<s>
Experimental	B-General_Concept
uncertainty	I-General_Concept
analysis	I-General_Concept
is	O
a	O
technique	O
that	O
analyses	O
a	O
derived	O
quantity	O
,	O
based	O
on	O
the	O
uncertainties	O
in	O
the	O
experimentally	O
measured	O
quantities	O
that	O
are	O
used	O
in	O
some	O
form	O
of	O
mathematical	O
relationship	O
(	O
"	O
model	O
"	O
)	O
to	O
calculate	O
that	O
derived	O
quantity	O
.	O
</s>
<s>
For	O
example	O
,	O
an	O
experimental	B-General_Concept
uncertainty	I-General_Concept
analysis	I-General_Concept
of	O
an	O
undergraduate	O
physics	O
lab	O
experiment	O
in	O
which	O
a	O
pendulum	O
can	O
estimate	O
the	O
value	O
of	O
the	O
local	O
gravitational	O
acceleration	O
constant	O
g	O
.	O
The	O
relevant	O
equation	O
for	O
an	O
idealized	O
simple	O
pendulum	O
is	O
,	O
approximately	O
,	O
</s>
<s>
where	O
T	O
is	O
the	O
period	O
of	O
oscillation	B-Algorithm
(	O
seconds	O
)	O
,	O
L	O
is	O
the	O
length	O
(	O
meters	O
)	O
,	O
and	O
θ	O
is	O
the	O
initial	O
angle	O
.	O
</s>
<s>
There	O
are	O
three	O
quantities	O
that	O
must	O
be	O
measured	O
:	O
(	O
1	O
)	O
the	O
length	O
of	O
the	O
pendulum	O
,	O
from	O
its	O
suspension	O
point	O
to	O
the	O
center	O
of	O
mass	O
of	O
the	O
“	O
bob	O
;	O
”	O
(	O
2	O
)	O
the	O
period	O
of	O
oscillation	B-Algorithm
;	O
(	O
3	O
)	O
the	O
initial	O
displacement	O
angle	O
.	O
</s>
<s>
Next	O
,	O
the	O
period	O
of	O
oscillation	B-Algorithm
T	O
could	O
suffer	O
from	O
a	O
systematic	O
error	O
if	O
,	O
for	O
example	O
,	O
the	O
students	O
consistently	O
miscounted	O
the	O
back-and-forth	O
motions	O
of	O
the	O
pendulum	O
to	O
obtain	O
an	O
integer	O
number	O
of	O
cycles	O
.	O
</s>
<s>
Of	O
concern	O
here	O
is	O
a	O
consistent	O
,	O
systematic	O
,	O
nonrandom	O
error	O
in	O
the	O
measurement	O
of	O
the	O
period	O
of	O
oscillation	B-Algorithm
of	O
the	O
pendulum	O
.	O
</s>
<s>
Another	O
motivation	O
for	O
this	O
form	O
of	O
sensitivity	O
analysis	O
occurs	O
after	O
the	O
experiment	O
was	O
conducted	O
,	O
and	O
the	O
data	B-General_Concept
analysis	I-General_Concept
shows	O
a	O
bias	O
in	O
the	O
estimate	O
of	O
g	O
.	O
Examining	O
the	O
change	O
in	O
g	O
that	O
could	O
result	O
from	O
biases	O
in	O
the	O
several	O
input	O
parameters	O
,	O
that	O
is	O
,	O
the	O
measured	O
quantities	O
,	O
can	O
lead	O
to	O
insight	O
into	O
what	O
caused	O
the	O
bias	O
in	O
the	O
estimate	O
of	O
g	O
.	O
This	O
analysis	O
can	O
help	O
to	O
isolate	O
such	O
problems	O
as	O
measurement	O
mistakes	O
,	O
problems	O
with	O
apparatus	O
,	O
incorrect	O
assumptions	O
about	O
the	O
model	O
,	O
etc	O
.	O
</s>
<s>
The	O
symbol	O
∂z	O
/	O
∂x1	O
represents	O
the	O
"	O
partial	O
derivative	B-Algorithm
"	O
of	O
the	O
function	O
z	O
with	O
respect	O
to	O
one	O
of	O
the	O
several	O
variables	O
x	O
that	O
affect	O
z	O
.	O
</s>
<s>
For	O
the	O
present	O
purpose	O
,	O
finding	O
this	O
derivative	B-Algorithm
consists	O
of	O
holding	O
constant	O
all	O
variables	O
other	O
than	O
the	O
one	O
with	O
respect	O
to	O
which	O
the	O
partial	O
is	O
being	O
found	O
,	O
and	O
then	O
finding	O
the	O
first	B-Algorithm
derivative	I-Algorithm
in	O
the	O
usual	O
manner	O
(	O
which	O
may	O
,	O
and	O
often	O
does	O
