NORMA TÉCNICA COLOMBIANA
GTC 51 1997-11-26
GUÍA PARA LA EXPRESIÓN DE INCERTIDUMBRE EN LAS MEDICIONES
E:
GUID GUIDE E TO TO THE THE EXPR EXPRES ESSI SION ON OF UN UNCE CERT RTAI AINT NTY Y IN IN MEASUREMENT.
CORRESPONDEN ENC CIA:
esta esta nor norma es es (EQV) a la Gui Guidde to to the Expression of Uncertainty in Measurement publicada por la (ISO).
DESCRIPTORES:
control me metrológico; ca calibración; instrumento de medida; incertidumbre; productos calibrados; metrología.
I.C.S.: 17.020.00 Editada por el Instituto Colombiano de Normas Técnicas y Certificación (ICONTEC) Apar Aparta tado do 1423 142377 Bog Bogot otá, á, D.C. D.C. - Tel Tel.. 607 60788 8888 88 - Fax Fax 2221 222143 4355 Prohibida su reproducción
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PRÓLOGO El Instituto Colombiano de Normas Técnicas y Certificación, ICONTEC, ICONTEC, es el organismo nacional de normalización, según el Decreto 2269 de 1993. ICONTEC es una entidad de carácter privado, sin ánimo de lucro, cuya Misión es fundamental para brindar brindar soporte y desarrollo desarrollo al productor productor y protección al consumidor, consumidor, Colabora con el el sector gubernamental y apoya al sector privado privado del país, para lograr lograr ventajas competitivas competitivas en los mercados interno y externo. La representación de todos los sectores involucrados en el proceso de Normalización Técnica está garantizada por los Comités Técnicos y el período de Consulta Pública, este último caracterizado por la participación del público en general. La GTC 51 fue ratificada por el Consejo Directivo el 1997-11-26. Esta guía está sujeta a ser actualizada permanentemente con el objeto de que responda en todo momento a las necesidades y exigencias actuales. A continuación se relacionan las empresas que colaboraron en el estudio de esta guía a través de su participación en el Comité C omité Técnico 000002 Metrología. ASEA BROWN BOVERI CERAMITA S.A. COLCERÁMICA S.A. COLGATE PALMOLIVE COLTAVIRA ECSI S.A. ELECTROPORCELANAS GAMMA S.A. EQUIPOS Y CONTROLES INDUSTRIALES FÁBRICA NACIONAL DE MUÑECOS GRASAS S.A. GRIVEL S.A. INGENIERÍA DE SERVICIOS
LUMINEX S.A. METRÓN LTDA. NACIONAL DE CHOCOLATES PROCABLES PRUGEN LTDA. SCHLAGE LOCK DE COLOMBIA S.A. SHELL COLOMBIA S.A. SUPERINTENDENCIA DE INDUSTRIA Y COMERCIO THERMOMETRIC UNIVERSIDAD JAVERIANA
Además de las anteriores, en Consulta Pública el Proyecto se puso a consideración de las siguientes empresas. AROTEQ COL. ASOCRETO BÁSCULAS PROMETÁLICOS S.A. CHICLE ADAMS S.A. COATS CADENA S.A. COLOMBIANA UNIVERSAL DE PAPELES
GRIVAL S.A. ICOLLANTAS S.A. INCAUCA S.A. INCOLBESTOS INDUSTRIAS E INVERSIONES SAMPER POLIGAR
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ICONTEC cuenta con un Centro de Información Información que pone a disposición de los interesados interesados normas internacionales, regionales y nacionales. DIRECCIÓN DE NORMALIZACIÓN
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GUÍA TÉCNICA COLOMBIANA
GTC 51
TABLA DE CONTENIDO
PREFACIO 0.
INTRODUCCIÓN
1.
ALCANCE
2.
DEFINICIONES
2.1 2.1
TÉRM TÉ RMIN INOS OS GEN GENER ERAL ALES ES SOB SOBRE RE MET METRO ROLO LOGÍ GÍA A
2.2
EL TÉ TÉR RMIN MINO “IN “INC CER ERTI TID DUMBRE”
2.3 2.3
TÉRM TÉ RMIN INOS OS ES ESPE PECÍ CÍFI FICO COS S PAR PARA ESTA ESTA GUÍA GUÍA
3.
CONCEPTOS BÁSICOS
3.1
MEDICIÓN
3.2 3.2
ERRO ER RORE RES, S, EF EFEC ECTO TOS S Y COR CORRE RECC CCIO IONE NES S
3.3
INCERTIDUMBRE
3.4 3.4
CON CONSIDE SIDERA RACI CION ONES ES PR PRÁ ÁCT CTIC ICAS AS
4.
EVA EV ALU LUA ACIÓ CIÓN DE DE LA LA IN INCE CERT RTID IDUM UMB BRE ES ESTÁ TÁND NDAR AR
4.1
MODELOS DE LA MEDICIÓN
4.2
EVALU EVA LUAC ACIÓN IÓN DE LA INC INCER ERTID TIDUM UMBR BRE E ESTÁ ESTÁND NDAR AR TIPO TIPO A
4.3
EVALU EVA LUAC ACIÓN IÓN DE LA INC INCER ERTID TIDUM UMBR BRE E ESTÁ ESTÁND NDAR AR TIPO TIPO B ÓN GRÁ GRÁ
ÓN D
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GUÍA TÉCNICA COLOMBIANA
GTC 51
5.2 5.2
ARGU AR GUME MEN NTOS TOS CORR CORREL ELAC ACIO IONA NADO DOS S
6.
DETE DE TERM RMIN INAC ACIÓ IÓN N DE DE LA LA IINC NCER ERTI TIDU DUMB MBRE RE EX EXPA PAND NDID IDA A
6.1
INTRODUCCIÓN
6.2
INC INCERTID TIDUMBR MBRE EX EXPA PAN NDIDA IDA
6.3 6.3
ELEC EL ECCI CIÓN ÓN DE DEL L FAC FACTO TOR R DE DE COB COBER ERTU TURA RA
7.
INFORME DE DE LA LA IN INCERTIDUMBRE
7.1
GUÍA GENERAL
7.2
GUÍA ESPECÍFICA
8.
RESUME RESU MEN N DEL DEL PR PROC OCED EDIM IMIE IENT NTO O PAR PARA A LA LA EVA EVALU LUAC ACIÓ IÓN N Y EX EXPR PRES ESIÓ IÓN N DE LA INCERTIDUMBRE ANEXOS
A.
RECO RE COME MEND NDAC ACIO IONE NES S DEL DEL GRUP GRUPO O DE DE TRA TRABA BAJO JO Y DEL DEL CIPM CIPM
A.1
RECO RE COME MEND NDAC ACIÓ IÓN N INCINC-11 (198 (1980) 0)
A.2
RECO RE COME MEND NDAC ACIÓ IÓN N 1 (CI(CI-19 1981 81))
A.3
RECO RE COME MEND NDAC ACIÓ IÓN N 1 (CI(CI-19 1986 86))
B.
TÉR TÉ RMINO INOS MET ETR ROLÓG LÓGICO ICOS GEN GENER ERA ALES
B.1
ORIGEN IGEN DE LA LAS DEF DEFIN INIC ICIO ION NES
B.2
DEFINICIONES TÉ
DÍ
BÁ
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GUÍA TÉCNICA COLOMBIANA
GTC 51
D.
VALO VA LOR R “VE “VERD RDAD ADER ERO” O”,, ERR ERROR OR E INC INCER ERTI TID DUMBR UMBRE E
D.1
EL MEN ENSU SUR RANDO (M (MED EDIB IBL LE)
D.2
LA MAGNITU ITUD REALIZADA
D.3
EL VA VALOR LOR “VERD “VERDAD ADERO ERO”” Y EL VA VALOR LOR “COR “CORREG REGIDO IDO””
D.4
EL ERROR
D.5
INCERTIDUMBRE
D.6 D.6
REPR RE PRES ESEN ENTA TACI CION ONES ES GRÁ RÁFI FICA CAS S
E.
MOTI MOTIVA VACI CION ONES ES Y BAS BASES ES PAR PARA AL LA AR REC ECOM OMEN ENDA DACI CIÓN ÓN INCINC-11 (198 (1980) 0)
E.1 E.1
“SEG “SEGUR URO” O”,, “AL “ALEA EATO TORI RIO” O” Y “SI “SIST STEM EMÁT ÁTIC ICO” O”
E.2
JUSTIFIC JUSTI FICAC ACIÓ IÓN N PA PARA RA LA LAS S EVA EVALU LUAC ACIO IONE NES S REA REALES LES DE LA INCERTIDUMBRE
E.3
JUST JU STIFI IFICA CACI CIÓN ÓN PAR PARA A EL TRA TRATA TAMI MIEN ENTO TO IDÉN IDÉNTIC TICO O DE TOD TODAS AS L LAS AS COMPONENTES DE LA INCERTIDUMBRE
E.4
LA D DESV ESVIA IACI CIÓN ÓN ESTÁ ESTÁND NDAR AR COM COMO O UNA UNA MEDID MEDIDA A DE LA INC INCER ERTID TIDUM UMBR BRE E
E.5
UNA COMP UNA COMPAR ARAC ACIÓN IÓN DE DOS DOS PUN PUNTO TOS S DE VISTA VISTA EVA EVALU LUAC ACIÓN IÓN Y EXPRESIÓN DE LA INCERTIDUMBRE
F.
GUÍA GUÍA PR PRÁC ÁCTI TICA CA PA PARA RA LA EV EVAL ALUA UACI CIÓN ÓN DE LA LAS S COM COMPO PONE NENT NTES ES DE LA INCERTIDUMBRE
F.1
EVALUAC EVALU ACIÓ IÓN N DE DE LAS LAS COMP COMPON ONEN ENTES TES OBTEN OBTENID IDAS AS A PA PART RTIR IR DE DE OBSERVACIONES REPETIDAS: EVALUACIÓN DE LA INCERTIDUMBRE ESTÁNDAR TIPO A
F.2
COMPO COMPONE NENT NTES ES EVA EVALU LUAD ADAS AS MEDIAN MEDIANTE TE OTRO OTROS S MÉTOD MÉTODOS. OS. EVALUACIÓN DE LA INCERTIDUMBRE ESTÁNDAR TIPO B
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GUÍA TÉCNICA COLOMBIANA
GTC 51
G.3 G.3
LA DIS DISTR TRIB IBUC UCIÓ IÓN N T Y LOS LOS GRAD GRADOS OS DE DE LIBE LIBERT RTAD AD
G.4 G.4
GRAD GRADOS OS DE LIBE LIBERT RTAD AD EF EFEC ECTI TIVO VOS S
G.5 G.5
OTRAS CO CONSID SIDER ERA ACION IONES
G.6 G.6
RES ESU UMEN Y CON CONCLU LUSI SIO ONES
H.
