<s>
In	O
mathematics	O
and	O
its	O
applications	O
,	O
the	O
root	B-General_Concept
mean	I-General_Concept
square	I-General_Concept
of	O
a	O
set	O
of	O
numbers	O
(	O
abbreviated	O
as	O
RMS	O
,	O
or	O
rms	O
and	O
denoted	O
in	O
formulas	O
as	O
either	O
or	O
)	O
is	O
defined	O
as	O
the	O
square	O
root	O
of	O
the	O
mean	O
square	O
(	O
the	O
arithmetic	O
mean	O
of	O
the	O
squares	O
)	O
of	O
the	O
set	O
.	O
</s>
<s>
The	O
RMS	O
is	O
also	O
known	O
as	O
the	O
quadratic	B-General_Concept
mean	I-General_Concept
(	O
denoted	O
)	O
and	O
is	O
a	O
particular	O
case	O
of	O
the	O
generalized	O
mean	O
.	O
</s>
<s>
In	O
estimation	O
theory	O
,	O
the	O
root-mean-square	B-General_Concept
deviation	I-General_Concept
of	O
an	O
estimator	O
is	O
a	O
measure	O
of	O
the	O
imperfection	O
of	O
the	O
fit	O
of	O
the	O
estimator	O
to	O
the	O
data	O
.	O
</s>
<s>
The	O
RMS	B-General_Concept
value	I-General_Concept
of	O
a	O
set	O
of	O
values	O
(	O
or	O
a	O
continuous-time	O
waveform	O
)	O
is	O
the	O
square	O
root	O
of	O
the	O
arithmetic	O
mean	O
of	O
the	O
squares	O
of	O
the	O
values	O
,	O
or	O
the	O
square	O
of	O
the	O
function	O
that	O
defines	O
the	O
continuous	O
waveform	O
.	O
</s>
<s>
The	O
RMS	B-General_Concept
value	I-General_Concept
of	O
a	O
continuous	O
function	O
or	O
signal	O
can	O
be	O
approximated	O
by	O
taking	O
the	O
RMS	O
of	O
a	O
sample	O
consisting	O
of	O
equally	O
spaced	O
observations	O
.	O
</s>
<s>
Additionally	O
,	O
the	O
RMS	B-General_Concept
value	I-General_Concept
of	O
various	O
waveforms	O
can	O
also	O
be	O
determined	O
without	O
calculus	O
,	O
as	O
shown	O
by	O
Cartwright	O
.	O
</s>
<s>
If	O
the	O
waveform	O
is	O
a	O
pure	O
sine	O
wave	O
,	O
the	O
relationships	O
between	O
amplitudes	B-Application
(	O
peak-to-peak	O
,	O
peak	O
)	O
and	O
RMS	O
are	O
fixed	O
and	O
known	O
,	O
as	O
they	O
are	O
for	O
any	O
continuous	O
periodic	O
wave	O
.	O
</s>
<s>
For	O
a	O
zero-mean	O
sine	O
wave	O
,	O
the	O
relationship	O
between	O
RMS	O
and	O
peak-to-peak	O
amplitude	B-Application
is	O
:	O
</s>
<s>
Waveforms	O
made	O
by	O
summing	O
known	O
simple	O
waveforms	O
have	O
an	O
RMS	B-General_Concept
value	I-General_Concept
that	O
is	O
the	O
root	O
of	O
the	O
sum	O
of	O
squares	O
of	O
the	O
component	O
RMS	B-General_Concept
values	I-General_Concept
,	O
if	O
the	O
component	O
waveforms	O
are	O
orthogonal	B-Algorithm
(	O
that	O
is	O
,	O
if	O
the	O
average	O
of	O
the	O
product	O
of	O
one	O
simple	O
waveform	O
with	O
another	O
is	O
zero	O
for	O
all	O
pairs	O
other	O
than	O
a	O
waveform	O
times	O
itself	O
)	O
.	O
</s>
<s>
Alternatively	O
,	O
for	O
waveforms	O
that	O
are	O
perfectly	O
positively	O
correlated	O
,	O
or	O
"	O
in	O
phase	O
"	O
with	O
each	O
other	O
,	O
their	O
RMS	B-General_Concept
values	I-General_Concept
sum	O
directly	O
.	O
</s>
<s>
So	O
,	O
the	O
RMS	B-General_Concept
value	I-General_Concept
,	O
IRMS	O
,	O
of	O
the	O
function	O
I(t )	O
is	O
the	O
constant	O
