula

where N is the number of data points, y(i) is the i-th measurement, and y ̂(i) is its corresponding prediction.

Note: RMSE is NOT scale invariant and hence comparison of models using this measure is affected by the scale of the data. For this reason, RMSE is commonly used over standardized data.


Why is Root Mean Square Error (RMSE) Important?

In machine learning, it is extremely helpful to have a single number to judge a model’s performance, whether it be during training, cross-validation, or monitoring after deployment. Root mean square error is one of the most widely used measures for this. It is a proper scoring rule that is intuitive to understand and compatible with some of the most common statistical assumptions.

Note: By squaring errors and calculating a mean, RMSE can be heavily affected by a few predictions which are much worse than the rest. If this is undesirable, using the absolute value of residuals and/or calculating median can give a better idea of how a model performs on most predictions, without extra influence from unusually poor predictions.


How C3 AI Helps Organizations Use Root Mean Square Error (RMSE)

The C3 AI platform provides an easy way to automatically calculate RMSE and other evaluation metrics as part of a machine learning model pipeline. This extends into automated machine learning, where C3 AI® MLAutoTuner can automatically optimize hyperparameters and select model based on RMSE or other measures.




Root mean square deviation (RMSD)

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Root Mean Square Error In R, The root mean square error (RMSE) allows us to measure how far predicted values are from observed values in a regression analysis.

In other words, how concentrated the data around the line of best fit.

RMSE = √[ Σ(Pi – Oi)2 / n ]


  • Σ symbol indicates “sum”
  • Pi is the predicted value for the ith observation in the dataset
  • Oi is the observed value for the ith observation in the dataset
  • n is the sample size

Naive Bayes Classification in R » Prediction Model »

Root Mean Square Error in R.

Method 1: Function

Let’s create a data frame with predicted values and observed values.

data <- data.frame(actual=c(35, 36, 43, 47, 48, 49, 46, 43, 42, 37, 36, 40), predicted=c(37, 37, 43, 46, 46, 50, 45, 44, 43, 41, 32, 42)) data    actual predicted 1      35        37 2      36        37 3      43        43 4      47        46 5      48        46 6      49        50 7      46        45 8      43        44 9      42        43 10     37        41 11     36        32

We will create our own function for RMSE calculation

sqrt(mean((data$actual - data$predicted)^2)) 2.041241

The root mean square error is 2.041241.

Market Basket Analysis in R » What Goes With What »

Method 2: Package

rmse() function available from the Metrics package, Let’s make use of the same.

rmse(actual, predicted)

library(Metrics) rmse(data$actual, data$predicted) 2.041241

The root mean square error is 2.041241.


Mean square error is a useful way to determine the extent to which a regression model is capable of integrating a dataset.

The larger the difference indicates a larger gap between the predicted and observed values, which means poor regression model fit. In the same way, the smaller RMSE that indicates the better the model.

Based on RMSE we can compare the two different models with each other and be able to identify which model fits the data better.

Decision Trees in R » Classification & Regression »

The post How to Calculate Root Mean Square Error (RMSE) in R appeared first on finnstats.



Mean error root square

The regression line predicts the average y value associated with a given x value. Note that is also necessary to get a measure of the spread of the y values around that average. To do this, we use the root-mean-square error (r.m.s. error).

To construct the r.m.s. error, you first need to determine the residuals. Residuals are the difference between the actual values and the predicted values. I denoted them by $\hat{y}_i -y_i$, where $y_i$ is the observed value for the ith observation and $\hat{y}_i$ is the predicted value.

They can be positive or negative as the predicted value under or over estimates the actual value. Squaring the residuals, averaging the squares, and taking the square root gives us the r.m.s error. You then use the r.m.s. error as a measure of the spread of the y values about the predicted y value.

\begin{displaymath}RMS Errors= \sqrt{\frac{\sum_{i=1}^n (\hat{y_i}-y_i)^2}{n}}\end{displaymath}

As before, you can usually expect 68% of the y values to be within one r.m.s. error, and 95% to be within two r.m.s. errors of the predicted values. These approximations assume that the data set is football-shaped.

Squaring the residuals, taking the average then the root to compute the r.m.s. error is a lot of work. Fortunately, algebra provides us with a shortcut (whose mechanics we will omit).

