Standard deviation about a value other than the mean
$begingroup$
We generally define standard deviation to be the deviation of the data points about the mean of the whole set, but why about mean? Couldn't we do it about the RMS value? Or any other value?
So I was doing some calculation, on the basis of which it seems to me that the standard deviation is a minimum about the mean.
Definition of standard deviation is ( about $x_0$)
begin{align*}
langle（x-x_0)^2rangle
&= langle x^2 - 2 x,x_0+ x_0 ^2 rangle \
&= langle x^2 rangle - langle x rangle ^2 + [ langle xrangle ^2 - 2 langle x rangle x_0+ x_0^2] \
&=(Delta x)^2 + [ langle x rangle - x_0 ]^2 .
end{align*}
So deviation is minimum about the average value!
Is this the reason we always calculate average values in statistical mechanics ?
statistics
asked Jan 6 at 19:09
user101134user101134
62
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migrated from physics.stackexchange.com Jan 9 at 11:02
This question came from our site for active researchers, academics and students of physics.
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show 4 more comments
$begingroup$
We generally define standard deviation to be the deviation of the data points about the mean of the whole set, but why about mean? Couldn't we do it about the RMS value? Or any other value?
So I was doing some calculation, on the basis of which it seems to me that the standard deviation is a minimum about the mean.
Definition of standard deviation is ( about $x_0$)
begin{align*}
langle（x-x_0)^2rangle
&= langle x^2 - 2 x,x_0+ x_0 ^2 rangle \
&= langle x^2 rangle - langle x rangle ^2 + [ langle xrangle ^2 - 2 langle x rangle x_0+ x_0^2] \
&=(Delta x)^2 + [ langle x rangle - x_0 ]^2 .
end{align*}
So deviation is minimum about the average value!
Is this the reason we always calculate average values in statistical mechanics ?
statistics
asked Jan 6 at 19:09
user101134user101134
62
$endgroup$
migrated from physics.stackexchange.com Jan 9 at 11:02
This question came from our site for active researchers, academics and students of physics.
$begingroup$
Would Cross Validated be a better home for this question?
$endgroup$
– Qmechanic
Jan 6 at 19:28
$begingroup$
If it weren't about the mean, it wouldn't be a "deviation". If it were w.r.t to $0$, it would be the 2nd raw moment. There really is no other reason to pick another center other than those two.
$endgroup$
– JEB
Jan 6 at 19:33
$begingroup$
Is what the reason we always calculate average values?
$endgroup$
– InertialObserver
Jan 6 at 22:27
$begingroup$
JEB 7 , we know that standard deviation is about average value, but from the. Definition I can say that if I change x average by any other value it would simply mean the deviation about that perticular value.
$endgroup$
– user101134
Jan 7 at 2:57
$begingroup$
@user101134 if you were doing probability distribution functions, you would soon come to realise that taking deviations with respect to the mean makes more sense. Because the density functions peak over the mean (for example in Gaussian, or the Poisson).
$endgroup$
– KV18
Jan 7 at 9:07
|
show 4 more comments
$begingroup$
We generally define standard deviation to be the deviation of the data points about the mean of the whole set, but why about mean? Couldn't we do it about the RMS value? Or any other value?
So I was doing some calculation, on the basis of which it seems to me that the standard deviation is a minimum about the mean.
Definition of standard deviation is ( about $x_0$)
begin{align*}
langle（x-x_0)^2rangle
&= langle x^2 - 2 x,x_0+ x_0 ^2 rangle \
&= langle x^2 rangle - langle x rangle ^2 + [ langle xrangle ^2 - 2 langle x rangle x_0+ x_0^2] \
&=(Delta x)^2 + [ langle x rangle - x_0 ]^2 .
end{align*}
So deviation is minimum about the average value!
Is this the reason we always calculate average values in statistical mechanics ?
statistics
asked Jan 6 at 19:09
user101134user101134
62
$endgroup$
We generally define standard deviation to be the deviation of the data points about the mean of the whole set, but why about mean? Couldn't we do it about the RMS value? Or any other value?
So I was doing some calculation, on the basis of which it seems to me that the standard deviation is a minimum about the mean.
Definition of standard deviation is ( about $x_0$)
begin{align*}
langle（x-x_0)^2rangle
&= langle x^2 - 2 x,x_0+ x_0 ^2 rangle \
&= langle x^2 rangle - langle x rangle ^2 + [ langle xrangle ^2 - 2 langle x rangle x_0+ x_0^2] \
&=(Delta x)^2 + [ langle x rangle - x_0 ]^2 .
end{align*}
So deviation is minimum about the average value!
Is this the reason we always calculate average values in statistical mechanics ?
statistics
statistics
asked Jan 6 at 19:09
user101134user101134
62
asked Jan 6 at 19:09
user101134user101134
62
asked Jan 6 at 19:09
user101134user101134
62
asked Jan 6 at 19:09
user101134user101134
62
asked Jan 6 at 19:09
user101134user101134
62
62
migrated from physics.stackexchange.com Jan 9 at 11:02
This question came from our site for active researchers, academics and students of physics.
migrated from physics.stackexchange.com Jan 9 at 11:02
This question came from our site for active researchers, academics and students of physics.
