Root Mean Square Error
UGeun Jang∗
Department of Energy Resources Engineering, Seoul National University
Abstract
Keywords:
1. Root Mean Square
In mathematics, the root mean square (abbreviated RMS or rms), also known as the
quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is
especially useful when variates are positive and negative, e.g., sinusoids. RMS is used
in various fields, including electrical engineering.
It can be calculated for a series of discrete values or for a continuously varying function. The name comes from the fact that it is the square root of the mean of the squares
of the values. It is a special case of the generalized mean with the exponent p = 2.
1.1. Definition
The RMS value of a set of values (or a continuous-time waveform) is the square root
of the arithmetic mean (average) of the squares of the original values (or the square of
the function that defines the continuous waveform).
In the case of a set of n values {x1 , x1 , · · · , xn }, the RMS value is given by :
r
xrms =
1 2
x1 + x22 + · · · + xn2 .
n
(1)
∗
Corresponding author
Email address: orange224@gmail.com (UGeun Jang)
URL: http://ifreq.wordpress.com (UGeun Jang)
Preprint submitted to my journal
February 1, 2012
The corresponding fomular for a continuous function (or waveform) f (t) defined over
the interval T 1 ≤ t ≤ T 2 is
s
frms =
1
T2 − T1
Z
T2
f (t)
2
dt .
(2)
T1
The RMS over all time of a periodic function is equal to the RMS of one period of
the function. The RMS value of a continuous function or signal can be approximated
by taking the RMS of a series of equally spaced samples. Additionally, the RMS value
of various waveforms can also be determined without calculus, as shown by Cartwright
(2007).
2. Root Mean Square Error
When two data sets - one set from theoretical prediction and the other from actual
measurement of some physical variable, for instance - are compared, the RMS of the
pairwise differences of the two data sets can serve as a measure how far on average the
error is from 0. The mean of the pairwise differences does not measure the variability
of the difference, and the variability as indicated by the standard deviation is around the
mean instead of 0. Therefore, the RMS of the differences is a meaningful measure of the
error.
3. Root Mean Square Deviation
The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values predicted by a model or an
estimator and the values actually observed from the thing being modeled or estimated.
RMSD is a good measure of accuracy. These individual differences are also called residuals, and the RMSD serves to aggregate them into a single measure of predictive power.
3.1. Formula
The RMSD of an estimator θ̂ with respect to the estimated parameter θ is defined as
the square root of the mean square error:
r q
2 RMSD θ̂ = MSE θ̂ = E θ̂ − θ .
2
(3)
For an unbiased estimator, the RMSD is the square root of the variance, known as the
standard error.
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 distance between two oblong objects, expressed as random vectors
 
 x1,1 
 
 x1,2 
θ1 =  . 
 .. 
 
x1,n
and
 
 x2,1 
 
 x2,2 
θ2 =  .  .
 .. 
 
x2,n
(4)
The formula becomes:
RMSD (θ1 , θ2 ) =
p
MSE (θ1 , θ2 ) =
s
P
=
i
x1,i − x2,i
n2
q E (θ1 , θ2 )2
(5)
2
.
References
Cartwright, K. V., 2007. Determining the effective or rms voltage of various waveforms without calculus.
The Technology Interface 8 (1).
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