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
 |  |  | (1) |
 |  |  | (2) |
 |  |  | (3) |
where 
denotes the mean of the values 
.
For a variate 
from a continuous distribution 
,
![x_(RMS)=sqrt((int[P(x)]^2dx)/(intP(x)dx)),](https://mathworld.wolfram.com/images/equations/Root-Mean-Square/NumberedEquation1.svg) | (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).](https://mathworld.wolfram.com/images/equations/Root-Mean-Square/NumberedEquation2.svg) | (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.
