Mediciones Electrónicas
PROPAGACIÓN DEL ERROR ALUMNO: CÓDIGO: PROFESOR: ING. JOSE MESTAS RAMOS 2015-1
Propagación del error Introducción: La propaga!"# $% %rror%& %& $% a'g(#a )a#%ra !#%*!+a,'% p%ro &! &% p(%$%# o#+ro'ar )!#!)!ar/ &!g(!%#$o a'g(#a& pa(+a& para o)o 'a orr%+a +o)a $% 'a a#+!$a$ $% )%$!$a& #%%&ar!a&/ 'a %'%!"# a$%(a$a $%' %(!po para %' '!)a r%g!"#. E!&+%# a'g(#a& +#!a& para a'('ar 'a propaga!"# $%' %rror %# 'a &()a/ $!*!&!"# po+%#!a!"# %# )%$!!o#%&/ 'a& (a'%& &%r3# +%)a $% %&+% +ra,a4o.
Fundamento Teórico: La propaga!"# $% %rror o +a),!# ''a)a$o !#%r+!$(),r% %&+3 a&o!a$a a 'o& &!g(!%#+%& a+or%&: • • • •
La #a+(ra'%a $% 'a )ag#!+($ (% &% )!$% E' !#&+r()%#+o $% )%$!!"# E' o,&%r*a$or La& o#$!!o#%& %+%r#a&
Ca$a (#o $% %&+o& a+or%& o#&+!+(% por &%para$o (#a (%#+% $% !#%r+!$(),r% o#+r!,(% %# )aor o )%#or gra$o a 'a !#%r+!$(),r% +o+a' $% 'a )%$!$a. La +ar%a $% $%+%+ar %*a'(ar 'a& !#%r+!$(),r%& #o %& &!)p'% % !)p'!a o#o%r $!*%r&o& a&p%+o& $% 'a )%$!!"#. E# pr!#!p!o/ %& po&!,'% 'a&!6ar 'a& (%#+%& $% !#%r+!$(),r%& %# $o& o#4(#+o& ,!%# $!%r%#!a$o&/ 'a& (% &% $%,%# a: • •
Error%& a!$%#+a'%& o a'%a+or!o& Error%& &!&+%)3+!o&
Incertidumbre en medidas reproducibles
2
C(a#$o a' r%a'!ar (#a &%r!% $% )%$!$a& $% (#a )!&)a )ag#!+($ &% o,+!%#%# 'o& )!&)o& r%&('+a$o&/ #o &% p(%$% o#'(!r (% 'a !#%r+!$(),r% &%a %ro7 'o (% &(%$% %& (% 'o& %rror%& (%$a# o('+o& a (% &o# )%#or%& (% 'a !#%r+!$(),r% a&o!a$a a' apara+o $% )%$!!"#. E# %&+% a&o/ p(%$% %&+a,'%%r&% (# r!+%r!o &!)p'% 8+!': (a#$o 'a& )%$!$a& &o# r%pro$(!,'%&/ &% a&!g#a (#a !#%r+!$(),r% !g(a' a 'a )!+a$ $% 'a $!*!&!"# )3& p%(%9a $%' !#&+r()%#+o/ 'a (a' &% o#o% o)o r%&o'(!"#. Por %4%)p'o/ a' )%$!r o# (# !#&+r()%#+o gra$(a$o %# )!'!'!+ro& r%p%+!$a& *%%& %' *o'()%# $% (# r%!p!%#+% &% o,+!%#% &!%)pr% ;.0 )'/ 'a !#%r+!$(),r% &%r3 0.5 )'. Lo (% &!g#!6a (% 'a )%$!!"# %&+3 %#+r% <.5 a ;.5 )'/ a &+% &% '% o#o% o)o !#+%r*a'o $% o#6a#a $% 'a )%$!!"# &( +a)a9o %& %' $o,'% $% 'a !#%r+!$(),r%. E&+o g%#%ra')%#+% &% ap'!a (a#$o &% +ra+a $% apara+o& $% )%$!!"# +a'%& o)o r%g'a&/ +ra#&por+a$or%&/ ,a'a#a&/ pro,%+a&/ )a#")%+ro&/ +%r)")%+ro&/ %+
Medidas de tendencia central La )%$!$a )3& o)8# $% 'a +%#$%#!a %#+ra' $% (#a )(%&+ra o o#4(#+o $% )%$!!o#%& %&+3 $a$a por %' pro)%$!o o )%$!a ar!+)+!a. S!# %),argo/ a'g(#a& *%%& %&+% *a'or #o ,a&+a %& #%%&ar!o a'('ar o+ra& *ar!a,'%& %&+a$=&+!a& (% a($%# a a#a'!ar %' r%&('+a$o $% (#a )%$!!"#. E&+a& *ar!a,'%& %&+a$=&+!a& &o# 'a )%$!a 'a )o$a.
