Root-Mean-Square
المؤلف:
Hoehn, L. and Niven, I.
المصدر:
"Averages on the Move." Math. Mag. 58
الجزء والصفحة:
...
30-6-2019
1952
Root-Mean-Square
For a set of 
numbers or values of a discrete distribution 
, ..., 
, the root-mean-square (abbreviated "RMS" and sometimes called the quadratic mean), is the square root of mean of the values 
, namely
where 
denotes the mean of the values 
.
For a variate 
from a continuous distribution 
,
![x_(RMS)=sqrt((int[P(x)]^2dx)/(intP(x)dx)),](http://mathworld.wolfram.com/images/equations/Root-Mean-Square/NumberedEquation1.gif) |
(4)
|
where the integrals are taken over the domain of the distribution. Similarly, for a function 
periodic over the interval 
], the root-mean-square is defined as
![f_(RMS)=sqrt(1/(T_2-T_1)int_(T_1)^(T_2)[f(t)]^2dt).](http://mathworld.wolfram.com/images/equations/Root-Mean-Square/NumberedEquation2.gif) |
(5)
|
The root-mean-square is the special case 
of the power mean.
Hoehn and Niven (1985) show that
 |
(6)
|
for any positive constant 
.
Physical scientists often use the term root-mean-square as a synonym for standard deviation when they refer to the square root of the mean squared deviation of a signal from a given baseline or fit.
REFERENCES:
Hoehn, L. and Niven, I. "Averages on the Move." Math. Mag. 58, 151-156, 1985.
Kenney, J. F. and Keeping, E. S. "Root Mean Square." §4.15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 59-60, 1962.
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