Root Mean Square Error CalculatorFree online math calculator — instant results, no sign-up
Calculate root mean square error between paired observations.
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About the Root Mean Square Error Calculator
The Root Mean Square Error Calculator measures the difference between two lists of paired observations by taking the square root of their average squared difference. Use it to compare predictions with observed values or two sets of measurements matched in the same order. For the default lists 2, 4, 6, 8, 10 and 1, 2, 3, 4, 5, the RMSE is 3.3166247904.
- Values
- Enter the first list of observations, defaulting to 2, 4, 6, 8, 10, in the same measurement units as the paired list.
- Paired values or weights
- Enter the corresponding second observations, defaulting to 1, 2, 3, 4, 5, in the same units as Values; for this RMSE calculation, these entries are paired values, not weights.
Formula and method
RMSE = sqrt((1/n) × Σ(i=1 to n)(x_i − y_i)^2). With the defaults: RMSE = sqrt(((2 − 1)^2 + (4 − 2)^2 + (6 − 3)^2 + (8 − 4)^2 + (10 − 5)^2)/5) = sqrt(55/5) = sqrt(11) ≈ 3.3166247904.
- RMSERoot mean square error, expressed in the same units as the input observations.
- nNumber of matched observation pairs; n = 5 for the default inputs.
- iPosition of a pair in the two lists, from 1 through n.
- x_iObservation at position i in Values.
- y_iObservation at position i in Paired values or weights.
- ΣSum of the squared differences over all matched pairs.
Match the entries by position and subtract each second-list value from its first-list partner. The default differences are 1, 2, 3, 4, 5, whose squares are 1, 4, 9, 16, 25. Sum these squares to get 55, divide by the five pairs to obtain a mean squared error of 11, and take the square root to obtain RMSE ≈ 3.3166247904.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Values2, 4, 6, 8, 10
- Paired values or weights1, 2, 3, 4, 5
Results
- RMSE3.3166247904
How to use the Root Mean Square Error Calculator
- 1Enter 2, 4, 6, 8, 10 in Values to reproduce the default example, or enter your own first observation list.
- 2Enter 1, 2, 3, 4, 5 in Paired values or weights for the default example, or enter your corresponding second list.
- 3Check that both lists contain the same number of numeric entries, use the same units, and match observations in the same order.
- 4Run the calculation to square each paired difference, average the squares, and take the square root.
- 5Read the RMSE result; the default inputs produce 3.3166247904 in the input measurement units.
Reading your result
The RMSE output row gives a nonnegative measure of discrepancy between the paired lists in their original units. The default result, 3.3166247904, is sqrt(11) rounded to ten decimal places; it is not a percentage or a simple average absolute difference. Zero means every pair matches exactly, while a smaller RMSE indicates closer agreement when comparisons use the same units and evaluation data.
Limitations
This calculation assumes equal-length, nonempty numeric lists with correctly aligned pairs and compatible units. Each pair contributes equally; the second input does not supply statistical weights despite its label. Squaring makes RMSE sensitive to large errors, and the result does not indicate whether predictions tend to be too high or too low. There is no universal threshold for a good RMSE, and values measured on different scales are not directly comparable. This formula divides by n, not by a degrees-of-freedom adjustment used in some regression error estimates.
Common questions
Does Paired values or weights accept weights for this RMSE calculation?+
For this calculation, it supplies the second observation in each pair. The default entries 1, 2, 3, 4, 5 are subtracted from 2, 4, 6, 8, 10 respectively; they do not weight the errors.
Why is the default RMSE 3.3166247904 rather than 3?+
The absolute differences are 1, 2, 3, 4, 5, whose ordinary average is 3. RMSE instead averages their squares to get 11 and then takes sqrt(11), giving approximately 3.3166247904.
Will swapping the two lists change the RMSE?+
No, provided the same pairs are preserved. Swapping the lists reverses each difference's sign, but squaring removes that sign, leaving the RMSE unchanged.
Can I sort each list separately before calculating RMSE?+
Not unless the sorted positions still represent the intended pairs. RMSE matches entries by position, so sorting each list independently can change which observations are compared and produce a misleading result.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.