$begingroup$
Would Cross Validated be a better home for this question?
$endgroup$
– Qmechanic
Jan 6 at 19:28
$begingroup$
If it weren't about the mean, it wouldn't be a "deviation". If it were w.r.t to $0$, it would be the 2nd raw moment. There really is no other reason to pick another center other than those two.
$endgroup$
– JEB
Jan 6 at 19:33
$begingroup$
Is what the reason we always calculate average values?
$endgroup$
– InertialObserver
Jan 6 at 22:27
$begingroup$
JEB 7 , we know that standard deviation is about average value, but from the. Definition I can say that if I change x average by any other value it would simply mean the deviation about that perticular value.
$endgroup$
– user101134
Jan 7 at 2:57
$begingroup$
@user101134 if you were doing probability distribution functions, you would soon come to realise that taking deviations with respect to the mean makes more sense. Because the density functions peak over the mean (for example in Gaussian, or the Poisson).
$endgroup$
– KV18
Jan 7 at 9:07
|
show 4 more comments
$begingroup$
Would Cross Validated be a better home for this question?
$endgroup$
– Qmechanic
Jan 6 at 19:28
$begingroup$
If it weren't about the mean, it wouldn't be a "deviation". If it were w.r.t to $0$, it would be the 2nd raw moment. There really is no other reason to pick another center other than those two.
$endgroup$
– JEB
Jan 6 at 19:33
$begingroup$
Is what the reason we always calculate average values?
$endgroup$
– InertialObserver
Jan 6 at 22:27
$begingroup$
JEB 7 , we know that standard deviation is about average value, but from the. Definition I can say that if I change x average by any other value it would simply mean the deviation about that perticular value.
$endgroup$
– user101134
Jan 7 at 2:57
$begingroup$
@user101134 if you were doing probability distribution functions, you would soon come to realise that taking deviations with respect to the mean makes more sense. Because the density functions peak over the mean (for example in Gaussian, or the Poisson).
$endgroup$
– KV18
Jan 7 at 9:07
$begingroup$
Would Cross Validated be a better home for this question?
$endgroup$
– Qmechanic
Jan 6 at 19:28
$begingroup$
Would Cross Validated be a better home for this question?
$endgroup$
– Qmechanic
Jan 6 at 19:28
$begingroup$
If it weren't about the mean, it wouldn't be a "deviation". If it were w.r.t to $0$, it would be the 2nd raw moment. There really is no other reason to pick another center other than those two.
$endgroup$
– JEB
Jan 6 at 19:33
$begingroup$
If it weren't about the mean, it wouldn't be a "deviation". If it were w.r.t to $0$, it would be the 2nd raw moment. There really is no other reason to pick another center other than those two.
$endgroup$
– JEB
Jan 6 at 19:33
$begingroup$
Is what the reason we always calculate average values?
$endgroup$
– InertialObserver
Jan 6 at 22:27
$begingroup$
Is what the reason we always calculate average values?
$endgroup$
– InertialObserver
Jan 6 at 22:27
$begingroup$
JEB 7 , we know that standard deviation is about average value, but from the. Definition I can say that if I change x average by any other value it would simply mean the deviation about that perticular value.
$endgroup$
– user101134
Jan 7 at 2:57
$begingroup$
JEB 7 , we know that standard deviation is about average value, but from the. Definition I can say that if I change x average by any other value it would simply mean the deviation about that perticular value.
$endgroup$
– user101134
Jan 7 at 2:57
$begingroup$
@user101134 if you were doing probability distribution functions, you would soon come to realise that taking deviations with respect to the mean makes more sense. Because the density functions peak over the mean (for example in Gaussian, or the Poisson).
$endgroup$
– KV18
Jan 7 at 9:07
$begingroup$
@user101134 if you were doing probability distribution functions, you would soon come to realise that taking deviations with respect to the mean makes more sense. Because the density functions peak over the mean (for example in Gaussian, or the Poisson).
$endgroup$
– KV18
Jan 7 at 9:07
|
show 4 more comments
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$begingroup$
Would Cross Validated be a better home for this question?
$endgroup$
– Qmechanic
Jan 6 at 19:28
$begingroup$
If it weren't about the mean, it wouldn't be a "deviation". If it were w.r.t to $0$, it would be the 2nd raw moment. There really is no other reason to pick another center other than those two.
$endgroup$
– JEB
Jan 6 at 19:33
$begingroup$
Is what the reason we always calculate average values?
$endgroup$
– InertialObserver
Jan 6 at 22:27
$begingroup$
JEB 7 , we know that standard deviation is about average value, but from the. Definition I can say that if I change x average by any other value it would simply mean the deviation about that perticular value.
$endgroup$
– user101134
Jan 7 at 2:57
$begingroup$
@user101134 if you were doing probability distribution functions, you would soon come to realise that taking deviations with respect to the mean makes more sense. Because the density functions peak over the mean (for example in Gaussian, or the Poisson).
$endgroup$
– KV18
Jan 7 at 9:07