EJEMPLOS
H.1
CALIB CA LIBRA RACIO CIONE NES S DE BL BLOQ OQUE UES S CA CALIB LIBRE RE PAR PARAL ALELO ELO
H.2
MEDIC MEDICION IONES ES SIMU SIMULT LTÁN ÁNEA EA DE DE RESIS RESISTE TENC NCIA IA Y REA REACT CTAN ANCI CIA A
H.3 H.3
CALI CA LIBR BRAC ACIÓ IÓN N DE DE UN UN TER TERMÓ MÓME METR TRO O
H.4 H.4
MED MEDICIÓ ICIÓN N DE RA RAD DIAC IACTIVI TIVIDA DAD D
H.5
ANÁLISIS ISIS DE VARIAN IANZA
H.6
MEDIC MEDICION IONES ES EN UN UNA A ESCA ESCALA LA DE RE REFER FEREN ENCIA CIA:: DUR DUREZA EZA
J.
GLOSA LOSAR RIO DE LO LOS SÍM SÍMB BOLOS LOS PR PRINC INCIPA IPALES
K.
BIBLIOGRAFÍA
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GUÍA TÉCNICA COLOMBIANA
GTC 51
PREFACIO
En 1978, al reconocerse la falta de consenso alrededor de la expresión de incertidumbres en las mediciones, la más alta autoridad en metrología en el mundo, el Comité lnternational des Poids et Mesures (CIPM) (CIPM) sugirió al Bureau Bureau International International des Poids et Mesures (BIPM) (BIPM) que abordara el problema conjuntamente conjuntamente con los laboratorios laboratorios nacionales y que que hiciera una una recomendación. El BIPM preparó un cuestionario detallado el cual cubría los temas involucrados y lo hizo llegar a 32 laboratorios nacionales de metrología interesados en el tema (y, para información, a cinco organizaciones internacionales). A principios de 1979 se recibieron las respuestas de 21 laboratorios [1] 1. Casi todos estaban de acuerdo en que era importante contar con un procedimiento aceptado internacionalmente para expresar las incertidumbres en mediciones y para combinar los componentes individuales de la incertidumbre en una única incertidumbre incertidumbre total. Pero, no se alcanzó un consenso acerca del método que debía ser utilizado. El BIPM convocó, entonces, a una reunión con el propósito de diseñar un procedimiento uniforme y mayoritariamente aceptable para la especificación de las incertidumbres, al cual asistieron expertos de 11 laboratorios nacionales. Este Grupo de Trabajo para la Expresión de Incertidumbres generó la Recomendación INC-1 (1980), Expresión de Incertidumbres Experimentales [2]. El CIPM aprobó dicha recomendación en 1981 [31] y la reafirmó en 1986 [4]. La tarea de desarrollar una guía detallada basada en la Recomendación del Grupo de Trabajo (que es un bosquejo general más que una receta detallada) fue delegada por el CIPM en la International Organization for Standarization (ISO), puesto que la ISO está más al tanto de las necesidades que surgen del amplio espectro de intereses de la industria y el comercio. La responsabilidad responsabilidad fue asignada al Grupo Asesor Técnico Técnico en Metrología (TAG (TAG 4) dado que
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Desarrollar un documento guía basado en la Recomendación del Grupo de Trabajo para la Expresión de Incertidumbres del BIPM que proporcione reglas acerca de la expresión de incertidumbres incertidumbres en la medición para ser usado en normalización, normalización, calibración, calibración, acreditación de laboratorios laboratorios y servicios de metrología; El propósito de tal guía es: -
prop propor orci cion onar ar info inform rmac ació iónn com compl plet etaa ace acerc rcaa de de cóm cómoo se se lle llega ga a la la exp expre resi sión ón de incertidumbres, y
-
prop propor orci cion onar ar una una bas basee para para la comp compar arac ació iónn inte intern rnac acio iona nall de de los los res resul ulta tado doss de las mediciones.
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GUÍA TÉCNICA COLOMBIANA
GTC 51
GUÍA PARA LA EXPRESIÓN DE INCERTIDUMBRES EN LAS MEDICIONES
0.
INTRODUCCIÓN
0.1 Cuando se reporte el resultado de una medición de una magnitud física es obligatorio proporcionar alguna indicación cuantitativa de la calidad del resultado, de manera tal que el usuario pueda apreciar su contabilidad. Sin esta indicación, los resultados de las mediciones no pueden ser comparados, ni entre ellas mismas ni con respecto a valores de referencia dados en una especificación o norma. Por lo tanto, es necesario que exista un procedimiento fácil de usar y aceptado de manera general para caracterizar la calidad del resultado de una medición, esto es, para evaluar y expresar su incertidumbre. 0.2 El concepto de incertidumbre como un atributo cuantificable es relativamente nuevo en la historia de las mediciones, a pesar de que los conceptos de error y análisis de error han sido parte de la práctica de la ciencia de la medición por largo tiempo. Actualmente se acepta de manera general que cuando todas las componentes, conocidas o supuestas, del error han sido evaluadas y se han aplicado las correcciones apropiadas, aún persiste una incertidumbre acerca del estado de corrección del resultado expresado, esto es, persiste una duda acerca de qué tan bien representa el resultado de la medida al valor de la cantidad que está siendo medida. 0.3
De la misma manera que el uso casi universal del Sistema Internacional de unidades
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GUÍA TÉCNICA COLOMBIANA
GTC 51
La cantidad utilizada para expresar la incertidumbre debe ser: -
internamente consistente: debe
poder obtenerse directamente a partir de los componentes que contribuyen a ella, asimismo, debe ser independiente de la forma en que dichas componentes son agrupadas y del método en que éstas se descomponen en subcomponentes.
-
transferible: debe
ser posible utilizar directamente la incertidumbre es evaluada para un resultado, como una componente al evaluar la incertidumbre de otra medida en la cual es utilizado el primer resultado.
Adicionalmente, en muchas aplicaciones industriales y comerciales, así como en las áreas de salud salud y seguridad, frecuentemente es necesario proporcionar un intervalo, centrado en el resultado de la medición, que contenga una fracción considerable de la distribución de valores que pueden ser razonablemente atribuidos a la cantidad que se está midiendo. Por tanto, el método ideal para evaluar y expresar la incertidumbre en la medición debe ser capaz de proporcionar, directamente, tal tipo de intervalo, en particular, uno con una probabilidad de cobertura o nivel de confianza que corresponda de forma realista con lo requerido. 0.5 La metodología en la que se basa este documento guía 2 se describe en la Recomendación INC-1 (1980) [2] del Grupo de Trabajo para la Expresión de Incertidumbres, que fue habilitado por el BIPM en respuesta a una petición del CIPM (véase el Prefacio). Esta metodología satisface todos los requisitos descritos arriba. Este no es el caso para la mayoría de los métodos usados comúnmente. La recomendación INC-1 (1980) fue aprobada y ratificada por el CIPM en sus propias Recomendaciones 1 (CI-1981) [3] y 1 (CIPM-1982) [4]; la traducción al español de tales Recomendaciones del CIPM se reproducen en el Anexo A (véanse los numerales A.2 y A.3 A. 3 respectivamente). Dado que la Recomendación INC-1 (1980) es la base para este documento, su traducción al español se reproduce en 0.7 y el texto en francés, que es el legalmente autorizado, se reproduce en el numeral A.1. 0.6 Un sumario sucinto del procedimiento recomendado en este documento guía para evaluar y expresar las incertidumbres en la medición se da en el capítulo 8, en tanto que en el Anexo H se presentan, detalladamente, varios ejemplos. En otros anexos se tratan los siguientes temas: términos generales de metrología (véase el Anexo B); términos y conceptos
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GUÍA TÉCNICA COLOMBIANA 1)
GTC 51
La incert incertidu idumbr mbree en en el el resu resulta ltado do de una medici medición ón consta consta,, gene general ralmen mente, te, de varias componentes que pueden ser agrupadas en dos categorías, dependiendo de la manera en que se estime su valor numérico: A)
aque aquellllas as que que se se eval evalúa úann por por méto método doss esta estadí díst stic icos os..
B)
aque aquellllas as que que se eval evalúa úann por por otro otross medi medios os.. No siempre existe una correspondencia simple entre las categorías A y B y la clasificación en incertidumbres “aleatorias” y “sistemáticas” que se usaba anteriormente El término “incertidumbre sistemática” puede ser confuso y debe ser evitado. Cualquier informe detallado de la incertidumbre debe constar de una lista completa de las componentes, especificando en cada caso el método usado para la obtención de su valor numérico.
2)
Los Los comp compon onen ente tess en la la cate catego gorí ríaa A se car carac acte teri riza zann medi median ante te las las var varia ianz nzas as 2 estimadas si (o las “desviaciones estándar” estimadas s )i y el número de grados de libertad νi. En caso de ser necesario, debe darse el valor de las covarianzas.
3)
Los Los comp compon onen ente tess en la la cate catego gorí ríaa B debe debenn ser ser cara caracte cteri riza zado doss medi median ante te las las 2 , cantidades u j las cuales pueden ser consideradas como aproximaciones a las 2
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GUÍA TÉCNICA COLOMBIANA
GTC 51
GUÍA PARA LA EXPRESIÓN DE INCERTIDUMBRES EN LAS MEDICIONES
1.
ALCANCE
1.1 Esta Esta guía guía estab establec lecee reglas reglas gen genera erales les par paraa la evalu evaluaci ación ón y la expre expresió siónn de incertidumbres incertidumbres en la medición, las cuales pueden seguirse a diferentes niveles de exactitud y en muchos campos desde el piso de ventas hasta la investigación. Por tanto, se pretende que los principios principios de esta guía sean aplicables aplicables a una amplia gama gama de mediciones, mediciones, incluyendo aquellas requeridas para: el
l de de l
alid alidad ad
l a
mi
de la
lida lidadd en en l
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GUÍA TÉCNICA COLOMBIANA
GTC 51
1.4 Esta guía proporciona reglas generales para evaluar y expresar la incertidumbre en la medición, más que instrucciones técnicas detalladas y específicas. En ella no se discute cómo la incertidumbre del resultado de una medición particular, una vez evaluada, puede ser utilizada para diferentes propósitos, por ejemplo, para deducir conclusiones acerca de la compatibilidad de ese resultado particular con algunos otros similares, para establecer los límites de tolerancia en un proceso de manufactura, o para decidir si un cierto curso de acción puede ser seguido sin riesgos. Puede ser, por tanto, necesario desarrollar normas particulares, basadas en esta guía , para enfrentar los problemas peculiares de los campos específicos de medición o para tratar los varios usos de las expresiones cuantitativas de la incertidumbre. Nota. Pueden existir situaciones en las que se considere que el concepto de incertidumbre de la medición no es totalmente aplicable. como en el caso en que se determina la precisión de un método de ensayo (véase la referencia [5]. por ejemplo).
