Fgure : (a) precse but naccurate data. (b) accurate but mprecse data. [] Fg..(p.) Fgure : Accuracy vs Precson []p.0-0 () 05. m 0.35 m 05. ± 0.35m 05.

Size: px

Start display at page:

Download "Fgure : (a) precse but naccurate data. (b) accurate but mprecse data. [] Fg..(p.) Fgure : Accuracy vs Precson []p.0-0 () 05. m 0.35 m 05. ± 0.35m 05."

さやにばし
5 years ago
Views:

1 9 3 Error Analyss [] Danel C. Harrs, Quanttatve Chemcal Analyss, Chap.3-5. th Ed [] J. R. Taylor (, 000. An Introducton to Error Analyss, nd Ed. 997 Unv. Sc. Books) [3] 00 ( [] 973 Posson [5] 99 [] P. R. Bevngton and D. K. Robnson, Data reducton and error analyss for the physcal scences, WCB/McGraw-Hll, nd Ed. 99. [7]. C. Barford accuracy vs precson accuracy : Measure of how close the result of an experemt comes to the true value. precson : Measure of how carefully the result s determned wthout reference to any true value. (systematc errors) : random errors) : x (best estmate) x best (error) δx( 0) x(measured) = x best ± δx () relatve uncertanty = δx x best () and/or I-7

2 Fgure : (a) precse but naccurate data. (b) accurate but mprecse data. [] Fg..(p.) Fgure : Accuracy vs Precson []p.0-0 () 05. m 0.35 m 05. ± 0.35m 05. ± 0.m () 05.3 ± 0.0m 05. ± 0.0m, 05.3 ± m 05 ± m. x.xxx) 0 y , , , , (.35 ± 0.000) (.33 ± 0.000) = 3 0 ± 3 0 ± ± BSE (sample) (parent populaton) samplng) I-75

推測 statstcal nference 母集団 p a r e n t p o p u la t o n 標本

s(sd:standard devaton) s = (x x) (5) = / 0 + 50 x s,

7 µ, σ x(= x /) G µ,σ/ ( x) ( ) G µ,σ (x) G µ,σ (x) = πσ e (x

3 推測 statstcal nference 母集団 p a r e n t p o p u la t o n 標本 sample 抽出 s a m p l n g x x, x,,,, x (average or mean)[ ] x, ( x = x + x x (sample varance)s 3 = x (3) = s = (x x) () = s(sd:standard devaton) s = (x x) (5) = / x s, µ(populaton mean) ( σ(popolaton varance) ( ) x f ( x) f, µ, σ x(= x /) G µ,σ/ ( x) ( ) G µ,σ (x) G µ,σ (x) = πσ e (x µ) /σ () Fg. 3 G µ,σ (x) 50 0 top %, 5 top.5%, 70 top.3%, 0 top0.5% / ϵ ϵ ϵ n r n r (n r)ϵ n C r (/) r (/) n r n (n r)ϵ 5 I-7

4 X=0, σ=0 G X,σ (x ) 0. X=, σ= x G 0, (x ) x Fgure 3: µ ± σ µ x µ t x erf(t) [] 5 3 p.5) 9 (x, y) p(x, y) x y h(x, y) h h = dxdyh(x, y)p(x, y) (7) x y p(x, y) = f (x)g(y) () f (x) = dyp(x, y) (9) g(y) = dxp(x, y) (0),h(x, y) = x + y x y x + y = dxdy(x + y) f (x)g(y) = = x + y dxx f (x) dyg(y) } {{ } = + dyyg(y) dx f (x) } {{ } = (), x + x + x x = x + x + x x () x = µ( ) x ( x x ) x x + x + x x = µ = µ (3) I-77

5 h(x, y) = [(x + y) x + y ] x y [(x + y) x + y ] = dxdy[(x + y) x + y ] f (x)g(y) = dxx f (x) + dyy g(y) + dxx f (x) dyyg(y) x + y + x + y = x + y + x y x y x y = x x + y y () σ x+y = σ x + σ y σ x = dx(x x ) f (x) = x x, σ y = (5) dy(y y ) g(y) = y y () [( x + x + x x x x = σ (, ) x + x + x x ] = x x x x (7) σ x = σ = σ () 0 x,,,, z δx,,,, δz q(x,,,,, z) x,,,, z q δq = ( q ) x δx ( ) q z δz (9) δq q x δx q z δz (0) q(x; δx, y + δy) q(x, y) + ( ) q δx + x ( ) q δy +... () y 0 ( q x) δx 3 x,y x + y σ x + σ y () 500 ± 0 g, 00 ± 0 g g ± g [, p9] q = x+y = 00, q/ x =, q/ y =, δq = (δx) + (δy) = = 0 (. ), Ans. 00 ± 0g ().0 ± 0.5 g g ± g 00 ± 0 g.0 ± 0.5 g 0g (3) (7±) k, k =Constant : q = kx, q/ x = k, δq = kδx, Ans. 7k ± k. () (5±) (±), (5) (0±)/(0±) x ± δx σ ( x (confdence level).3%, 90%, 95% (confdence nterval),.3%, 90%, 95% x z σ µ x + z µ, z = (.3%), z =.5(90%), z =.90(95%) () [5, -9] I-7

