Download pdf - Metrologia Enotita 8-12

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u

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m

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0.27 0.28 0.29 0.30 0.319

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time series (sec)0.4

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Velocity fluctuations (m/s)

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=f (1, 2 , 3 ,, N ) (6.1)

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, 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). .

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( (, , ), . . Q. n , Q , qj

q qj

(j=1, 2, ... , n)

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(experimental variance) s2(q) , :

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s q q qn

(experimental standard deviation). (experimentalq (experimentalstandard deviation of the mean), :

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1:

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x :

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u x

|+|=|-|=,:

2 21( )u x

, ,

( )3i

u x

IV IV. . , , , .: , , , + , , 100% + 100%. , , , . (+ + )/2

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VI ' VI. , V. , 90, 95 99 % . , , . , , . , u(xi), xi . , , 1.64, 1.96 2.58.

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: : :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