2.
DEFINICIONES
2.1 2.1
TÉRM TÉ RMIN INOS OS GENE GENERA RALE LES S SOBR SOBRE E METR METROL OLOG OGÍA ÍA
En el Anexo B se da la definición de algunos términos generales de metrología relevantes a esta guía, tales como “magnitud “magnitud medible”, “mensurando “mensurando (medible)” (medible)” y “error de medición”. medición”. Estas definiciones se han tomado del Vocabulario Internacional de términos básicos y generales en Metrología (abreviado como VIM) [6] (NTC 2194 Segunda actualización). Adicionalmente, en el Anexo C se dan las definiciones de algunos términos estadísticos básicos tomados principalmente de la Norma Internacional ISO 3534-1 [7]. A partir del
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GTC 51
2.2.3 La definición formal del término “incertidumbre de medición” que se ha desarrollado para utilizarse en esta guía y en el VIM [6] (VIM párrafo 3.9) es la siguiente: -
incertidumbre (de la medición)
-
pará paráme metr troo aso asoci ciad adoo con con el resu resultltad adoo de de una una medi medici ción ón que que car carac acte teri riza za la dispersión de los valores, que en forma razonable se le podría atribuir a una magnitud por medir.
Notas: 1)
El pará parámet metro ro pued puedee ser, ser, por por ejem ejemplo plo,, una una desv desviac iación ión estánd estándar ar (o (o un múltip múltiplo lo dado dado de ella ella), ), o la semilongitud de un intervalo que tenga un nivel de confianza determinado.
2)
En gene general ral,, la incert incertidu idumbr mbree de una medici medición ón comp compren rende de much muchas as compon component entes. es. Alguna Algunass de de estas estas componentes pueden ser evaluadas a partir de la distribución estadística de los resultados de una serie de mediciones y puede ser caracterizada por desviaciones estándar experimentales. Las otras componentes, que también pueden ser caracterizadas utilizando desviaciones estándar, se determinan a partir de distribuciones de probabilidad supuestas. basadas en la experiencia u otra información.
3)
Se enti entiend endee que que el resu resultad ltadoo de la la medic medición ión es la la mejor mejor esti estimac mación ión del valor valor de de la magn magnitu itudd por por medir medir,, y que todos los componentes de incertidumbre contribuyen a la dispersión, incluyendo aquellos resultantes de efectos sistemáticos. Tales como los componentes asociados con correcciones y patrones de referencia.
2.2.4 La definición de incertidumbre de medición dada en el numeral 2.2.3 es una definición operacional que se enfoca en el resultado de la medición y su incertidumbre evaluada. Sin
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Nota. Una discusión más a fondo relacionada con estos términos, puede encontrarse en: para el numeral 2.3.3, véase los numerales 3.3.3 y 4.3; para el numeral 2.3.4. véase el Capítulo 5 y las ecuaciones (10) y (13); y para el numeral 2.3.5, véase el Capítulo 6.
2.3.1 Incertidumbre estándar Incertidumbre del resultado de una medición expresada como una desviación estándar. 2.3.2 2.3.2 Evalua Evaluació ción n (de incert incertidu idumbr mbre) e) Tipo Tipo A Método para evaluar la incertidumbre mediante el análisis estadístico de una serie de observaciones. 2.3.3 2.3.3 Evalua Evaluació ción n (de incert incertidu idumbr mbre) e) Tipo Tipo B Método para evaluar la incertidumbre por otro medio que no sea el análisis estadístico de una serie de observaciones. 2.3.4 2.3.4 Incert Incertidu idumbr mbree estánda estándarr combinad combinadaa Incertidumbre Incertidumbre estándar del resultado de una medición cuando el resultado se obtiene a partir de los valores de algunas otras magnitudes, igual a la raíz cuadrada positiva de una suma de términos, siendo estos términos las varianzas y covarianzas de estas otras magnitudes ponderadas de acuerdo cómo el resultado de la medición varía con respecto a cambios en estas magnitudes. 2.3
Inc
idumbr idumbr
andida andida
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GUÍA TÉCNICA COLOMBIANA 3.
GTC 51
CONCEPTOS BÁSICOS
Una discusión adicional de los conceptos básicos puede ser encontrada en el Anexo D, el cual se centra en las ideas de valor “verdadero”, error e incertidumbre, e incluye ilustraciones gráficas de estos conceptos; y en el Anexo E, el cual explora la motivación y base estadística para la recomendación INC-1 (1980) en la cual esta guía se basa. El Anexo J es un glosario de los principales símbolos matemáticos usados a través de la guía . 3. 1
MEDICIÓN
3.1.1 El objetivo de una medición (véase el numeral B.2.5) es determinar el valor (véase el numeral B.2.2) del mensurando (véase el numeral B.2.9), esto es, el valor de la cantidad particular (véase el numeral B.2.1, nota 1) a ser medida. Una medición entonces comienza con una especificación apropiada del mensurando, el método de medición (véase el numeral B.2.7), y el procedimiento de medición (véase el numeral B.2.8).
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3.1.4 En muchos casos, el resultado de una medición se determina sobre la base de una serie de observaciones obtenida bajo condiciones de repetibilidad (véase el numeral B.2.15, Nota 1). 3.1.5 Se asume que las variaciones en observaciones repetidas se presentan debido a magnitudes de influencia (véase el numeral numeral B.2.10) que pueden pueden afectar el resultado de de la medición y no se mantienen completamente completamente constantes. 3.1.6 El modelo matemático de la medición que transforma el conjunto de observaciones repetidas en el resultado de la medición es de importancia crítica porque, además de las observaciones, generalmente incluye varias magnitudes de influencia que son conocidas de una manera inexacta. Esta falta de conocimiento contribuye a la incertidumbre de los resultados de la medición, así como las variaciones de las observaciones repetidas y cualquier incertidumbre incertidumbre asociada con el modelo matemático en sí mismo. 3.1.7 Esta guía trata al mensurando como un escalar (una magnitud simple). La extensión a un conjunto de mediciones relacionadas determinadas simultáneamente en la misma medición requiere
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3.2.3 El error sistemático, como el error aleatorio, no puede ser eliminado, pero a menudo puede, también, ser reducido. Si un error sistemático se presenta como consecuencia de un efecto reconocido de una magnitud de influencia en el resultado de una medición, de aquí en adelante llamado efecto sistemático, el efecto puede ser cuantificado cuantificado y, si es significativo significativo en relación a la exactitud requerida de la medición, una corrección (véase el numeral B.2.23) o un factor factor de correcci corrección ón (véase el numeral B.2.24) puede ser aplicado para compensar el efecto. Se supone que después de la corrección, la esperanza o valor esperado del error originado por un efecto sistemático es cero. Nota. La incertidumbre de una corrección aplicada al resultado de una medición para compensar por un efecto sistemático no es el error sistemático, frecuentemente llamado sesgo de la medición, debido al efecto como es llamado algunas veces. Esta es, por el contrario, una medida de la incertidumbre del resultado debida al conocimiento incompleto del valor requerido de la corrección. El error originado por la compensación imperfecta de un efecto sistemático no puede ser conocido conocido exactamente. Los términos términos “error” e “incertidumbre” “incertidumbre” deben ser usados apropiadamente y debe tenerse cuidado para distinguir uno del otro.
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Nota. El resultado de una medición (después de la corrección) puede estar muy cerca, al valor del mensurando y de una forma que no puede conocerse (y entonces tener un error despreciable), y aun así tener una gran incertidumbre. Entonces la incertidumbre del resultado de una medición no debe ser confundida con el error desconocido remanente
3.3.2 En la práctica, existen muchas fuentes posibles de incertidumbre en una medición, incluyendo: a)
defin efinic iciión inco incom mplet pletaa de del mens mensuurando ndo;
b)
real realiz izac ació iónn imp imper erfe fect ctaa de de la la def defin inic ició iónn del del mens mensur uran ando do;;
c)
muestr muestreos eos no repres represent entati ativos vos la muestr muestraa med medida ida puede puede no repre represen sentar tar el mensurando definido;
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GUÍA TÉCNICA COLOMBIANA
GTC 51
las componentes en sí mismas elimina esta ambigüedad. Al mismo tiempo, ello no impide colocar en grupos designados a las componentes individuales que han sido evaluadas por los dos métodos diferentes con la finalidad de que sean usadas para un propósito particular (véase el numeral 3.4.3).
3.3.4 El propósito de la clasificación Tipo A y Tipo B es para indicar las dos diferentes maneras de evaluar componentes de incertidumbre y es por conveniencia de discusión solamente; la clasificación no significa que exista alguna diferencia en la naturaleza de los componentes que resultan de cada uno de los dos tipos de evaluación. Ambos tipos de evaluación están basados en distribuciones de probabilidad (véase el numeral C.2.3), y las componentes de incertidumbre resultantes de cualquier tipo son cuantificadas por varianzas y desviaciones estándar. 3.3.5 La varianza estimada u2 que caracteriza a una componente de incertidumbre obtenida de una evaluación tipo A se calcula mediante series de observaciones repetidas y es la varianza estimada estadística familiar s2 (véase el numeral 4.2). La desviación estándar 2
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GUÍA TÉCNICA COLOMBIANA 3.4 3.4
GTC 51
CONSI ONSIDE DER RAC ACIO ION NES PR PRÁ ÁCT CTIC ICAS AS
3.4.1 Si todas las cantidades de las cuales el resultado de una medición depende, se varían, su incertidumbre puede ser evaluada por métodos estadísticos. Sin embargo, debido a que esto es raramente posible en la práctica dadas las limitaciones en el tiempo y los
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GUÍA TÉCNICA COLOMBIANA
GTC 51
3.4.6 El valor estimado de un mensurando obtenido del resultado de una medición es, algunas veces, expresado en términos del valor adoptado de un patrón de medición en lugar de en términos de la unidad relevante del Sistema Internacional de Unidades ( SI). SI). En estos casos la incertidumbre atribuida al resultado de la medición puede ser significativamente menor que cuando este resultado es expresado en las unidades SI relevantes. (En efecto, el
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Notas: 1)
Para Para simp simplif lifica icarr la notaci notación ón en esta esta guía, guía, se se utili utiliza za el mism mismoo símbo símbolo lo para para la magn magnitu itudd físic físicaa (el (el mensurando) y para la variable aleatoria (véase el numeral 4.2.1) que representa el posible resultado de una observación de esa magnitud. Cuando se establece que X i tiene una distribución de probabilidad particular. el símbolo se usa en ese último sentido; se presupone que la magnitud física misma puede
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Nota. Sin embargo, la ecuación (1) puede ser tan elemental con Y = X 1 - X 2. Esta expresión modela, por ejemplo, la comparación de dos determinaciones de la misma magnitud X .