,	O
involve	O
the	O
chain	O
rule	O
)	O
.	O
</s>
<s>
Eq(5 )	O
is	O
a	O
linear	O
function	O
that	O
approximates	B-Algorithm
,	O
e.g.	O
,	O
a	O
curve	O
in	O
two	O
dimensions	O
(	O
p	O
=	O
1	O
)	O
by	O
a	O
tangent	O
line	O
at	O
a	O
point	O
on	O
that	O
curve	O
,	O
or	O
in	O
three	O
dimensions	O
(	O
p	O
=	O
2	O
)	O
it	O
approximates	B-Algorithm
a	O
surface	O
by	O
a	O
tangent	O
plane	O
at	O
a	O
point	O
on	O
that	O
surface	O
.	O
</s>
<s>
In	O
the	O
application	O
of	O
these	O
partial	O
derivatives	B-Algorithm
,	O
note	O
that	O
they	O
are	O
functions	O
that	O
will	O
be	O
evaluated	O
at	O
a	O
point	O
,	O
that	O
is	O
,	O
all	O
the	O
parameters	O
that	O
appear	O
in	O
the	O
partials	O
will	O
have	O
numerical	O
values	O
.	O
</s>
<s>
and	O
now	O
the	O
task	O
is	O
to	O
find	O
the	O
partial	O
derivatives	B-Algorithm
in	O
this	O
equation	O
.	O
</s>
<s>
Plugging	O
these	O
derivatives	B-Algorithm
into	O
Eq(9 )	O
,	O
</s>
<s>
This	O
information	O
is	O
very	O
valuable	O
in	O
post-experiment	O
data	B-General_Concept
analysis	I-General_Concept
,	O
to	O
track	O
down	O
which	O
measurements	O
might	O
have	O
contributed	O
to	O
an	O
observed	O
bias	O
in	O
the	O
overall	O
result	O
(	O
estimate	O
of	O
g	O
)	O
.	O
</s>
<s>
Next	O
,	O
consider	O
the	O
fact	O
that	O
,	O
as	O
the	O
students	O
repeatedly	O
measure	O
the	O
oscillation	B-Algorithm
period	O
of	O
the	O
pendulum	O
,	O
they	O
will	O
obtain	O
different	O
values	O
for	O
each	O
measurement	O
.	O
</s>
<s>
In	O
the	O
figure	O
there	O
are	O
10000	O
simulated	O
measurements	O
in	O
the	O
histogram	B-Algorithm
(	O
which	O
sorts	O
the	O
data	O
into	O
bins	O
of	O
small	O
width	O
,	O
to	O
show	O
the	O
distribution	O
shape	O
)	O
,	O
and	O
the	O
Normal	O
PDF	O
is	O
the	O
solid	O
line	O
.	O
</s>
<s>
The	O
positive	O
square	O
root	O
of	O
the	O
variance	O
is	O
defined	O
to	O
be	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
and	O
it	O
is	O
a	O
measure	O
of	O
the	O
width	O
of	O
the	O
PDF	O
;	O
there	O
are	O
other	O
measures	O
,	O
but	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
symbolized	O
by	O
the	O
Greek	O
letter	O
σ	O
"	O
sigma	O
,	O
"	O
is	O
by	O
far	O
the	O
most	O
commonly	O
used	O
.	O
</s>
<s>
In	O
Figure	O
2	O
there	O
are	O
again	O
10000	O
measurements	O
of	O
T	O
,	O
which	O
are	O
then	O
used	O
in	O
Eq(2 )	O
to	O
estimate	O
g	O
,	O
and	O
those	O
10000	O
estimates	O
are	O
placed	O
in	O
the	O
histogram	B-Algorithm
.	O
</s>
<s>
That	O
g-PDF	O
is	O
plotted	O
with	O
the	O
histogram	B-Algorithm
(	O
black	O
line	O
)	O
and	O
the	O
agreement	O
with	O
the	O
data	O
is	O
very	O
good	O
.	O
</s>
<s>
Thus	O
,	O
even	O
when	O
using	O
arguably	O
the	O
simplest	O
nonlinear	O
function	O
,	O
the	O
square	O
of	O
a	O
random	O
variable	O
,	O
the	O
process	O
of	O
finding	O
the	O
mean	O
and	O
variance	O
of	O
the	O
derived	O
quantity	O
is	O
difficult	O
,	O
and	O
for	O
more	O
complicated	O
functions	O
it	O
is	O
safe	O
to	O
say	O
that	O
this	O
process	O
is	O
not	O
practical	O
for	O
experimental	O
data	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
Figure	O
3	O
shows	O
a	O
histogram	B-Algorithm
of	O
10000	O
samples	O
of	O
z	O
,	O
with	O
the	O
PDF	O
given	O
above	O
