Root mean square error calculator

Root mean square error calculator DEFAULT

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

MSE Calculator

Omni's MSE calculator is here for you whenever you need to quickly determine the sum of squared errors (SSE) and mean squared error (MSE) when searching for the line of best fit. You can also use this tool if you are wondering how to calculate MSE by hand, since it can show you the results of intermediate calculations.

Not sure what MSE is? Need just the formula for MSE, or rather looking for a precise mathematical definition of MSE and an explanation of the reasoning behind it? You're in the right place! Scroll down to learn everything you need about MSE in statistics! An example of MSE calculated step-by-step is also included for your convenience!

What is MSE in statistics?

In statistics, the mean squared error (MSE) measures how close predicted values are to observed values. Mathematically, MSE is the average of the squared differences between the predicted values and the observed values. We often use the term residuals to refer to these individual differences.

We most often define the predicted values as the values obtained from simple linear regression, or just as the arithmetic mean of the observed values - in the latter case, all the predicted values are equal.

💡 In simple linear regression, the line of best fit found via the method of least squares is exactly the line that minimizes MSE!

We now have a basic idea of what MSE is, so it's time to quickly explain how to find MSE with the help of our mean square error calculator.

How to use this MSE calculator?

It can't be any simpler! To use our MSE calculator most efficiently, follow these steps:

  1. Choose the mode of the mean square error calculator - should the predicted values be automatically set as the average of the observed values, or do you want to enter custom values?

  2. Next, input your data. You can enter up to 30 values - the fields will appear as you go.

  3. The MSE and SSE of your observations are already there!

  4. Do you want to see some of the details of the calculations? Turn the option to !

    Tip: This option allows you to use this calculator to generate examples of MSE!

  5. You can also increase the precision of calculations - just click the and adjust the value.

As nice as it is to use Omni's MSE calculator, it may happen that you'll have to compute MSE or SSE by hand. In the next section, we'll provide you with all the formulas you need.

How to find MSE and SSE?

Let be the observed values and be the predicted values.

The equation for MSE is the following:


where runs from to .

If we ignore the factor in front of the sum, we arrive at the formula for SSE:


where runs from to . In other words, the relationship between SSE and MSE is the following:


Matrix formula for MSE

Let us consider the column-vector with coefficients defined as

for . That is, is the vector of residuals.

Using , we can say that MSE is equal to times the squared magnitude of , or times the dot product of by itself:


Alternatively, we can rewrite this MSE equation as follows:


where is the transpose of , i.e., a row-vector, and the operation between and is matrix multiplication.

The above formulas lead us immediately to the following expression for SSE:


Why do we take squares in MSE?

Wouldn't it be simpler and more intuitive to add the differences between actual data and predictions without squaring them first?

No, there are good reasons for taking the squares!

Namely, the predicted values can be greater than or less than the observed values. And when we add together positive and negative differences, individual errors may cancel each other out. As a result, we can get the sum close to (or even equal to) zero even though the terms were relatively large. This could lead us to a false conclusion that our prediction is accurate since the error is low.

In contrast, when we take a square of each difference, we get a positive number, and each individual error increases the sum. In other words, squaring makes both positive and negative differences contribute to the final value in the same way. Thanks to squaring, we can say that that the smaller the value of MSE, the better model.

In particular, if the predicted values coincided perfectly with observed values, then MSE would be zero. This, however, nearly never happens in practice: MSE is almost always strictly positive because there's almost always some noise (randomness) in the observed values.

As you can see, we really can't take simple differences. However, squares are not the only option! In the next section, we will tell you, among other things, about MAE, which uses absolute values instead of squares to achieve exactly the same effect - get rid of negative signs of differences.

Alternatives to MSE in statistics

As we've seen in the formulas, the units of MSE are the square of the original units, exactly like in the case of variance. To return to the original units, we often take the square root of MSE, obtaining the root mean squared error (RMSE):


This is analogous to taking the square root of variance in order to get the standard deviation.