4.1.3 El conjunto de argumentos X 1 , X 2 , ..., X N pueden dividirse en las siguientes categorías:
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GUÍA TÉCNICA COLOMBIANA
GTC 51
4.1.6 Cada valor estimado de un argumento xi y incertidumbre estándar asociada u( xi) se obtienen a partir de una distribución de los posibles valores del argumento X i. Esta distribución de probabilidad puede estar basada en frecuencia, es decir, basada en una serie de observaciones X i,k i,k de X i, puede ser una distribución a priori. Las evaluaciones tipo A de las componentes de la incertidumbre estándar están basadas en distribuciones de
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GUÍA TÉCNICA COLOMBIANA
GTC 51
B.2.11 Resultado de una una medición [VlM3.1] Valor atribuido a un mensurando, obtenido mediante una medición Notas: 1)
Cuan Cuando do se da un resu resultltad ado, o, debe debe acla aclara rars rsee si si est estee se se ref refie iere re a: -
la indicación el resultado no corregido el resultado corregido
2)
si se han promediado varios datos.
Una decl declara aració ciónn compl completa eta del del resul resultad tadoo de una una medic medición ión incl incluye uye inform informaci ación ón acer acerca ca de de la ince incerti rtidum dumbre bre de medición.
B.2.12 Resultado no corregido [VIM 3.3] 3.3] Resultado de una medición antes de ser corregido por un error sistemático B.2.13 Resultado corregido corregido [VIM 3.4] Resultado de una medición después de ser corregido por un error sistemático B.2.14 Exactitud de la medición medición [VIM 3.5] Cercanía de concordancia entre el resultado de una medición y el valor verdadero del mensurando Notas: 1)
“exa “exact ctit itud ud”” es es un un co concep ncepto to cuali ualita tati tivvo.
2)
el término precisión no debería ser usado en vez de “exactitud”“
Comentario de la guía : véase en la guía guía el comentario de de numeral B.2.3. B.2.15 Repetibilidad (de resultados resultados de mediciones) [VIM 3.6] Cercanía de concordancia entre los resultados de mediciones sucesivas del mismo mensurando mensurando que se han de llevar a cabo bajo las mismas condiciones de medición
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Notas: 1)
Esta Estass con condi dici cion ones es son son lla llama mada dass con condi dici cion ones es de repe repetib tibililid idad ad..
2)
Las Las con condi diccion iones de repet epetib ibililid idad ad inc incluy luyen: en:
3)
-
el mismo procedimiento de medición
-
el mismo observador
-
el mism mismoo ins instr trum umen ento to de de medi medici ción ón,, usad usadoo baj bajoo las las mism mismas as cond condic icio ione ness
-
la misma ubicación
-
repetic tición so sobre un un co corto pe período de de titiempo.
La repet repetibi ibilid lidad ad puede puede ser ser expre expresad sadaa cuantit cuantitati ativam vament entee en térmi términos nos de de las caract caracterí erísti sticas cas de dispe dispersi rsión ón de los resultados.
B.2.16 Reproducibilidad (de los resultados resultados de mediciones) [VIM 3.7] Cercanía de concordancia entre los resultados de mediciones del mismo mensurando que se han llevado a cabo bajo diferentes condiciones de medición Notas: 1)
Una decl declara aració ciónn válid válidaa de repr reprodu oducib cibili ilidad dad requ requier ieree de la la espec especific ificaci ación ón del del cambi cambioo en las las cond condici icione ones. s.
2)
Las Las cond condic icio ione ness modi modifi fica cada dass pued pueden en incl inclui uir: r: -
principio de medición
-
método de medición
-
observador
-
instrumento de medición
-
patrón de referencia
-
ubicación
-
condiciones de uso
-
t i em p o .
3)
La repr reprodu oducib cibili ilidad dad pued puedee ser ser expre expresa sada da cuan cuantit titati ativam vament entee en térm término inoss de las las cara caracte cterís rístic ticas as de de dispersión de los resultados.
4)
Se entien entiende de que los result resultado adoss aquí aquí son, son, genera generalme lmente nte,, resu resulta ltados dos correg corregido idos. s.
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GUÍA TÉCNICA COLOMBIANA
GTC 51
B.2.17 desviación estándar estándar experimental [VIM 3.8] Para una serie de n mediciones del mismo mensurando, la magnitud dispersión de los resultados, está dada por la fórmula: n
s (q ( qk )
=
∑= (qk − q)
k 1
s(qk )
que caracteriza la
2
n−1
en donde qk es el resultado de la k -ésima -ésima medición y resultados considerados
q
es la media aritmética de los
n
Notas: 1)
Considerando la la se serie de de n valores como una muestra de una distribución q es una estimación no sesgada de la media µq, y s2(qk ) es una estimación no sesgada de la varianza σ2 de tal distribución.
2)
La expresión s(qk) / √n es una estimación de la desviación estándar de la distribución de q y es llamada la desviación estándar experimental experimental de la media .
3)
La “des “desvia viació ciónn están estándar dar experi experimen mental tal de de la media” media” es inco incorre rrecta ctamen mente te llama llamada da algu algunas nas veces veces erro errorr estándar de la media .
Comentario de la guía : Algunos de los símbolos usados en el VIM han sido cambiados para asegurar la consistencia con la notación usada en el capitulo 4.2 de esta guía . B.2.18 Incertidumbre (de una una medición) [VIM 3.9] Parámetro, asociado con el resultado de una medición, que caracteriza la dispersión de los valores que podrían ser atribuidos razonablemente al mensurando Notas: 1)
El pued puedee ser ser por por ejemp ejemplo, lo, una una desv desviac iación ión estánd estándar ar (o (o un múltip múltiplo lo dado dado de esta esta), ), o el el semii semiinte nterva rvalo lo de de un intervalo que tiene un nivel de confianza declarado.
2)
La ince incerti rtidum dumbre bre de una una medi medició ciónn inclu incluye, ye, en genera general,l, much muchas as compon component entes. es. Alguna Algunass de de esta estass componentes pueden ser evaluadas a partir de la distribución estadística de los resultados de series de mediciones y pueden ser caracterizadas caracterizadas mediante desviaciones desviaciones estándar experimentales. Las otras componentes, las cuales pueden ser caracterizadas también mediante desviaciones estándar, son evaluadas a partir de distribuciones supuestas de probabilidad basadas en la experiencia o en otra información.
3)
Se enti entiend endee que que el result resultado ado de la la medic medición ión es la la mejor mejor estima estimació ciónn del del valor valor del mensur mensurand ando, o, y que todas las componentes de incertidumbre, incluyendo aquellas que surgen por efectos sistemáticos, tales como las componentes asociadas con correcciones y patrones de referencia, contribuyen a la dispersión.
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Comentario de la guía : Se hace notar en el VIM que esta definición y las notas son idénticas a las que están en esta esta guía (véase el numeral numeral 2.2.3) B.2.19 Error (de una medición) medición) [VIM 3.10] Resultado de una medición menos el valor verdadero del mensurando Notas: 1)
Dado Dado que que un valor valor verd verdade adero ro no no puede puede ser determ determina inado, do, en la la práct práctica ica se usa un valo valorr verd verdade adero ro convencional (véase (véase [VIM]1.19 [B.2.3] y 1.20 [B.2.4])
2)
Cuando Cuando sea necesa necesario rio disting distinguir uir “error “error”” de “error “error relati relativo" vo".. el prim primero ero es llam llamado ado alguna algunass veces veces error absoluto de medición . Este no debería ser confundido con el valor absoluto del error , el cual es el módulo del error.
Comentario de la guía : Si el resultado de una medición depende de los valores de otras magnitudes diferentes al mensurando, los errores de los valores medidos de estas cantidades contribuyen al error del resultado de la medición. Véase en la guía el comentario B.2.22 y B.2.3. B.2.20 Error relativo relativo [VIM 3.12] 3.12] Error de medición dividido por el valor verdadero del mensurando Nota. Dado que el valor verdadero no puede determinarse, en la práctica se utiliza un valor verdadero convencional convencional (véanse [VIM] 1.19 [B.2.3] y 1.20 [ B.2.4]).
Comentario de la guía : véase en la guía guía el comentario del del numeral B.2.3 B.2.21 Error aleatorio aleatorio [VIM 3.13] 3.13] Resultado de una medición menos la media que podría resultar de un número infinito de mediciones del mismo mensurando llevados a cabo bajo condiciones de repetibilidad Notas: 1) 2)
El erro errorr ale aleat ator orio io es igua iguall al al err error or meno menoss el el err error or sist sistem emát átic icoo Debido Debido a que que sólo sólo puede puede hace hacerse rse un un número número fini finito to de medi medicio ciones nes,, es única únicamen mente te posib posible le deter determin minar ar un estimado del error aleatorio
Comentario de la guía: véase en la guía el comentario del numeral B.2.22 B.2.22 Error sistemático sistemático [VIM 3.14] Media que resulta de un número infinito de mediciones del mismo mensurando llevadas a
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Notas: 1)
El pará parámet metro ro pued puedee ser, ser, por por ejem ejemplo plo,, una una desvi desviaci ación ón está estánda ndarr (o un múlt múltipl iploo dado dado de de esta) esta),, o el el semiintervalo, un intervalo que tiene un nivel de confianza declarado.
2)
La ince incertid rtidumb umbre re de de una una medi medició ciónn incl incluye uye,, en genera general,l, much muchas as compon component entes. es. Alguna Algunass de de esta estass componentes puede, ser evaluadas a partir de la distribución estadística de los resultados de series de mediciones y pueden ser caracterizadas mediante desviaciones estándar experimentales. Las otras componentes, las cuales pueden ser. caracterizadas también mediante desviaciones estándar, son evaluadas a partir de distribuciones supuesta probabilidad basadas en la experiencia o en otra información.
3)
Se enti entiend endee que que el result resultado ado de la la medic medición ión es la la mejor mejor estima estimació ciónn del del valor valor del mensur mensurand ando, o, y que todas las, componentes de incertidumbre, incluyendo aquellas que surgen por efectos sistemáticos, tales como las componentes asociadas con correcciones y patrones de referencia, contribuyen a la dispersión.
Comentario de la guía : se hace notar en el VIM que esta definición y las notas son idénticas a las que están en es guía (véase el numeral numeral 2.2.3) B.2.23 Corrección [VIM 3.15] Valor añadido algebraicamente al resultado no corregido de una medición para compensar el error sistemático Notas: 1)
La corr correc ecci ción ón es igua iguall al al neg negat ativ ivoo del del erro errorr sis siste temá mátitico co esti estima mado do
2)
Dado Dado que que el error error sist sistemá emátic ticoo no puede puede ser ser cono conocid cidoo perfec perfectam tament ente, e, la la comp compens ensaci ación ón no no puede puede ser completa.