El promedio $% (#a )(%&+ra o o#4(#+o $% )%$!!o#%& %&+3 $a$o por:
´= X
( x1 + x 2 + x 3+ … + x n) N
=
∑ x
i
N
>
La mediana %& %' *a'or $% 'a )%$!!"# (% $!*!$% 'a )(%&+ra %# $o& )!+a$%&: (#a )!+a$ &o# a(%''a& )%$!!o#%& )%#or%& a 'a )%$!a#a 'a o+ra )!+a$ %& %' o#4(#+o $% )%$!!o#%& )aor%& (% 'a )%$!a#a. S(po#!%#$o (% 'a )(%&+ra %&+3 or$%#a$a $% )%#or a )aor/ 'a )%$!a#a %&+3 $a$o por: mediana = X N +1 2
C(a#$o 'a )(%&+ra +!%#% (# #8)%ro !)par $% %'%)%#+o&. S! 'a )(%&+ra +!%#% (# #8)%ro par $% )%$!!o#%&/ 'a )%$!a#a %&+3 $a$a por X N + X N mediana =
2
2
+1
2
La moda %& 'a )%$!!"# (% o(rr% o# )aor r%(%#!a. E# (# o#4(#+o $% )%$!!o#%& p(%$% ?a,%r )3& $% (#a )o$a. C(a#$o %' o#4(#+o $% $a+o& %& &!)+r!o/ %' pro)%$!o 'a )%$!a#a o!#!$%#/ &= a$%)3&/ 'o& $a+o& +!%#%# (#a &o'a )o$a/ &% $!% (% 'o& $a+o& &o# (#!)o$a'%& 'a )%$!a#a/ 'a )o$a %' pro)%$!o +!%#%# %' )!&)o *a'or. C(a#$o 'a )%$!a#a #o Co!#!$% o# %' pro)%$!o/ 'o& $a+o& %&+3# arga$o& o &%&ga$o& ?a!a 'a !(!%r$a o ?a!a 'a $%r%?a $%' pro)%$!o.
Medidas de dispersión La +%#$%#!a %#+ra' #o %& &(6!%#+% para $%+%r)!#ar %' r%&('+a$o $% (#a )%$!!"#. Por %4%)p'o/ 'o& &!g(!%#+%& o#4(#+o& $% $a+o& Muestra 1: 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50
Muestra 2:20,25,30,35,40,45,45,50,55,60,65,70
T!%#%# %' )!&)o *a'or para %' pro)%$!o/ 'a )%$!a#a 'a )o$a. S!# %),argo 'o& $a+o& %# 'a )(%&+ra 2 %&+3# )3& $!&p%r&o& (% %# 'a )(%&+ra 1. La $!&p%r&!"# $% (# o#4(#+o $% )%$!!o#%& &% p(%$% )%$!r $% $!%r%#+%& )a#%ra&. Lo& !#$!a$or%& )3& (+!'!a$o& para r%pr%&%#+ar 'a $!&p%r&!"# $% (# o#4(#+o $% $a+o& &o# 'a $%&*!a!"# )%$!a 'a $%&*!a!"# %&+3#$ar.
La desviación media $% (#a )(%&+ra %&+3 $a$a por
La desviación estndar $% 'a )(%&+ra %&+3 $a$a por:
C(a#$o &% o,+!%#% (#a )%$!!"# $% (#a )(%&+ra $% $a+o&/ %' *a'or %#+ra' $% 'a )%$!!"# &% r%pr%&%#+a o# %' pro)%$!o $% 'o& $a+o& %' %rror o !#%r+!$(),r% &% r%pr%&%#+a o# 'a $%&*!a!"# )%$!a (a#$o &% +ra+a $% 'a,ora+or!o& !#+ro$(+or!o&/ o# 'a $%&*!a!"# %&+3#$ar para (# +ra+a)!%#+o+o $% $a+o& )3& r!g(ro&o.