Another (slightly less popular) measure of the quality of prediction is the mean absolute error (MAE), where, instead of squaring the differences between observed and predicted values, we take the absolute differences between them:


where runs from to . When the predicted values are all equal to the mean of observed values, we arrive at the mean absolute deviation.

MSE example

Phew, we're finally done with the definition of MSE and all the formulas. It's high time we looked at an example!

Assume we have the following data:


  1. We see there are sixteen numbers, so .

  2. Next, we compute the average:


    Hence, .

  3. We compute the differences between each observation and the mean and also their squares:

xx - μ(x - μ)²
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  1. We sum the numbers from the 3rd column:

    , to get their SSE:


  2. To find MSE, we divide SSE by the sample length :


  3. To find RMSE, we take the square root of MSE:



How do I calculate MSE by hand?

To calculate MSE by hand, follow these instructions:

  1. Compute differences between the observed values and the predictions.
  2. Square each of these differences.
  3. Add all these squared differences together.
  4. Divide this sum by the sample length.
  5. That's it, you've found the MSE of your data!

How do I calculate SSE from MSE?

If you're given MSE, just one simple step separates you from finding SSE! The only thing you need to know is the sample length . Then apply this formula:

and enjoy your newly-computed SSE!

How do I calculate RMSE from MSE?

To calculate RMSE from MSE, you need to remember that RMSE is the abbreviation of the root mean sum of errors, so, as its name indicates, RMSE is just the square root of MSE:


How do I calculate RMSE from SSE?

In order to correctly calculate RMSE from SSE, recall that RMSE is the square root of MSE, which, in turn, is SSE divided by the sample length . Combining these two formulas, we arrive at the following direct relationship between RMSE and SSE:


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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.




Standard deviation of residuals or Root-mean-square error (RMSD)

Root Mean Square Error

What is Root Mean Square Error (RMSE)?

Root Mean Square Error (RMSE) measures how much error there is between two data sets. In other words, it compares a predicted value and an observed or known value. The smaller an RMSE value, the closer predicted and observed values are.

It’s also known as Root Mean Square Deviation and is one of the most widely used statistics in GIS.

Different than Mean Absolute Error (MAE), we use RMSE in a variety of applications when comparing two data sets.

Here’s an example of how to calculate RMSE in Excel with 10 observed and predicted values. But you can apply this same calculation to any size data set.

Root Mean Square Error Example

For example, we can compare any predicted value with an actual measurement (observed value).

  • Predicted value
  • Observed value

Root mean square error takes the difference for each observed and predicted value.

You can swap the order of subtraction because the next step is to take the square of the difference. This is because the square of a negative value will always be a positive value. But just make sure that you keep the same order throughout.

After that, divide the sum of all values by the number of observations. Finally, we get a RMSE value. Here’s what the RMSE Formula looks like:

rmse formula

How to Calculate RMSE in Excel

Here is a quick and easy guide to calculate RMSE in Excel. You will need a set of observed and predicted values:

Excel Spreadsheet In cell A1, type “observed value” as a header. For cell B1, type “predicted value”. In C2, type “difference”.

2. Place values in columns

Check settings If you have 10 observations, place observed elevation values in A2 to A11. In addition, populate predicted values in cells B2 to B11 of the spreadsheet

3. Find the difference between observed and predicted values

Decimal Degrees In column C2, subtract observed value and predicted value. Repeat for all rows below where predicted and observed values exist.

Now, these values could be positive or negative.

4. Calculate the root mean square error value

Save to fileIn cell D2, use the following formula to calculate RMSE:

Cell D2 is the root mean square error value. And save your work because you’re finished.

If you have a smaller value, this means that predicted values are close to observed values. And vice versa.

What’s Next?

RMSE quantifies how different a set of values are. The smaller an RMSE value, the closer predicted and observed values are.

If you’ve tested this RMSE guide, you can try to master some other widely used statistics in GIS:


Calculator square root mean error

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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.


Root Mean Square Error


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