B.2.24 Factor de corrección [VIM 3.16] Factor numérico por el cual se multiplica al resultado de una medición para compensar al error sistemático Nota. Dado que el error sistemático no puede ser perfectamente conocido, la compensación no puede ser completa.
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GUÍA TÉCNICA COLOMBIANA
GTC 51 ANEXO C
Términos y conceptos estadísticos básicos
C.1 C.1
ORIG ORIGEN EN DE LA LAS S DE DEFI FINI NICI CION ONES ES
Las definiciones de los términos estadísticos básicos dados en este anexo están tomados de la Norma Internacional ISO 3534-1 [7]. Esta debe ser la primera fuente consultada para las definiciones de términos que no se incluyan aquí. Algunos de estos términos y los conceptos en que se sustentan se discuten en C.3 luego de la presentación de sus definiciones formales en C.2 con la finalidad de facilitar el posterior uso de esta guía . Sin embargo, C.3, que también también incluye las definiciones de de algunos términos relacionados, relacionados, no está basada basada directamente sobre ISO 3534-1. C.2
DEFINICIONES
Al igual que el capítulo 2 y en el anexo B, el encerrar ciertas palabras de algunos términos entre paréntesis significa que las palabras podrían omitirse si su ausencia no es causa de confusión. Los términos C.2.1 a C.2.14 se definen haciendo uso de las propiedades de las poblaciones. Las definiciones de los términos C.2.15 a C.2.31 están relacionadas a un conjunto de observaciones observaciones (véase la referencia [71]). C.2.1 Probabilida Probabilidad d [ISO 3534-1,1 3534-1,1.1] .1] Un número real en la escala 0 a 1 asociado a un evento aleatorio. Nota. Este puede estar relacionado a una frecuencia relativa de ocurrencia a largo plazo o a un grado de credibilidad de que un evento ocurrirá. Para un alto grado de credibilidad. la probabilidad es cercana a 1.
C.2.2 Variable Variable aleator aleatoria; ia; variac variación ión [ISO [ISO 3534-1, 3534-1, 1.2] 1.2]
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GTC 51
C.2.3 Distribución de probabilidad probabilidad (de una variable variable aleatoria) aleatoria) [ISO [ISO 3534-1, 3534-1, 1.3] Una función que da la probabilidad de que una variable aleatoria tome cualquier valor dado o pertenezca a un conjunto dado de valores Nota. La probabilidad sobre el conjunto total de valores de la variable aleatoria es igual a 1.
C.2.4 Función Función de distribuci distribución ón [ISO [ISO 3534-1, 3534-1, 1.4] 1.4] Una función que da, para cada valor de x, la probabilidad de que la variable aleatoria X sea menor o igual que x: F(x) = Pr(X ≤ x)
C.2.5 Función Función de densidad densidad de probabilid probabilidad ad (para (para una variable variable continua) continua) [ISO [ISO 3534-1,1.5] 3534-1,1.5] La derivada (cuando existe) de la función de distribución: f(x) = dF(x)/ d x
f(x)d x es el “elemento de probabilidad”: Nota. f(x)d
f(x) d x = Pr(x < X < x + d x)
C.2.6 Función Función de probab probabilida ilidad d de masa masa [ISO 35343534-1, 1, 1.6] Una función que da, para cada valor xi de una variable aleatoria discreta X , la probabilidad pi de que la variable aleatoria sea igual a xi:
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GTC 51
C.2.9 Esperanza (de una variable aleatoria aleatoria o de una distribución de probabilidad); probabilidad); valor esperado; media [ISO 3534-1, 1.18] 1)
Par Para un una va variab iable alea aleato torria disc discre reta ta X que toma valores xi con probabilidades la esperanza, si existe, es:
pi,
µ = E(X) =Σ pi xi en la cual la sumatoria corre sobre todos los valores xi que puede tomar X . 2)
Par Para un una va variab iable aleat leatooria con contin tinua X con función de densidad de probabilidad f(x), la esperanza, si existe, es m
=
E(x)
= ∫ x f (x) dx
en donde la integral se extiende sobre el (los) intervalo(s) de variación de X . C.2.10 Variable aleatoria centrada centrada [ISO 3534-1, 1.21] Una variable aleatoria para la cual la esperanza es cero. Nota. Si la variable aleatoria X tiene una esperanza igual a µ, la variable aleatoria centrada correspondiente es ( X X - µ)
C.2.11 Varianza (de una variable aleatoria o de una distribución de probabilidad) [ISO 3534-1, 1.22] La esperanza del cuadrado de la variable aleatoria centrada ([ISO 3534-1] 1.21 [C.2.10]):
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C.2.13 Momento central1 de orden q [ISO q [ISO 3534-1, 1.28] En una distribución distribución univariada, univariada, la esperanza esperanza de la q-ésima potencia de la variable aleatoria centrada ( X - µ): q E [(X − µ ) ]
Nota. El momento central de orden 2 es la varianza ([ISO 3534-1] 1.22 [C.2.11]) de la variable variable aleatoria X .
C.2.14 Distribución normal; distribución distribución de Laplace-Gauss USO 3534-1, 1.37) La distribución de probabilidad de una variable aleatoria continua X , cuya función de densidad de probabilidad es
f(x)
=
1 x − m exp − s 2p 2 s 1
2
para - ∞ < x < + ∞ Nota. µ es la esperanza y σ es la desviación estándar de la distribución normal.
C.2.15 Característica Característica [ISO 3534-1, 2.2] Una propiedad que ayuda a identificar o diferenciar entre objetos de una población dada. Nota. La característica puede ser cuantitativa (por variables) o cualitativa (por atributos).
C.2.16 Población [ISO 3534-1, 3534-1, 2.3]
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C.2.18 Distribución de frecuencia frecuencia [ISO 3534-1, 2.15] La relación empírica entre los valores de una característica y sus frecuencias o sus frecuencias relativas. Nota. La distribución puede presentarse gráficamente como un histograma ([ISO 3534-1] 2.17), diagrama de barras ([ISO 3534-1] 2.18), polígono de frecuencias acumulativo ([ISO 3534-1] 2.19), o como una tabla de dos vías ([ISO 3534-1] 2.22).
C.2.19 Media aritmética; promedio promedio [ISO 3534-1, 2.26] 2.26] La suma de valores dividida por el número de valores Notas: 1)
El térm término ino “media “media”” se usa genera generalme lmente nte cuando cuando se hace hace refere referenci nciaa a un un parám parámetr etroo de poblac población ión y el término “promedio” cuando se hace referencia al resultado de los cálculos sobre datos obtenidos en una muestra.
2)
El prom promedi edioo de una muestr muestraa simpl simplee aleat aleatori oriaa tomad tomadaa de una poblac población ión es un un estim estimado adorr no sesgad sesgadoo de la media de esta población. Sin embargo, suelen usarse otros estimadores, tales como la media geométrica o armónica, o la mediana o moda.
C.2.20 Varianza [3534-1,2,33 [3534-1,2,33]] Una medida de la dispersión, la cual es la suma de los cuadrados de las desviaciones de los datos con respecto del del promedio, promedio, dividida por una cantidad igual igual al número de las observaciones menos uno. EJEMPLO. Para n observaciones xi, x2 ....... xn con promedio
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GTC 51
Comentario de la guía: la varianza definida aquí se designa más apropiadamente como “el estimado de la muestra de la varianza de la población.” La varianza de una muestra se define usualmente como el momento central de orden 2 de la muestra (véanse los numerales C.2.13 y C.2.22). C.2.1 Desviació Desviación n estánda estándarr [ISO 3534-1, 3534-1, 2.34] 2.34] La raíz cuadrada positiva de la varianza. Nota. La desviación estándar de la muestra es un estimador sesgado de la desviación estándar de la población.
C.2.22 Momento central central de orden orden q [ISO q [ISO 3534-1, 2.37] En una distribución de característica única, la media aritmética de la q-ésima potencia de la diferencia entre los valores observados y su promedio x: 1 ( X i − X )q n i
∑
Donde n es el número de observaciones. Nota. El momento central de orden 1 es igual a cero.
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GUÍA TÉCNICA COLOMBIANA
GTC 51
C.2.25 Estimador [ISO 3534-1, 3534-1, 2.50] Una estadística usada para estimar un parámetro de población. C.2.26 Estimado [ISO 3534-1, 3534-1, 2.51] El valor de un estimador obtenido como resultado de una estimación. C.2.27 Intervalo de confianza bilateral bilateral [ISO 3534-1, 2.57] Cuando T 1 y T 2 son dos funciones de los valores observados tales que, siendo θ un parámetro de población a estimar, la probabilidad Pr(T 1 ≤ θ ≤ T 2) es al menos menos igual igual a (1 - α) [don [donde de (1 - α) es un número fijo, positivo y menor a 1], el intervalo entre T 1 y T 2 es un intervalo de confianza bilateral (1 - α) para θ. Notas: 1)
Los límites T 1 y T 2 del intervalo de confianza son estadísticas ([ISO 3534-1] 2.45 [C.2.23]) y como tales. generalmente toman diferentes valores de muestra a muestra.
2)
En una una seri seriee larga larga de mues muestra tras, s, la la frecu frecuenc encia ia rela relativ tivaa de casos casos en en los los cuale cualess el valor valor verd verdade adero ro del del parámetro de población θ es cubierto por el intervalo de confianza es mayor que o igual que (1 - α).
C.2.28 Intervalo de confianza confianza unilateral [ISO 3534-1, 2.58] Cuando T es una función de los valores observados tal que, siendo θ un parámetro de
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GUÍA TÉCNICA COLOMBIANA
GTC 51
C.2.30 Intervalo de cobertura estadística estadística [ISO 3534-1, 2.61] Un intervalo para el cual puede decirse, con un nivel de confianza dado, que contiene al menos una proporción específica de la población. Notas: 1)
Cuando Cuando ambos ambos lími límites tes se se defin definen en medi mediant antee estad estadíst ística icas, s, el el interv intervalo alo es bila bilater teral. al. Cuan Cuando do uno uno de de los los dos dos límites no es finito o es la frontera de la variable, el intervalo es unilateral.
2)
También También se se le llama llama “int “interv ervalo alo de toler toleranc ancia ia esta estadís dístic tica”. a”. Este Este términ términoo no deberí deberíaa usars usarsee porqu porquee puede puede causar confusión con “intervalo de tolerancia” el cual se define en ISO 3534-2.