also	O
graphed	O
;	O
the	O
agreement	O
is	O
excellent	O
.	O
</s>
<s>
In	O
this	O
simulation	O
the	O
x	O
data	O
had	O
a	O
mean	O
of	O
10	O
and	O
a	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
2	O
.	O
</s>
<s>
If	O
,	O
as	O
is	O
usually	O
the	O
case	O
,	O
the	O
PDF	O
of	O
the	O
derived	O
quantity	O
has	O
not	O
been	O
found	O
,	O
and	O
even	O
if	O
the	O
PDFs	O
of	O
the	O
measured	O
quantities	O
are	O
not	O
known	O
,	O
it	O
turns	O
out	O
that	O
it	O
is	O
still	O
possible	O
to	O
estimate	O
the	O
mean	O
and	O
variance	O
(	O
and	O
,	O
thus	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
)	O
of	O
the	O
derived	O
quantity	O
.	O
</s>
<s>
(	O
For	O
a	O
derivation	B-Algorithm
of	O
Eq(13 )	O
and	O
(	O
14	O
)	O
,	O
see	O
this	O
section	O
,	O
below	O
.	O
)	O
</s>
<s>
Again	O
,	O
as	O
was	O
the	O
case	O
with	O
the	O
bias	O
calculations	O
,	O
the	O
partial	O
derivatives	B-Algorithm
are	O
evaluated	O
at	O
a	O
specific	O
point	O
,	O
in	O
this	O
case	O
,	O
at	O
the	O
mean	O
(	O
average	O
)	O
value	O
,	O
or	O
other	O
best	O
estimate	O
,	O
of	O
each	O
of	O
the	O
independent	O
variables	O
.	O
</s>
<s>
A	O
more	O
elegant	O
way	O
of	O
writing	O
the	O
so-called	O
"	O
propagation	B-General_Concept
of	I-General_Concept
error	I-General_Concept
"	O
variance	O
equation	O
is	O
to	O
use	O
matrices	B-Architecture
.	O
</s>
<s>
First	O
define	O
a	O
vector	O
of	O
partial	O
derivatives	B-Algorithm
,	O
as	O
was	O
used	O
in	O
Eq(8 )	O
above	O
:	O
</s>
<s>
Next	O
,	O
to	O
find	O
an	O
estimate	O
of	O
the	O
variance	O
for	O
the	O
pendulum	O
example	O
,	O
since	O
the	O
partial	O
derivatives	B-Algorithm
have	O
already	O
been	O
found	O
in	O
Eq(10 )	O
,	O
all	O
the	O
variables	O
will	O
return	O
to	O
the	O
problem	O
.	O
</s>
<s>
The	O
variances	O
(	O
or	O
standard	B-General_Concept
deviations	I-General_Concept
)	O
and	O
the	O
biases	O
are	O
not	O
the	O
same	O
thing	O
.	O
</s>
<s>
Here	O
,	O
only	O
the	O
time	O
measurement	O
was	O
presumed	O
to	O
have	O
random	O
variation	O
,	O
and	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
used	O
for	O
it	O
was	O
0.03	O
seconds	O
.	O
</s>
<s>
To	O
make	O
clearer	O
what	O
happens	O
as	O
the	O
random	O
error	O
in	O
a	O
measurement	O
variable	O
increases	O
,	O
consider	O
Figure	O
4	O
,	O
where	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
time	O
measurements	O
is	O
increased	O
to	O
0.15	O
s	O
,	O
or	O
about	O
ten	O
percent	O
.	O
</s>
<s>
Rather	O
than	O
the	O
variance	O
,	O
often	O
a	O
more	O
useful	O
measure	O
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
σ	O
,	O
and	O
when	O
this	O
is	O
divided	O
by	O
the	O
mean	O
μ	O
we	O
have	O
a	O
quantity	O
called	O
the	O
relative	O
error	O
,	O
or	O
coefficient	O
of	O
variation	O
.	O
</s>
<s>
If	O
,	O
as	O
is	O
often	O
the	O
case	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
estimated	O
g	O
should	O
be	O
needed	O
by	O
itself	O
,	O
this	O
is	O
readily	O
obtained	O
by	O
a	O
simple	O
rearrangement	O
of	O
Eq(18 )	O
.	O
</s>
<s>
This	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
usually	O
quoted	O
along	O
with	O
the	O
"	O
point	O
estimate	O
"	O
of	O
the	O
mean	O
value	O
:	O
for	O
the	O
simulation	O
