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μετρολογια μετρολογια μετρολογιαμετρολογια μετρολογια ω μετρολογιαω μετρολογια
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18
19
20
13
14
15
16
17
18
u
(
m
/
s
)
0.27 0.28 0.29 0.30 0.319
10
11
12
time series (sec)0.4
0.1
0.2
0.3
P
D
F
2009 2010 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 140.0
Velocity fluctuations (m/s)
8
, ,
, .
VIM (International vocabulary of basic and general terms inmetrology, 1993), :
, . , .
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.
, , .
, . : ,
:
, (GUM)
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( ()
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( )
( )
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=> =>
, :
( , )
, ,
, 1, 2 , 3 ,, i , , N1 2 3 i N f :
=f (1, 2 , 3 ,, N ) (6.1)
, ( ), , , , , , (.. ). , f . i
. , . , f
, , , , .
, ,
i , .
1. . , , ,
. , , , .
2. , , .
, ,
, y , (6.1) x1, x2,.., x 1, 2 , 3 ,, N . :
y=f(x1, x2,.., x ) (6.2)
, , , x1, x2,.., x :
xi
n , xi, (parent population).
, ,
(variance) , (variance) , (standard deviation),
.
(standard uncertainty of measurement) y y , u(y) .
xi, Xi, i i u(xi). .
, ,
( (, , ), . . Q. n , Q , qj
q qj
(j=1, 2, ... , n)
1
1 nj
jq q
n
1jn
q :
, ,
(experimental variance) s2(q) , :
2 2
1
1( ) ( )1
n
jj
s q q qn
(experimental standard deviation). (experimentalq (experimentalstandard deviation of the mean), :
q
22 ( )( ) s qs q ( )s q
n
(experimental standard deviation of the mean). , :
( )u q q
( ) ( )u q s q
, ,
( ) ( )u q s q
. . ,:
, .
, ,
. , , .
I. Xi , , , , , , xi. u(xi), , . . ,
.
II. Xi , , , xi u(xi), .
, ,
III X III. Xi , , + , , (.. , ), X Xi ( ). :
1:
1 ( )2i
x :
2 21( ) ( )12i
u x
|+|=|-|=,:
2 21( )u x
, ,
( )3i
u x
IV IV. . , , , .: , , , + , , 100% + 100%. , , , . (+ + )/2
2 21( )6i
u x
V. xi, , , u(xi),
, ,
.
VI ' VI. , V. , 90, 95 99 % . , , . , , . , u(xi), xi . , , 1.64, 1.96 2.58.
, ,
: : :u
: u => 68%
+
: u 68% :u => 58% :u => 65%
+ +
, ,
(uncorrelated) f( ) y, y=f(x1, x2,.., x ), :
2 2( ) ( )N
iu y u y (6.3)1i
ui(y) (i=1, 2,..., ) y y xi
( ) ( )i i iu y c u x (6.4) ci xi, f Xi , xi: xi:
ii i
f fcx X (6.5)
, ,
,...,i i N Ni i X x X xx X
ci, xi y. f (6.5), , y xi, +u(xi) -u(xi) ci y 2u(xi). y , , .. xi u(xi).
u(xi) , (6.4), u(y) , . , .
:
2( )u y u (6 6)( )c yu y u (6.6)
, ,
, , , , , . n n . O . (6.1), . u(y) (RSS rootsum square) . . 6 1 6.1
( )
Xi xi u(xi) ci
ui (y)
X1 x1 u(x1) c1 u1 (y)X ( ) ( )X2 x2 u(x2) c2 u2(y): : : : :
XN xN u(xN) cN uN (y)
, ,
XN xN u(xN) cN uN (y)Y y u(y)
()
( ( ).
( )cU k u y (6.7)( )c y ( ) , k k=2. 95%. . . .. , , (3) , (3), , . , .
, ,
.
. k o u(y) y. , u(y) . . , , , . ', , veff , o ui(y).i
, ,
k :
() u(y) y.
() veff , u(y) W l h S tt th it Welch Satterthwaite:
4 ( )u yv (6 8)41
( )eff N ii i
vu y
v
(6.8)
ui(y) (i=1,2,,N), , y xi, , i, ui(y).
, ,
u(q) , i=n-1. . ,
. , , ,
, , - , , +, , , . , u(x ) u(xi) vi .
( ) k 5 2 () k 5.2. t .
:
Y = y U (6 9)
, ,
Y y U (6.9)
5.2: kpv (kpv ,+ kpv),pv k 5. : kpv ( kpv , kpv), pv,k v (student)
pv,k
68.27 90 95 95.45 99 99.73 68.27 90 95 95.45 99 99.731 1.84 6.31 12.71 13.97 63.66 235.802 1.32 2.92 4.30 4.53 9.92 19.213 1.20 2.35 3.18 3.31 5.84 9.224 1.14 2.13 2.78 2.87 4.60 6.625 1.11 2.02 2.57 2.65 4.03 5.516 1.09 1.94 2.45 2.52 3.71 4.907 1.08 1.89 2.36 2.43 3.50 4.538 1.07 1.86 2.31 2.37 3.36 4.289 1.06 1.83 2.26 2.32 3.25 4.09
10 1 05 1 81 2 23 2 28 3 17 3 9610 1.05 1.81 2.23 2.28 3.17 3.9611 1.05 1.80 2.20 2.25 3.11 3.8512 1.04 1.78 2.18 2.23 3.05 3.7613 1.04 1.77 2.16 2.21 3.01 3.6914 1.04 1.76 2.14 2.20 2.98 3.6415 1 03 1 75 2 13 2 18 2 95 3 5915 1.03 1.75 2.13 2.18 2.95 3.5916 1.03 1.75 2.12 2.17 2.92 3.5417 1.03 1.74 2.11 2.16 2.90 3.5118 1.03 1.73 2.10 2.15 2.88 3.4819 1.03 1.73 2.09 2.14 2.86 3.4520 1 03 1 72 2 09 2 13 2 85 3 4220 1.03 1.72 2.09 2.13 2.85 3.4225 1.02 1.71 2.06 2.11 2.79 3.3330 1.02 1.70 2.04 2.09 2.75 3.2735 1.01 1.70 2.03 2.07 2.72 3.2340 1.01 1.68 2.02 2.06 2.70 3.2045 1 01 1 68 2 01 2 06 2 69 3 18
, ,
45 1.01 1.68 2.01 2.06 2.69 3.1850 1.01 1.68 2.01 2.05 2.68 3.16
100 1.005 1.660 1.984 2.025 2.626 3.077 1.000 1.645 1.960 2.000 2.576 3.000
, . , , , , . . , . . , (6.3) . , (6.1) Xi . , :
( )Y f X X X X X X (6 10), ,
1 1 2 2( , ,..., )Y f X X X X X X (6.10)
Taylor N . Taylor :
2 2 1 1
2 1
( ) ( )( ) ( ) ...1! 2! ( 1)!
n n
nn
df X d f X d f Xf X X f X RdX dX dX n
(6.11)
N (6.11)
1 2 1 21 2
( , ,..., , ) ( ... )Ndf df dfY f X X XdX dX dX
2 2 22 2 2
1 22 2 21 2
3
1 [ ( ) ( ) ... ( ) ] ...2!
1
NN
f f fX X XX X X
f
3 3
131
1 [ ( ) ...] ...3!
f XX (6.12)
, ,
. , , Y :
max
N
jfY Y Y X
X
(6.13)
1j
j jX , DC , :
P=I2R (6.14)
:
Pmax=2IR+I2R (6.15)
max( ) 2P I R
P I R (6.16)
, ,
P I R