C.2.31 Grados de libertad [ISO [ISO 3534-1, 2.85) En general, el número de términos en una suma menos el número de restricciones sobre los términos de la suma. C.3 C.3
ELAB EL ABOR ORAC ACIÓ IÓN N DE DE TÉRM TÉRMIN INOS OS Y CON CONCE CEPT PTOS OS
C.3. C.3.11 Espe Espera ranz nzaa La esperanza de una fusión g (z) sobre una función de densidad de probabilidad p(z) de la variable aleatoria z se define mediante
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GUÍA TÉCNICA COLOMBIANA
GTC 51
−
z
=
1 n z n i= 1 i
∑
C.3. C.3.22 Vari Varian anza za La varianza de una variable aleatoria es la esperanza de su desviación cuadrática alrededor de su esperanza. Por lo tanto la varianza de la variable aleatoria z con función de densidad de probabilidad p ( z) está dada por: s 2 (z)
= ∫ (z − µ z ) 2 p (z) dz
Donde µ z es la esperanza de zi. La varianza σ2( z) puede estimarse mediante s 2 (z ) i
n − = n −1 1 ∑ ( zi − z )2 i= 1
Donde y los zi son n observaciones independientes de z.
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GUÍA TÉCNICA COLOMBIANA
GTC 51
C.3.3 C.3.3 De Desvi sviac ación ión están estándar dar La desviación estándar es la raíz cuadrada positiva de la varianza. Mientras que una incertidumbre estándar Tipo A se obtiene sacando la raíz cuadrada de la varianza estadísticamente calculada, calculada, al determinar determinar la incertidumbre incertidumbre Tipo B frecuentemente es más más conveniente primero evaluar una desviación estándar equivalente no estadística y luego obtener la varianza equivalente elevando al cuadrado la desviación estándar. C.3. C.3.44 Co Cova vari rian anza za La covarianza covarianza de dos variables variables aleatorias aleatorias es una medida de su dependencia dependencia mutua. mutua. La covarianza de las variables aleatorias y y z se define mediante cov( y, z ) = cov( z, y ) = E ([ y-E(y) ][ z-E(z) ])
de donde resulta: cov ( y,z ) = cov( z,y ) =
∫∫ ( y - µ ) ( z - µ ) p ( y, z) d y d z
=
∫∫ y z p ( y, z) d y d z - µ µ
y
z
v
z
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GUÍA TÉCNICA COLOMBIANA
GTC 51
C.3.6 C.3.6 Coefic Coeficien iente te de correl correlaci ación ón El coeficiente de correlación es una medida de la dependencia relativa mutua de dos variables, igual a la razón de sus covarianzas y la raíz cuadrada positiva del producto de sus varianzas. Entonces
ρ ( y ,
z)
= ρ(z ,
y)
=
r (y , z ) i i
= r (zi ,
y) i
=
v( y , z) v( y , y) v ( z , z)
= σ( yv)( y,σz)(z)
con estimaciones s( y , z ) i i
=
s( y , z ) i i ( ) ( )
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GUÍA TÉCNICA COLOMBIANA
GTC 51
C.3. C.3.88 La dis distr trib ibuc ució ión n t ; la distribución de Student La distribución t o de Student es la distribución distribución de probabilidad probabilidad de una una variable variable aleatoria continua t cuya función de densidad de probabilidad es:
p(t,v)
=
1
Γ v +2 1
1 + t 2 Γ v2 v
πv
- ∞
<
t
< +
−(v − 1) / 2
α
en donde Γ es la función función gamma gamma y v > 0. El valor esperado esperado de de la distribución t es cero cero y su su
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GUÍA TÉCNICA COLOMBIANA
GTC 51 ANEXO D
Valor "verdadero", error e incertidumbre El término valor verdadero verdadero (véase el numeral numeral B.2.3) B.2.3) tradicionalmente tradicionalmente se ha usado en publicaciones publicaciones que tratan el tema de incertidumbre, pero no en esta guía por las razones razones que se presentan en este anexo. Debido a que los términos mensurando, “error” e “incertidumbre” “incertidumbre” se mal interpretan frecuentemente, frecuentemente, en este anexo se presenta, además, una discusión adicional sobre las ideas en que se fundamentan estos conceptos, para complementar la discusión que se da en el capítulo 3. Se presentan dos figuras para ilustrar por qué el concepto de incertidumbre que se adopta en esta guía se basa en en el resultado resultado de la medición y su incertidumbre evaluada en lugar de basarse en las magnitudes desconocidas valor “verdadero” y error.
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GUÍA TÉCNICA COLOMBIANA
GTC 51
D.3.2 Como un ejemplo, supóngase que el mensurando es el espesor de una determinada lámina de material material a una temperatura temperatura específica. específica. La lámina se lleva a una temperatura temperatura cerca de la especificada y se mide su espesor con un micrómetro. El espesor del material en ese punto y a esa temperatura, temperatura, bajo la presión aplicada aplicada por el micrómetro, es es la magnitud realizada. D.3.3 Se determinan la temperatura de el material, en el momento de la medición, y la presión aplicada. El resultado no corregido de la medición de la magnitud realizada se corrige entonces tomando en cuenta la curva de calibración del micrómetro, la diferencia entre la temperatura de la lámina y la temperatura especificada, y la ligera compresión de la lámina por la presión aplicada. D.3.4 Al resultado corregido se le puede llamar la mejor estimación del valor “verdadero”, en donde “verdadero” se refiere al valor de la magnitud que se cree satisface plenamente la definición del mensurando; pero si el micrómetro se hubiera aplicado a alguna parte diferente de la lámina, entonces la magnitud realizada hubiera sido diferente y se tendría un
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GUÍA TÉCNICA COLOMBIANA D.4
GTC 51
ERROR
El resultado corregido de una medición no es el valor del mensurando, es decir, existe un error debido a una medición imperfecta de la magnitud realizada por variaciones aleatorias en las observaciones (efectos aleatorios). determinación inadecuada de correcciones por efectos sistemáticos y conocimiento incompleto de ciertos fenómeno, físicos (que son también efectos sistemáticos). Nunca podrán conocerse exactamente ni el valor de la magnitud realizada realizada ni el del mensurando; mensurando; lo único que que podemos conocer son sus valores estimados. En el ejemplo anterior, el grosor medido de la lámina puede tener un error, es decir, puede diferir del valor del mensurando (el grosor de la lámina), debido a que cada uno de los siguientes factores pueden combinarse para contribuir con un error desconocido al resultado de la medición: a)
dife difere renc ncia iass peq peque ueña ñass en en las las lect lectur uras as del del mic micró róme metr troo cua cuand ndoo se se apl aplic icaa repetidamente repetidamente a la misma magnitud realizada
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GUÍA TÉCNICA COLOMBIANA D.6 D.6
GTC 51
REP EPR RES ESEN ENTA TAC CIÓN IÓN GRÁF GRÁFIC ICA A
D.6.1 La Figura D.1 muestra algunas de las ideas que se discuten en el capítulo 3 de esta guía y en este anexo. anexo. Ilustra por qué esta guía hace énfasis énfasis en la incertidumbre incertidumbre y no en el error. El error exacto del resultado de una medición por lo general no se conoce y no se puede conocer. Lo único que se puede hacer es estimar tanto los valores de los argumentos,
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GUÍA TÉCNICA COLOMBIANA
GTC 51 ANEXO E
Motivaciones y bases para la Recomendación INC (1980)
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GUÍA TÉCNICA COLOMBIANA
GTC 51
declarase siempre) deben ser aplicados a la incertidumbre determinada mediante un método realista, y sólo después de que la incertidumbre ha sido así determinada, de manera que el intervalo definido por la incertidumbre expandida tenga el nivel de confianza requerido y la operación puede ser fácilmente invertida. E.2.3 Al realizar una medición, frecuentemente deben incorporarse en el análisis los
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GUÍA TÉCNICA COLOMBIANA
GTC 51 ANEXO H Ejemplos
Este anexo proporciona proporciona seis ejemplos, ejemplos, H.1 a H.6, los cuales han sido resueltos resueltos de manera muy detallada a fin de ilustrar los principios básicos presentados en esta guía para la evaluación y expresión de las incertidumbres en las mediciones. Estos ejemplos, junto con los incluidos en el texto principal y en algunos otros anexos, deben permitir a los usuarios de esta guía poner en práctica dichos dichos principios en su propio trabajo. trabajo. Debido a que los ejemplos tienen un propósito ilustrativo, han sido necesariamente simplificados. Más aún, puesto que tanto los ejemplos como los datos numéricos usados en ellos han sido elegidos principalmente para demostrar los principios de esta guía , ni unos ni otros deben necesariamente ser interpretados como la descripción de un proceso real de medición. Si bien los datos son usados como se presentan, a fin de prevenir errores de redondeo, se usan más dígitos de los que se muestran durante los cálculos intermedios. De esta manera, el resultado obtenido de un cálculo que involucro varias magnitudes puede diferir ligeramente del resultado implicado por los valores numéricos dados en el texto para esas magnitudes. Se indicó en las primeras partes de esta guía que la clasificación de los métodos usados para evaluar componentes de incertidumbre como Tipo A o Tipo B es hecha únicamente por conveniencia; no se requiere para la determinación de la incertidumbre estándar combinada o de la incertidumbre expandida del resultado de una medición debido a todos los componentes de incertidumbre, sin importar la manera en que sean evaluados, se tratan de la misma manera (véanse los numerales 3.3.4, 5.1.2, y E.3.7). Así, en los ejemplos, no se identifica específicamente al método usado para evaluar una componente particular de incertidumbre, incertidumbre, con el tipo de dicha componente. Sin embargo, será claro de la discusión si el componente es obtenido de una evaluación Tipo A o Tipo B. H.1
CALIB CA LIBRA RACI CIÓN ÓN DE BL BLOQ OQUE UES S CAL CALIB IBRE RE PA PARA RALE LELO LO
Este ejemplo demuestra que aún una medición aparentemente simple puede involucrar aspectos sutiles de evaluación de incertidumbre. H.1.1 H.1.1 El prob problem lemaa de de medic medición ión La longitud nominal de un bloque patrón de 50 mm se determina por comparación con un patrón conocido de la misma longitud nominal. El resultado obtenido de la comparación de los dos bloques es la diferencia d de sus longitudes: d = l (1 +
αθ ) - l s
(1+
αsθs)
(H.1)
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Donde: l
es el mensurando. esto es, la longitud a 20 °C del bloque a calibrar:
I s
es la longitud del patrón a 20 °C dada en su certificado de calibración:
α y αs son los coeficientes de de expansión térmica térmica del bloque a calibrar calibrar y del patrón, respectivamente;
θ y θs son las desviaciones en temperatura de la temperatura de referencia de 20 °C del bloque y del patrón, respectivamente. respectivamente.