current	O
that	O
yields	O
the	O
same	O
power	O
dissipation	O
as	O
the	O
time-averaged	O
power	O
dissipation	O
of	O
the	O
current	O
I(t )	O
.	O
</s>
<s>
Average	O
power	O
can	O
also	O
be	O
found	O
using	O
the	O
same	O
method	O
that	O
in	O
the	O
case	O
of	O
a	O
time-varying	O
voltage	O
,	O
V(t )	O
,	O
with	O
RMS	B-General_Concept
value	I-General_Concept
VRMS	O
,	O
</s>
<s>
In	O
the	O
common	O
case	O
of	O
alternating	O
current	O
when	O
I(t )	O
is	O
a	O
sinusoidal	O
current	O
,	O
as	O
is	O
approximately	O
true	O
for	O
mains	O
power	O
,	O
the	O
RMS	B-General_Concept
value	I-General_Concept
is	O
easy	O
to	O
calculate	O
from	O
the	O
continuous	O
case	O
equation	O
above	O
.	O
</s>
<s>
Because	O
of	O
their	O
usefulness	O
in	O
carrying	O
out	O
power	O
calculations	O
,	O
listed	O
voltages	O
for	O
power	O
outlets	O
(	O
for	O
example	O
,	O
120V	O
in	O
the	O
US	O
,	O
or	O
230V	O
in	O
Europe	O
)	O
are	O
almost	O
always	O
quoted	O
in	O
RMS	B-General_Concept
values	I-General_Concept
,	O
and	O
not	O
peak	O
values	O
.	O
</s>
<s>
Peak	O
values	O
can	O
be	O
calculated	O
from	O
RMS	B-General_Concept
values	I-General_Concept
from	O
the	O
above	O
formula	O
,	O
which	O
implies	O
V=	O
VRMS×	O
,	O
assuming	O
the	O
source	O
is	O
a	O
pure	O
sine	O
wave	O
.	O
</s>
<s>
In	O
the	O
physics	O
of	O
gas	O
molecules	O
,	O
the	O
root-mean-square	B-General_Concept
speed	O
is	O
defined	O
as	O
the	O
square	O
root	O
of	O
the	O
average	O
squared-speed	O
.	O
</s>
<s>
where	O
R	O
represents	O
the	O
gas	O
constant	O
,	O
8.314J/	O
( mol·K	O
)	O
,	O
T	O
is	O
the	O
temperature	O
of	O
the	O
gas	O
in	O
kelvins	B-Operating_System
,	O
and	O
M	O
is	O
the	O
molar	O
mass	O
of	O
the	O
gas	O
in	O
kilograms	O
per	O
mole	O
.	O
</s>
<s>
If	O
is	O
the	O
arithmetic	O
mean	O
and	O
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
a	O
population	O
or	O
a	O
waveform	O
,	O
then	O
:	O
</s>
<s>
From	O
this	O
it	O
is	O
clear	O
that	O
the	O
RMS	B-General_Concept
value	I-General_Concept
is	O
always	O
greater	O
than	O
or	O
equal	O
to	O
the	O
average	O
,	O
in	O
that	O
the	O
RMS	O
includes	O
the	O
"	O
error	O
"	O
/	O
square	O
deviation	O
as	O
well	O
.	O
</s>
<s>
Physical	O
scientists	O
often	O
use	O
the	O
term	O
root	B-General_Concept
mean	I-General_Concept
square	I-General_Concept
as	O
a	O
synonym	O
for	O
standard	B-General_Concept
deviation	I-General_Concept
when	O
it	O
can	O
be	O
assumed	O
the	O
input	O
signal	O
has	O
zero	O
mean	O
,	O
that	O
is	O
,	O
referring	O
to	O
the	O
square	O
root	O
of	O
the	O
mean	O
squared	O
deviation	O
of	O
a	O
signal	O
from	O
a	O
given	O
baseline	O
or	O
fit	O
.	O
</s>
<s>
Standard	B-General_Concept
deviation	I-General_Concept
being	O
the	O
RMS	O
of	O
a	O
signal	O
's	O
variation	O
about	O
the	O
mean	O
,	O
rather	O
than	O
about	O
0	O
,	O
the	O
DC	O
component	O
is	O
removed	O
(	O
that	O
is	O
,	O
RMS(signal )	O
=	O
stdev(signal )	O
if	O
the	O
mean	O
signal	O
is	O
0	O
)	O
.	O
</s>