The r.m.s error is also equal to $\sqrt{1-r^2}$ times the SD of y.

\begin{displaymath}RMS Error= \sqrt{1-r^2} SD_y\end{displaymath}

Thus the RMS error is measured on the same scale, with the same units as $y$.

The term is always between 0 and 1, since r is between -1 and 1. It tells us how much smaller the r.m.s error will be than the SD.

For example, if all the points lie exactly on a line with positive slope, then r will be 1, and the r.m.s. error will be 0. This means there is no spread in the values of y around the regression line (which you already knew since they all lie on a line).

The residuals can also be used to provide graphical information. If you plot the residuals against the x variable, you expect to see no pattern. If you do see a pattern, it is an indication that there is a problem with using a line to approximate this data set.

To use the normal approximation in a vertical slice, consider the points in the slice to be a new group of Y's. Their average value is the predicted value from the regression line, and their spread or SD is the r.m.s. error from the regression.

Then work as in the normal distribution, converting to standard units and eventually using the table on page 105 of the appendix if necessary.

Next:Regression Line Up:Regression Previous:Regression Effect and Regression   IndexSusan Holmes
Root mean square deviation (RMSD)

Root-mean-square deviation

Statistical measure

For the bioinformatics concept, see Root-mean-square deviation of atomic positions.

The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values (sample or population values) predicted by a model or an estimator and the values observed. The RMSD represents the square root of the second sample moment of the differences between predicted values and observed values or the quadratic mean of these differences. These deviations are called residuals when the calculations are performed over the data sample that was used for estimation and are called errors (or prediction errors) when computed out-of-sample. The RMSD serves to aggregate the magnitudes of the errors in predictions for various data points into a single measure of predictive power. RMSD is a measure of accuracy, to compare forecasting errors of different models for a particular dataset and not between datasets, as it is scale-dependent.[1]

RMSD is always non-negative, and a value of 0 (almost never achieved in practice) would indicate a perfect fit to the data. In general, a lower RMSD is better than a higher one. However, comparisons across different types of data would be invalid because the measure is dependent on the scale of the numbers used.

RMSD is the square root of the average of squared errors. The effect of each error on RMSD is proportional to the size of the squared error; thus larger errors have a disproportionately large effect on RMSD. Consequently, RMSD is sensitive to outliers.[2][3]


The RMSD of an estimator\hat{\theta} with respect to an estimated parameter \theta is defined as the square root of the mean square error:

\operatorname{RMSD}(\hat{\theta}) = \sqrt{\operatorname{MSE}(\hat{\theta})} = \sqrt{\operatorname{E}((\hat{\theta}-\theta)^2)}.

For an unbiased estimator, the RMSD is the square root of the variance, known as the standard deviation.

The RMSD of predicted values \hat y_t for times t of a regression'sdependent variable{\displaystyle y_{t},} with variables observed over T times, is computed for T different predictions as the square root of the mean of the squares of the deviations:

{\displaystyle \operatorname {RMSD} ={\sqrt {\frac {\sum _{t=1}^{T}({\hat {y}}_{t}-y_{t})^{2}}{T}}}.}

(For regressions on cross-sectional data, the subscript t is replaced by i and T is replaced by n.)

In some disciplines, the RMSD is used to compare differences between two things that may vary, neither of which is accepted as the "standard". For example, when measuring the average difference between two time series x_{1,t} and x_{2,t}, the formula becomes

{\displaystyle \operatorname {RMSD} ={\sqrt {\frac {\sum _{t=1}^{T}(x_{1,t}-x_{2,t})^{2}}{T}}}.}


Normalizing the RMSD facilitates the comparison between datasets or models with different scales. Though there is no consistent means of normalization in the literature, common choices are the mean or the range (defined as the maximum value minus the minimum value) of the measured data:[4]

\mathrm{NRMSD} = \frac{\mathrm{RMSD}}{y_\max -y_\min} or {\displaystyle \mathrm {NRMSD} ={\frac {\mathrm {RMSD} }{\bar {y}}}}.

This value is commonly referred to as the normalized root-mean-square deviation or error (NRMSD or NRMSE), and often expressed as a percentage, where lower values indicate less residual variance. In many cases, especially for smaller samples, the sample range is likely to be affected by the size of sample which would hamper comparisons.

Another possible method to make the RMSD a more useful comparison measure is to divide the RMSD by the interquartile range. When dividing the RMSD with the IQR the normalized value gets less sensitive for extreme values in the target variable.