H.1.2 H.1.2 El modelo modelo matemá matemátic tico o De la ecuación (H.1), el mensurando está dado por l (1 l= s
α sθ s ) + d (1 + αθ )
(H.2)
= ls + d + ls (α sθ s −αθ ) +... Si la diferencia de temperatura entre el bloque calibrado y el patrón está escrita como δθ = θ -θs y la diferencia entre sus coeficientes de expansión térmica como δα = α - αs la ecuación (H.2) se convierte en l = f (ls ,d, αs , θ , δ , δθ) = l s +d - ls[δα .θ +αs.δθ)]
(H.3)
Las diferencias δθ y δα , se consideran iguales a cero, mas no así sus incertidumbres; y se asume que δα,αs ,δθ y θ no están correlacionadas. (Si el mensurando se expresara en términos de las variables θ, θs , α y αs , sería necesario incluir la correlación entre θ y θs , y entre α y αs .) Se sigue entonces, de la ecuación (H.3), que la estimación del valor del mensurando l puede obtenerse de la expresión simple ls + d , donde ls es la longitud del patrón a 20 °C, dada en su certificado de calibración, y d se estima mediante la media aritmética de n = 5 observaciones repetidas e independientes. La incertidumbre estándar combinada uC ,(l) se obtiene mediante la aplicación de la ecuación (10) en el numeral 5.1.2 a la ecuación (H.3), como se discute en seguida.
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GUÍA TÉCNICA COLOMBIANA
GTC 51
H.1.3 H.1.3 Contri Contribuc bución ión de varia varianz nzas as Los aspectos pertinentes de este ejemplo discutidos en ésta y en los siguientes subcapítulos, se resumen en la Tabla H.1. Ya que esto asume que δα = 0 y δθ = 0, la aplicación de la ecuación (10) en el numeral 5.1.2 a la ecuación (H.3) da como resultado 2
2
2
2
c
s
d
αs
u (l ) = c u2(ls ) + c u2 (d) + c
u2(α s )
+ cθ2 u2 (θ ) + cδ2a u2(δα ) + cθ2α u2(δθ )
(H.4)
Donde cs =∂ f / ∂ ls = 1 - (δαθ +
αs . δθ) = 1
cd = ∂ f / ∂d = 1 cαs =∂ f / ∂αs= - l sδθ = 0 cθ = ∂ f/ ∂θ ∂θ = - l sδα = 0 cδα = ∂ f/ ∂δα ∂δα = - l sθ cδθ = ∂ f/ ∂δα ∂δα = - l sαs
y, por tanto 2 2 2 2 u (l) = u 2 (ls ) + u 2( d) + l θ 2u 2(δα ) + l α u2(δα ) c s s s
(H.5)
H.1.3.1 Incertidumbre de la calibración calibración del patrón, patrón, u (ls) = En el certificado de calibración se indica que la incertidumbre expandida del patrón es U = 0,075 µm, y se establece que fue obtenida usando un factor de cobertura de k = 3. La incertidumbre estándar es entonces u( I s) = (0,0 (0,075 75 µm) / 3 = 25 nm
H.1.3.2 Incertidumbre de de la medición de la diferencia diferencia de longitudes, longitudes,
u (d ).
La desviación estándar experimental ponderada característica de la comparación de l y ls se determinó a partir de la variabilidad de 25 observaciones repetidas independientes de la
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GUÍA TÉCNICA COLOMBIANA
GTC 51
u( d ) = s ( d ) = (l3 nm) 5 = 5,8 nm
De acuerdo al certificado de calibración del comparador usado para determinar la diferencia entre l y ls, su incertidumbre “debido a errores aleatorios” es ± 0,01 µm a un nivel de confianza del 95 por ciento y está basado en 6 mediciones repetidas, de esta manera la incertidumbre estándar, usando el factor t , t 95 grados de de libertad libertad 95 (5) = 2,57 para v =6 - 1 = 5 grados (véase el Anexo G. Tabla G.2), es u(d 1) = (0,02 µm) / 2,57 2,57 = 3,9 nm
La incertidumbre del comparador “debido a errores sistemáticos" está dada en el certificado como 0,02 mm al “nivel tres sigma.” La incertidumbre estándar por esta causa puede entonces ser tomada como u(d 2) = (0,02 µm/3) = 6,7 nm
La contribución total se obtiene de la suma de las varianzas estimadas: 2
2
2
2
u (d ) = u ( d ) + u (d 1) + u (d 2) = 93 nm
2
o u(d ) = 9,7 nm
H.1.3.3 Incertidumbre de los coeficientes de expansión expansión térmica, térmica, u(αs). El coeficiente de expansión térmica del bloque calibre está dado como αs = 11,5 x 10 -6°C-1 con una incertidumbre representada por una distribución rectangular con límites ± 2x 10 -6°C-1 La incertidumbre estándar es entonces [véase la ecuación (7) en el numeral 4.3.71] u(αs) = (2 x 10 -6°C-1) / 3 = 1,2 x 10 -6°C-1
∂αs = - l sδθ = 0 como se indicó en H.1.3, ésta incertidumbre no contribuye en Ya que cθ = θ f/ ∂α nada a la incertidumbre de l a primer orden. Tiene, sin embargo, una contribución a segundo orden que es discutida en el numeral H.1.7.
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Tabla H.1. Resumen de las componentes de la incertidumbre estándar Componentes de la incertidumbre estándar u( xi) u(ls) u(d) u( d ) u(d i) u(d 2) u (α s ) u (θ )
u(δα) u(δθ)
u( θ ) u ( Δ)
Fuente de de la la in incertidumbre
Valor de de la la incertidumbre estándar u( xi)
ci≡∂ f/ ∂ xi
Calibración del bloque calibre 25 nm Diferencia medida entre los 9,7 nm bloques calibre observaciones repetidas 5,8 nm efectos aleatorios del 3,9 nm comparador efectos sistemáticos en el 6,7 nm comparador Coeficiente de expansión 1,2 x 10-6 °C-1 térmica del bloque calibre Temperatura de la mesa de 0,41°C ensayo Temperatura media de la 0,2°C mesa variación cíclica de la 0,35°C temperatura del cuarto Diferencia en los coeficientes 0,58 x 10-6°°C -1 de expansión de los bloques Diferencia de la temperatura 0,029 °C de los bloques
U i ≡ ciu|xi| ( nm)
1 1
25 9,7
Grados de libertad
18 25 , 6 24 5 8
0
0
0
0
-ls θ
2,9
50
-l sαs
16,6
2
2 2 u (1) = ∑ u ( l) = 1002 nm2 c
uc (l )
i
= 32 nm
v (l ) = 16 eff
H.1.3.4 Incertidumbre de la desviación de de temperatura temperatura del bloque patrón, patrón, u(θ). La temperatura de la mesa de prueba se reporta como (19,9 ± 0,5) ° C; la temperatura en el momento de las observaciones individuales no fue registrada. Se dice que la desviación (offset) máxima observada, Δ = 0,5° representa la amplitud de una variación aproximadamente cíclica de la temperatura Utilizando un sistema termostático, no la incertidumbre de la temperatura media. El valor de la desviación de la temperatura media
θ
= 19,9 °C - 20°C = - 0,1 °C
se reporta diciendo que ella misma contiene una incertidumbre estándar debido a la incertidumbre en la temperatura media de la mesa de prueba ensayo igual a u( θ ) = 0,2 0, 2 °C en tanto que el ciclo de variación en el tiempo produce una distribución de temperatura en forma de U (arcosen), lo que produce una incertidumbre estándar de:
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GUÍA TÉCNICA COLOMBIANA
GTC 51
La desviación de temperatura θ puede considerarse igual a θ , y la incertidumbre incertidumbre estándar de θ se obtiene de u2(θ) = u2( θ ) + u2 (Δ) = 0,165 °C
de donde resulta u(θ) = 0,41 °C
∂θ = l sδα = 0 como se indicó en el numeral H.1.3, esta incertidumbre tampoco Ya. que cθ = ∂ f/ ∂θ contribuye en nada a la incertidumbre / a primer orden, sin embargo, tiene una contribución a segundo orden que se discute en H.1.7. H.1.3.5 Incertidumbre de la diferencia en los coeficientes coeficientes de expansión, expansión, u(δα). Los límites -6 -1 estimados de la variabilidad de δα son ± 1 x 10 °C , con la misma probabilidad de que δα tome cualquier valor dentro de estos límites. La incertidumbre estándar es u(δα) = (1 x 10 -6 °C-1) / √3 = 0,58 x 10 -6 °C-1
H.1.3.6 Incertidumbre en en la diferencia de las las temperaturas de los los bloques, u(δθ). Se espera que el patrón y el bloque a calibrar estén a la misma temperatura, pero la diferencia de temperatura puede estar con igual probabilidad en cualquier punto dentro del intervalo estimado -0,05 °C a + 0,05 °C. La incertidumbre estándar es: u(δθ) = (0,05 °C)/ 3 = 0,029 °C
H.1.4 Incertidumbre estándar combinada. La incertidumbre estándar combinada uc(l) se calcula a partir de la ecuación (H.5). Los términos individuales son obtenidos y substituidos en esta ecuación obteniéndose:
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GUÍA TÉCNICA COLOMBIANA
GTC 51
H.1. H.1.55 Re Resu sult ltad ado o final final El certificado de calibración para el bloque calibre indica que la longitud a 20 °C es ls = 50,000 623 mn. La media aritmética de las observaciones repetidas de la diferencia en longitudes entre el bloque desconocido y el patrón es 215 nm. Así, puesto que l = ls+d (véase el numeral H.1.2), la longitud l del bloque desconocido a 20 °C es 50,000 838 mm. De acuerdo con el numeral 7.2.2, el resultado final de la medición puede establecerse como: = -50,000 838 mm con una incertidumbre estándar combinada uc = 32 nm. La correspondiente correspondiente incertidumbre estándar combinada relativa es uc /l = 6,4 x 1-7. l
H.1.6 H.1.6 Incert Incertidu idumbr mbree expand expandida ida.. Supóngase que se requiere obtener una incertidumbre expandida U 99 99 = K 99 99uc(l ) que proporciona un intervalo con un nivel de confianza de, aproximadamente, 99 por ciento. El procedimiento procedimiento a usar esta resumido en el numeral G.6.4, y los grados de libertad requeridos requeridos están indicados en la Tabla H.1. Estos se obtuvieron como se muestra a continuación: 1)
[véase el numeral H.1.3.1]. El certificado de calibración establece que el número de libertad efectivos de la incertidumbre estándar combinada a partir de los cuales se obtuvo la incertidumbre incertidumbre expandida citada es veff (ls) =18
2)
[véase el numeral H.1.3.2]. A pesar de que d se obtuvo a partir de cinco observaciones repetidas los grados de libertad de u(d ) son v(d )= 25-1 25-1= 24, 24, dado que u(d) se obtuvo de una desviación estándar experimental ponderada basada en 25 observaciones (véase la nota del numeral H.3.6). Los grados de libertad de u(d 1) la incertidumbre debida a efectos aleatorios del comparador, son v(d i) = 6-1 = 5 puesto que d i se obtuvo de seis mediciones repetidas. La incertidumbre para los efectos sistemáticos en el comparador, igual a ± 0,02 µm, puede asumirse confiable en un 25 por ciento, y esta forma, los grados de libertad de la ecuación (G.3) en el numeral G.4.2 son v(d 2) = 8 (véase el ejemplo del numeral G.4.2) Los grados de libertad efectivos de u(d ) , , veff (d ) , , se obtienen entonces de la ecuación (G.2b) en el numeral G.4.1:
Incertidumbre de la calibración del patrón, u(ls)
Incertidumbre de la diferencia en las longitudes medidas, u(d)
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Incertidumbre Incertidumbre de la diferencia de los coeficientes de expansión, u(δα)
3)
[véase el numeral H.1.3.5]. Las cotas estimadas de ± 1 x 10 °C de la variabilidad de se consideran contables en un 10 por ciento. Esto da, de la ecuación (G.3) en el numeral G.4.2, v(δα) = 50.