this	O
would	O
be	O
9.81	O
±	O
0.41m/s2	O
.	O
</s>
<s>
For	O
example	O
,	O
to	O
see	O
if	O
the	O
relative	O
error	O
for	O
just	O
the	O
angle	O
measurement	O
was	O
correct	O
,	O
a	O
simulation	O
was	O
created	O
to	O
sample	O
the	O
angles	O
from	O
a	O
Normal	O
PDF	O
with	O
mean	O
30	O
degrees	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
5	O
degrees	O
;	O
both	O
are	O
converted	O
to	O
radians	O
in	O
the	O
simulation	O
.	O
</s>
<s>
Figure	O
5	O
shows	O
the	O
histogram	B-Algorithm
for	O
these	O
g	O
estimates	O
.	O
</s>
<s>
In	O
the	O
introduction	O
it	O
was	O
mentioned	O
that	O
there	O
are	O
two	O
ways	O
to	O
analyze	O
a	O
set	O
of	O
measurements	O
of	O
the	O
period	O
of	O
oscillation	B-Algorithm
T	O
of	O
the	O
pendulum	O
:	O
</s>
<s>
These	O
measurements	O
are	O
averaged	O
to	O
produce	O
the	O
estimated	O
mean	O
values	O
to	O
use	O
in	O
the	O
equations	O
,	O
e.g.	O
,	O
for	O
evaluation	O
of	O
the	O
partial	O
derivatives	B-Algorithm
.	O
</s>
<s>
where	O
s	O
is	O
the	O
estimated	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
nT	O
T	O
measurements	O
.	O
</s>
<s>
where	O
the	O
average	O
values	O
(	O
bars	O
)	O
and	O
estimated	O
standard	B-General_Concept
deviations	I-General_Concept
s	O
are	O
shown	O
,	O
as	O
are	O
the	O
respective	O
sample	O
sizes	O
.	O
</s>
<s>
Details	O
concerning	O
the	O
difference	O
between	O
the	O
variance	O
and	O
the	O
mean-squared	B-Algorithm
error	I-Algorithm
(	O
MSe	O
)	O
have	O
been	O
skipped	O
.	O
</s>
<s>
Finding	O
the	O
PDF	O
is	O
nontrivial	O
,	O
and	O
may	O
not	O
even	O
be	O
possible	O
in	O
some	O
cases	O
,	O
and	O
is	O
certainly	O
not	O
a	O
practical	O
method	O
for	O
ordinary	O
data	B-General_Concept
analysis	I-General_Concept
purposes	O
.	O
</s>
<s>
Having	O
the	O
expression	O
for	O
the	O
expected	O
value	O
of	O
z	O
,	O
which	O
will	O
involve	O
partial	O
derivatives	B-Algorithm
and	O
the	O
means	O
and	O
variances	O
of	O
the	O
random	O
variables	O
x	O
,	O
set	O
up	O
the	O
expression	O
for	O
the	O
expectation	O
of	O
the	O
variance	O
:	O
</s>
<s>
This	O
equation	O
is	O
needed	O
to	O
find	O
the	O
variance	O
approximation	O
,	O
but	O
it	O
is	O
useful	O
on	O
its	O
own	O
;	O
remarkably	O
,	O
it	O
does	O
not	O
appear	O
in	O
most	O
texts	O
on	O
data	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
It	O
is	O
essential	O
to	O
recognize	O
that	O
all	O
of	O
these	O
partial	O
derivatives	B-Algorithm
are	O
to	O
be	O
evaluated	O
at	O
the	O
mean	O
of	O
the	O
respective	O
x	O
variables	O
,	O
and	O
that	O
the	O
corresponding	O
variances	O
are	O
variances	O
of	O
those	O
means	O
.	O
</s>
<s>
These	O
expressions	O
are	O
based	O
on	O
"	O
Method	O
1	O
"	O
data	B-General_Concept
analysis	I-General_Concept
,	O
where	O
the	O
observed	O
values	O
of	O
x	O
are	O
averaged	O
before	O
the	O
transformation	O
(	O
i.e.	O
,	O
in	O
this	O
case	O
,	O
raising	O
to	O
a	O
power	O
and	O
multiplying	O
by	O
a	O
constant	O
)	O
is	O
applied.Type	O
I	O
bias	O
,	O
absolute	O
.........................................................................	O
Eq	O
Type	O
I	O
bias	O
,	O
relative	O
(	O
fractional	O
)	O
Eq	O
.........................................................	O
Mean	O