Re!la para e"presar una medida To$a )%$!$a a &%a r%pro$(!,'% o #o/ $%,% $% !r &%g(!$a por 'a (#!$a$ $% 'a *ar!a,'% (% &% )!$% &% %pr%&a $% 'a or)a
5
Mediciones directas e indirectas A 'a& a#+!$a$%& (% &% o,+!%#%# (+!'!a#$o (# !#&+r()%#+o $% )%$!$a &% '%& $%#o)!#a )%$!!o#%& $!r%+a&/ a 'a& )%$!!o#%& (% &% a'('a# a par+!r $% )%$!!o#%& $!r%+a& &% '%& $%#o)!#a )%$!!o#%& !#$!r%+a&. Por %4%)p'o %' *o'()%# (% o(pa (# '=(!$o %& (#a )%$!!"# $!r%+a &! &% )!$% o# (#a pro,%+a gra$(a$a/ &% o#&!$%ra o)o (#a )%$!!"# !#$!r%+a &! &% o,+!%#% $% 'a )%$!!"# $% 'a& $!)%#&!o#%& $%' r%!p!%#+% (% 'o o#+!%#%.
Propa!ación de la incertidumbre La& )%$!!o#%& $!r%+a&/ (% p(%$%# &%r r%pro$(!,'%& #o r%pro$(!,'%&/ +!%#%# a&o!a$a (#a !#%r+!$(),r%. La& )%$!!o#%& !#$!r%+a& +!%#%# a&o!a$a (#a !#%r+!$(),r% (% &% or!g!#a $% 'a propaga!"# $% 'a !#%r+!$(),r% $% 'a& )%$!!o#%& $!r%+a& $% 'a& (% (%ro# $%r!*a$a&.
Propa!ación de la incertidumbre en la suma # di$erencia S! 'a& )ag#!+($%& r &% )!$%# o# !#%r+!$(),r% @ @r r%&p%+!*a)%#+% &! &% (+!'!a# para a'('ar 'a $!%r%#!a B-r %#+o#%& 'a !#%r+!$(),r% a&o!a$a a 'a *ar!a,'% w %& 'a &()a $% 'a& !#%r+!$(),r%& a&o!a$a& a a r/ %& $%!r
Lo )!&)o %& !%r+o (a#$o &% a'('a 'a &()a Br
E&+% r%&('+a$o #o& !#$!a (% (a#$o &% o),!#a# $o& *ar!a,'%& )%$!a#+% (#a &()a o (#a r%&+a/ 'a& !#%r+!$(),r%& &!%)pr% &% &()a#. E4%)p'o/ 2 0.01 1.< 0.1 B >.<> 0.11.
Propa!ación de errores en el producto # en el cociente S! 'a& a#+!$a$%& r &% ?a# )%$!$o o# (#a !#%r+!$(),r% @ @r r%&p%+!*a)%#+% &! 'o& *a'or%& $% r &% (+!'!a# para a'('ar %' pro$(+o Br " %' o!%#+% BHr %#+o#%& 'a !#%r+!$(),r% a&o!a$a a / %&+3 $a$a por
E4%)p'o/ o#&!$r%&% 'a )('+!p'!a!"#
Co#&!$r%&% a?ora %' o!%#+%
E# %&+o& 8'+!)o& r%&('+a$o& p(%$%# *%r&% !ra& (% #o $a# (#a !#or)a!"# 8+!' %& #%%&ar!o (# r!+%r!o para %'!)!#ar'a&.
Propa!ación de errores en una potencia S%a
x = x ∓ ∆ x
n / q = x = x . x … . x
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Ap'!a#$o 'a r%g'a para %' pro$(+o: ∆ q ∆ x ∆ x ∆x ∆ x + + …+ =n . ≈ |q| | x| | x| | x| | x|
E' %rror r%'a+!*o $% (#a po+%#!a %& %' pro$(+o $% 'a po+%#!a por %' %rror r%'a+!*o $% 'a )ag#!+($: ∆q ∆ x = n. |q| | x|
Bibliografía:
.$!+(+or.o) .r%(r&o&+!.%$(a!o#.%& .$!&r(+a'a&)a+%)a+!a&.o)
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