{\displaystyle \mathrm {RMSDIQR} ={\frac {\mathrm {RMSD} }{IQR}}} where {\displaystyle IQR=Q_{3}-Q_{1}}

with {\displaystyle Q_{1}={\text{CDF}}^{-1}(0.25)} and Q_3 = \text{CDF}^{-1}(0.75) , where CDF−1 is the quantile function.

When normalizing by the mean value of the measurements, the term coefficient of variation of the RMSD, CV(RMSD) may be used to avoid ambiguity.[5] This is analogous to the coefficient of variation with the RMSD taking the place of the standard deviation.

{\displaystyle \mathrm {CV(RMSD)} ={\frac {\mathrm {RMSD} }{\bar {y}}}.}

Mean absolute error[edit]

Some researchers have recommended the use of the Mean Absolute Error (MAE) instead of the Root Mean Square Deviation. MAE possesses advantages in interpretability over RMSD. MAE is the average of the absolute values of the errors. MAE is fundamentally easier to understand than the square root of the average of squared errors. Furthermore, each error influences MAE in direct proportion to the absolute value of the error, which is not the case for RMSD.[2]


  • In meteorology, to see how effectively a mathematical model predicts the behavior of the atmosphere.
  • In bioinformatics, the root-mean-square deviation of atomic positions is the measure of the average distance between the atoms of superimposedproteins.
  • In structure based drug design, the RMSD is a measure of the difference between a crystal conformation of the ligand conformation and a docking prediction.
  • In economics, the RMSD is used to determine whether an economic model fits economic indicators. Some experts have argued that RMSD is less reliable than Relative Absolute Error.[6]
  • In experimental psychology, the RMSD is used to assess how well mathematical or computational models of behavior explain the empirically observed behavior.
  • In GIS, the RMSD is one measure used to assess the accuracy of spatial analysis and remote sensing.
  • In hydrogeology, RMSD and NRMSD are used to evaluate the calibration of a groundwater model.[7]
  • In imaging science, the RMSD is part of the peak signal-to-noise ratio, a measure used to assess how well a method to reconstruct an image performs relative to the original image.
  • In computational neuroscience, the RMSD is used to assess how well a system learns a given model.[8]
  • In protein nuclear magnetic resonance spectroscopy, the RMSD is used as a measure to estimate the quality of the obtained bundle of structures.
  • Submissions for the Netflix Prize were judged using the RMSD from the test dataset's undisclosed "true" values.
  • In the simulation of energy consumption of buildings, the RMSE and CV(RMSE) are used to calibrate models to measured building performance.[9]
  • In X-ray crystallography, RMSD (and RMSZ) is used to measure the deviation of the molecular internal coordinates deviate from the restraints library values.

See also[edit]


  1. ^Hyndman, Rob J.; Koehler, Anne B. (2006). "Another look at measures of forecast accuracy". International Journal of Forecasting. 22 (4): 679–688. CiteSeerX doi:10.1016/j.ijforecast.2006.03.001.
  2. ^ abPontius, Robert; Thontteh, Olufunmilayo; Chen, Hao (2008). "Components of information for multiple resolution comparison between maps that share a real variable". Environmental Ecological Statistics. 15 (2): 111–142. doi:10.1007/s10651-007-0043-y.
  3. ^Willmott, Cort; Matsuura, Kenji (2006). "On the use of dimensioned measures of error to evaluate the performance of spatial interpolators". International Journal of Geographical Information Science. 20: 89–102. doi:10.1080/13658810500286976.
  4. ^"Coastal Inlets Research Program (CIRP) Wiki - Statistics". Retrieved 4 February 2015.
  5. ^"FAQ: What is the coefficient of variation?". Retrieved 19 February 2019.
  6. ^Armstrong, J. Scott; Collopy, Fred (1992). "Error Measures For Generalizing About Forecasting Methods: Empirical Comparisons"(PDF). International Journal of Forecasting. 8 (1): 69–80. CiteSeerX doi:10.1016/0169-2070(92)90008-w.
  7. ^Anderson, M.P.; Woessner, W.W. (1992). Applied Groundwater Modeling: Simulation of Flow and Advective Transport (2nd ed.). Academic Press.
  8. ^Ensemble Neural Network Model
  9. ^ANSI/BPI-2400-S-2012: Standard Practice for Standardized Qualification of Whole-House Energy Savings Predictions by Calibration to Energy Use History

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