(	O
expected	O
value	O
)	O
Eq	O
.......................................................................	O
Type	O
II	O
bias	O
,	O
absolute	O
........................................................................	O
Eq	O
Type	O
II	O
bias	O
,	O
fractional	O
.......................................................................	O
Eq	O
Variance	O
,	O
absolute	O
...........................................................................	O
Eq	O
Standard	B-General_Concept
deviation	I-General_Concept
,	O
fractional	O
...........................................................	O
Eq	O
Comments	O
:	O
</s>
<s>
(	O
3	O
)	O
The	O
Type	O
II	O
bias	O
and	O
the	O
variance	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
all	O
decrease	O
with	O
increasing	O
sample	O
size	O
,	O
and	O
they	O
also	O
decrease	O
,	O
for	O
a	O
given	O
sample	O
size	O
,	O
when	O
x	O
's	O
standard	B-General_Concept
deviation	I-General_Concept
σ	O
becomes	O
small	O
compared	O
to	O
its	O
mean	O
μ	O
.	O
</s>
<s>
Be	O
aware	O
that	O
the	O
effectiveness	O
of	O
these	O
approximations	O
is	O
very	O
strongly	O
dependent	O
on	O
the	O
relative	O
sizes	O
of	O
μ	O
,	O
σ	O
,	O
and	O
b.Type	O
I	O
bias	O
,	O
absolute	O
.........................................................................	O
Eq	O
Type	O
I	O
bias	O
,	O
relative	O
(	O
fractional	O
)	O
Eq	O
.........................................................	O
Mean	O
(	O
expected	O
value	O
)	O
Eq	O
.......................................................................	O
Type	O
II	O
bias	O
,	O
absolute	O
........................................................................	O
Eq	O
Type	O
II	O
bias	O
,	O
fractional	O
.......................................................................	O
Eq	O
Variance	O
,	O
absolute	O
...........................................................................	O
Eq	O
Standard	B-General_Concept
deviation	I-General_Concept
,	O
fractional	O
...........................................................	O
Eq	O
Univariate	O
case	O
3NOTES	O
:	O
b	O
and	O
x	O
must	O
be	O
positive	O
.	O
</s>
<s>
Be	O
aware	O
that	O
the	O
effectiveness	O
of	O
these	O
approximations	O
is	O
very	O
strongly	O
dependent	O
on	O
the	O
relative	O
sizes	O
of	O
μ	O
,	O
σ	O
,	O
and	O
b.Type	O
I	O
bias	O
,	O
absolute	O
.........................................................................	O
Eq	O
Type	O
I	O
bias	O
,	O
relative	O
(	O
fractional	O
)	O
Eq	O
.........................................................	O
Mean	O
(	O
expected	O
value	O
)	O
Eq	O
.......................................................................	O
Type	O
II	O
bias	O
,	O
absolute	O
........................................................................	O
Eq	O
Type	O
II	O
bias	O
,	O
fractional	O
.......................................................................	O
Eq	O
Variance	O
,	O
absolute	O
...........................................................................	O
Eq	O
Standard	B-General_Concept
deviation	I-General_Concept
,	O
fractional	O
...........................................................	O
Eq	O
Multivariate	O
case	O
1NOTES	O
:	O
BVN	O
is	O
bivariate	O
Normal	O
PDF	O
.	O
</s>
