Root Mean Square - RMSE

What is RMSE?

Root mean square error or rmse is a frequently used measure of the difference between the numbers (population values and numbers) which is estimated by an estimator or mode. The root mean square is also known as root mean square deviation. The rmse details the standard deviation of the difference between the predicted and estimated values. Each of these differences is known as residuals when the calculations are completed over the data sample that was applied to determine, and known as prediction errors when estimated out of sample. The root mean square error or rmse accumulate the magnitude of the errors in estimating different times into a single measure of predictive power.


This article details about root mean square, root mean square formula, root mean square error meaning, root mean square definition, root mean square formula, etc.


Root Mean Square or rms Definition

Statistically, rmse is the square of the mean square, which is the arithmetic mean of the square of group value. Root mean square is also known as quadratic mean and is a specific situation of generalized mean whose exponent is 2. Root mean square isdefined as a varying function that relies on an integral of the square of the value which is immediate in a  cycle.


In other words, the root mean square of a group of a number is the square of the arithmetic mean or the squares of the functions which defines the constant waveform.


Root Mean Square Formula

For a group of n values including {x₁, x₂,x₃,…. xn}, the RMS is given by:


The formula for a continuous function f(x), defined for the interval \[x_{1}\] ≤ x ≤ \[x_{2}\] is given by: 


\[f_{rms} = \sqrt{\frac{1}{x_{2} - x_{1}}} \int_{x_{1}}^{x_{2}} |f(x)^{2} dx|\]


\[X_{rms} = \sqrt{\frac{a}{b}{\frac{(x_{1})^{2} + (x_{2})^{2} + (x_{n})^{2}}{n}}}\]


The RMS of a periodic function is always  similar to the RMS function of a single period. The continuous function’s RMS value can be estimated approximately by taking the RMS of a sequence of evenly spaced operations. Also, the RMS value of different waveforms can be calculated without calculus.


Root Mean Square Error Definition

Root mean square(rmse) is the standard deviation of the residuals ( estimated errors). Residuals are the approximation of how away from the regression line data points are. Rmse is a measure of how expanded these data are. In other words, rmse details you how intensive the data is around the line of best fit. Root means square error is primarily used in forecasting, climatology, regression analysis to verify experimental results.


Root Mean Square Error or rmse Formula 

The RMSE or root mean square deviation of an estimated model in terms of estimated value is stated as the square root of the mean square error. 


RMSE Formula = \[\sqrt{\sum_{i=1}^{n} (X_{obs, i} - X_{model, i})^{2}}\]


Here, Xobs, i is an observed value whereas Xmodel,i is known as modelled value at the time i.


What does root mean square value means?

Root means square value is defined as the square root of the mean value of a squared function. Root mean square value is primarily used as the effective d.c voltage ( or current) of an a.c. voltage ( or current). The root mean square value is further used in the computation of the average power of an AC waveform.


Solved Example

1. Find the root mean square of the following observation ; 5,4,8,1,?


Solutions: Using the root mean square formula: 


\[X_{rms}  = \sqrt{\frac{(x_{1})^{2} + (x_{2})^{2} + (x_{3})^{2}}{n}}\]


Root mean square = \[\sqrt{\frac{5^{2} + 4^{2} + 8^{2} + 1^{2}}{4}}\]


Root mean square = 5.14


2. Calculate the RMSE of the following data


6         7                8           9


Solution:


Step -1. Calculate the square of each no.


\[(x_{1})^{2} + (x_{2})^{2} + (x_{3})^{2}\]


= \[6^{2} + 7^{2} + 8^{2} + 9^{2}\]


= 36 + 49 + 64 + 81


= 230


Step 2. Calculate the mean square of each no.

\[\frac{1}{n} ((x_{1})^{2} + (x_{2})^{2} + (x_{3})^{2})\]


= ¼ (230)


= 27.5


\[X_{rms}  = \sqrt{\frac{(x_{1})^{2} + (x_{2})^{2} + (x_{3})^{2}}{n}}\]


= \[\sqrt{57.5}\]


= \[\frac{230}{4}\]


= 7.58


Quiz Time

1. RMS stands for

  1. Root mean square

  2. Root mean sum

  3. Root maximum sum

  4. Root minimum sum


2.  For a rectangular wave, the average current is ……….. Rms current.

  1. Greater than

  2. Less than

  3. Qual to

  4. Not related


3. The root mean square error is a measure of 

  1. Sample size

  2. Moving average period 

  3. Exponential Smoothing

  4. Forecast accuracy

FAQ (Frequently Asked Questions)

1. What is the Significance of the Root Mean Square Value?

Some of the importance of root mean square values are as follows:

  1. The important parameter of root mean square value is that it is used to detail the strength of an alternating circuit.

  2. Root mean square value of an AC Voltage/ current is equal to the DC voltage/ current that produces a similar heating effect when used across an identical resistor. Hence, it is also a measure of energy capacity in a given signal.

  3. When two unrelated signals are combined together, such as noise from two different sources, the rms value of their sum is equivalent to the square root of the sum of their rms values.

2. How does Root Mean Square Error is Defined in Terms of the Regression Line?

According to the regression line, the errors for the different values is simply the distance of a point above or below the line. We can calculate the ordinary size of these errors by taking the root mean square size for them.


These calculations give rise to the RMS error of the regression line which tells us how far above or below the points are. Generally, about 68% of the points made on a scatter diagram are within one RMS error of the regression line and above 95% are within two. This approximation value is known as 68% - 95% rule.