6 x(measured) = x ± z σ, z = (.3%), z =.5(90%), z =.90(95%) (3) z Fg.3 95% 90%.3% ( / confdence nterval) zσ x / (Fg.) Fg / / = Fgure : vs / σ s = (/) (x x), σ ( s = (/) (x x) s s s s = (x x) = (x µ + µ x) = [(x µ) + (µ x)(x µ)] + (µ x) = (x µ) (µ x) () x, s µ, σ s s = (x µ) (µ x) = σ σ = σ (5) x = σ / µ σ s u = s = (x x) = = (x x) () = u (unbased varance) 7 7 = (x x) x = (/) = x (x x) I-79

7 G µ,σ u T = ( x µ)/(u/ ) Student t m(m Student t f m (t) f m (t) = m+ Γ( ) ( (m+)/ mπγ( m ) + m) t (7) Γ Γ() =, Γ(n + ) = n!, Γ(/) = π, Γ(n + /) = (n)!/[ n n!] π Fg.5 Fg t (.3%) f (t).3% m m, f m (t) G 0, (t) Gaussan G(0,) m= m= m= f m (t ) t (.3%) t Fgure 5: Student t m m, f m (t) G 0, (t) x, u µ z% x t (z%) u µ x + t (z%) u () {x, x, x 3,..., x } 9 x(measured) = x ± t (z%) u (9) x = x, u = = (x x) t (z%) Eq.(7) 0 Table = W. S. Gosset Gosset ( Gosset Student The probable error of a mean, Bometrka, 90, -5. (7) χ ( F t, (7) 9 = 0 Appendx f m (t) Fortran I-0

8 .3% ±.u/ 5 = ±0.50u 95% ±.303u/ 3 = ±.u t (z%)/ Fg. Fg. Table : Student t t (z%) t (z%), z(%): (-) ().3 % 90 % 95 % 99 % t / / 3 t - (99%) t - (95%) t - (90%) t - (.3%) 0 0 Fgure : vs t (z%)/ Table σ.3%, 90%, 95% Table : % σ χ ( s 95% I-

9 Fgure 7: 95% student t ( (3) z =.90(95%), σ = u Fgure : 95% student t (3) z =.90(95%), σ = u s χ (0.05) s σ χ (0.975), s = (x x) (30) χ (0.05), χ (0.975) χ (0.05) =., χ (0.975) = 0. 3 ) A 35, 30, 35, 35, 357 g 30 g t T = ( x µ)/u/ 95% µ = 5, x = 35.(), µ = 30, u = [(35 35.()) + (30 35.()) +..]/(5 ) = 3.9, T = ( x µ)/u/ = 3. 5% 90% <.3 0.5% 3. >.0 ) S Table3 John Wllam Strutt 90 I-

10 Table 3: mass of constant volumes of gas (at constant temperature and pressure) solated by romovng oxygen from ar or generated by chemcal decomposton from ntrogen compounds.[, p70] From ar / g From chemcal decomposton /g G µa,σ A, G µb,σ B A,B σ A, σ B A, B x A, x B, u A, u B T AB ( A + B ) t T AB = ( x A x B ) (µ A µ B ) ( ), u A + AB = ( A )u A + ( B )u B = B u ( A ) + ( B ) AB A (x,a x A ) + B j (x j,b x B ) A + B (3) A,B T AB 95% n A = 7, x A =.30, u A = 0.000(3), B =, x B =.997, u B = u AB = 0.000, T AB = ( )/0.000/ (/7) + (/) = 0. 3(=7+-) % 5.5 : [R. B. Dean and W. J. Dxon, Anal. Chem., 3, 3 (95).] X = (max or mn) - / X Q 90% Table : 90% , -., -.7, -., -., -.3, -., -0.,.5, -. 0 Q X 3. (.) /.5 ( 3.) = 0.,.5 ( 0.) /.5 ( 3.) = Q.5 Q.5 -.? Fgure 9: R. A. Mllkan Regener Thompson (Cambrdge Unv.) 0 9 H. Fletcher F. Ehrenhaft 990 I-3

11 5 Appendx: Fortran program to calculate f m (t) c3579 c Student t dstrbuton mplct real* (a-h,o-z) p=acos(-.0d0) do m=, 00 f (mod(m,) == 0) then c Gamma (m/) routne for even m f (m == ) then gamm=.0d0 else gamm=.0d0 do =m/-,, - gamm=gamm*real() endf c Gamma ((m+)/) routne for even m n=m/ n=*n facn=.0d0 do =n,, - facn=facn*real() facn=.0d0 do =n,, - facn=facn*real() gamm=facn/.0d0**n/facn*sqrt(p) else c odd m c Gamma ((m+)/) routne for odd m f (m == ) then gamm=.0d0 else gamm=.0d0 do =(m+)/-,, - gamm=gamm*real() endf c Gamma (m/) routne for odd m f (m == ) then gamm=sqrt(p) else n=(m-)/ n=*n facn=.0d0 do =n,, - facn=facn*real() facn=.0d0 do =n,, - facn=facn*real() gamm=facn/.0d0**n/facn*sqrt(p) endf endf c t dep. calculaton do j=, t=-0.0d0+real(j-)/ d0*0.0d0 fmt=gamm/gamm/sqrt(p*real(m))* & (.0d0+t**/real(m))**(-(real(m)+.0d0)/.0d0) wrte (,*) t,fmt,m end F (χ ( ( M.Y. ) I-

統計的データ解析

統計的データ解析 ds45 xspec qdp guplot oocalc (Error) gg (Radom Error)(Systematc Error) x, x,, x ( x, x,..., x x = s x x µ = lm = σ µ x x = lm ( x ) = σ ( ) = - x = js j ( ) = j= ( j) x x + xj x + xj j x + xj = ( x x