4)
Incertidumbre de la diferencia de temperatura en los bloques, u(δθ)
-6
-1
[véase el numeral H.1.3.6]. El intervalo intervalo estimado de -0,05°C -0,05°C a + 0,05°C 0,05°C para la diferencia de temperatura δθ se cree contable solamente al 50 por ciento, lo cual utilizando la ecuación (G.3) en G.4.2, da v(δθ) = 2.
El cálculo de veff (l) de la ecuación (G.2b) en el numeral G.4.1 procede de la misma manera que el cálculo de veff (d ) en 2) de esta sección. De esta forma, de las ecuaciones (H.6b) y de (H.6c) y los valores dados para v de 1) a 4), se obtiene: v (l ) = eff (25 nm) 4 18
+
(32nm) 4 (9, 7 nm)4 (2, 9 nm) 4 + 50 25,6
+
(16, 6 m) 4 2
= 16,7
Para obtener la incertidumbre expandida requerida, este valor se trunca primero al entero menor mas cercano, veff (l) = 16. De aquí se sigue, a partir de la Tabla G.2 en el Anexo G, que t 99 99 (16) uc( I ) = 2,92x(32 nm) = 93 nm. De acuerdo con 99,(16) = 2,92 y, entonces, u99 = t 99 7.2.4, el resultado final de la medición puede anunciarse como: = (50,000 838 ± 0,000 093) mm, donde el número que sigue al símbolo ± es el valor numérico de una incertidumbre expandida U = kuc, con U determinada a partir de una incertidumbre incertidumbre estándar combinada uc = 32 nm y un factor de cobertura k = 2,92 basado en la distribución t para v = 16 grados de libertad, y que define la estimación del intervalo con un nivel de confianza del 99 por ciento. La incertidumbre expandida relativa correspondiente es -6 U / l = 1,9 x 10 l
H.1.7 H.1.7 Términ Términos os de segu segundo ndo orde orden n La nota en el numeral 5.1.2 indica que la ecuación (10), la cual es usada en este ejemplo para obtener la incertidumbre estándar combinada (l) debe aumentar cuando la no
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GUÍA TÉCNICA COLOMBIANA
GTC 51
pero solamente el primer término contribuye significativamente a
uc( I ):
lsu(δα) u(θ) = (0,05m)(0,58 m x 10 °C )(0,41°C) = 11,7 nm -6
-1
lsu(αs) u(δθ).= (0,05m)(1,2 x l0 -6°C-1)(0,029 °C) =1,7 nm
Los términos de segundo orden incrementan a H.2
u c( l )
de 32 nm a 34 nm.
Medici Medición ón simult simultáne áneaa de de resi resiste stenci nciaa y reacta reactanci nciaa
Este ejemplo muestra el tratamiento de múltiples mensurandos determinados simultáneamente simultáneamente en la misma medición, y la correlación de sus estimaciones. Se consideran solamente las variaciones aleatorias de las observaciones; en la práctica real, las incertidumbres de las correcciones para efectos sistemáticos contribuirían, también, a la incertidumbre de los resultados de medición. Los datos son analizados en dos formas diferentes obteniéndose obteniéndose en cada caso, esencialmente, los mismos valores numéricos. H.2.1 H.2.1 El probl problema ema de la medici medición ón La resistencia R y la reactancia X de un elemento de circuito se determinan midiendo la amplitud V de una diferencia de potencial alterno sinusoidal en sus terminales, la amplitud I de la corriente altema que pasa a través suyo, y el ángulo de corrimiento de fase φ de la diferencia de potencial altema con relación a la corriente altema. De esta manera, las tres cantidades que se miden son V, I y φ y los tres resultados resultados - los mensurandos mensurandos - son las tres 2 2 componentes de la impedancia R, X y Z. Ya que Z = R + X 2, existen solamente dos cantidades independientes. H.2.2 El modelo modelo matemátic matemático o y los datos datos Los mensurandos están relacionados a los argumentos mediante la ley de Ohm: R =
V I
cosφ ; X
= V I senφ ; Z = V I
(H.7)
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GUÍA TÉCNICA COLOMBIANA
GTC 51
La primera aproximación está resumida en la Tabla H.3. Los valores de los tres mensurandos R, X y Z se obtienen a partir de las relaciones dadas en la ecuación (H.7) usando los valores medios de V, I y φ de la Tabla H.2 para V, I y φ. Las incertidumbres estándar de R, X y Z se obtienen de la ecuación (16) en el numeral 5.2.2 puesto que, como se indicó anteriormente, los argumentos, V , I y φ están correlacionadas. correlacionadas. = V / Como ejemplo, considere Z = / I I . Identificando V con x1, I con X 2, y f con Z = V / / I I , la ecuación (16) en el numeral 5.2.2 da, para la incertidumbre estándar combinada de Z
− 2 1 V − − V 1 u (V )u( I 2 u c2 ( z ) u 2 (V + u I 2 r V , I ( ) + − ) ( ) − 2 I I I I 2
(H.8a)
u V − u I − 2 − − − − ( ) ( ) 2 2 2 u(V ) u ( I ) r (V , I ) = z + z = 2z V I V I
(H.8b)
2
o
−
−
−
2 2 uc,r 2 ( Z ) = u r (V ) + u r ( I ) −
2u r (V )
u r (l )
r
(V , I )
(H.8c)
r ” ” en la última expresión Donde u( V )= s( V ) , u( I )= s( I ), y el subíndice “ r expresión indica que que u es una incertidumbre relativa. La substitución de los valores apropiados de la Tabla H.2 en la ecuación (H.8a) da como resultado, entonces uc( Z ) = 0,236 Ω
Ya que los tres mensurandos o resultados dependen de los mismos argumentos, también están correlacionados. Los elementos de la matriz de covarianza que describe esta correlación puede ser escrita en general como:
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GUÍA TÉCNICA COLOMBIANA
GTC 51
y2 = X
x2 = I
y3 = Z
x3 = φ
N = 3
Los resultados de los cálculos de R, X y Z y de sus varianzas estimadas así como de sus coeficientes de correlación se dan en la Tabla H.3. H.2.4 Resultad Resultados: os: segunda segunda aproxima aproximación ción.. La segunda aproximación está resumida en la Tabla H.4. Ya que los datos han sido obtenidos como cinco conjuntos de observaciones de los tres argumentos V, I y φ, es posible calcular un valor para R, X y Z a partir de cada conjunto de datos de medición, y entonces tomar la media aritmética de los cinco valores individuales para obtener la mejor estimación de R, X y Z. La desviación estándar experimental de cada media (la cual es su incertidumbre estándar combinada) se calcula entonces a partir de los cinco valores individuales en la forma usual [ecuación (5) en el numeral 4.2.3]; y las covarianzas estimadas de las tres medidas se calculan aplicando la ecuación (17) en el numeral 5.2.3 directamente a los cinco valores individuales a partir de los cuales se obtuvo cada media. Tabla H.2 Valor de los argumentos V, I y φ obtenidos a partir de cinco conjuntos de observaciones simultáneas
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GUÍA TÉCNICA COLOMBIANA
GTC 51
de cada uno de los cinco conjuntos de observaciones. Las medias aritméticas, la incertidumbre estándar y los coeficientes de correlación estimado se calculan, entonces, directamente utilizando esos valores individuales. Los resultados numéricos obtenidos de esta forma difieren, de manera despreciable, despreciable, de los resultados dados en la Tabla H.3. En la terminología de la nota 4.1.4, la segunda aproximación es un ejemplo de como se n obtiene la estimación de y a partir de Y = ( ∑ Y k )/ n, en tanto que la primera aproximación k =1
f (X 1 ,X 2 ,.. X N ). Como se indicó en esa nota, es un ejemplo de como se obtiene y a partir de y = f(X en general, las dos aproximaciones darán resultados idénticos si f es una función lineal de sus argumentos (siempre y cuando se tomen en cuenta los coeficientes de correlación observados experimentalmente cuando se implementa la primera aproximación). Si f no es una función lineal, entonces los resultados de la primera aproximación serán diferentes de los de la segunda aproximación dependiendo del grado de no linealidad y las varianzas y covarianzas estimadas de las X i. Esto puede verse en la expresión
y
=
− −
−
f ( X1, X 2 ,...., X N ) ,
(H.10)
N N
∂ 2 f 2 i =1 j =1 ∂ X i∂ X j j
+ 1
∑∑
Donde el segundo término en el miembro derecho está compuesto por los términos de segundo orden en la expansión en serie de Taylor de f en términos de X i (véase también la
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Tabla H.3. Valores calculados de los mensurandos R, X y Z: primera aproximación Índice de mensurando l 1
Relación entre la estimación del Valor de la estimación Incertidumbre estándar mensurando yi y la estimación del de yi , la cual es el combinada u c( yi) del resultado de la medición resultado de la medición argumento xi uc( R) = 0,071 Ω y1 = R = 127,732Ω y1 = R= ( V / / I I ) cos φ uc( R) /R = 0,06x10-2
2
y2
= X = ( V / / I I ) sen φ
y2
= X = 219,847Ω
uc( X ) = 0,295Ω uc( X ) /X = 0,13 x 10-2
3
y3
= Z = V / / I I
y3
= Z = 254,260Ω
uc( Z ) = 0,236Ω uc( Z ) /Z =0,09 x 10-2
Coeficientes de correlación r(y l , y m) r ( y1 , y 2) = r ( R,X ) = - 0,588 0,588 r ( y1 , y 3) = r ( R,Z ) = - 0,485 0,485 r ( y2 ,y3) = r ( X,Z ) = 0,993
Tabla H.4. Valores calculados de los mensurandos R, X y Z; segunda aproximación
Conjunto número
Valores individuales de los mensurandos
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Si los datos de la Tabla H.2 se reinterpretan de manera que la segunda aproximación sea inapropiado, inapropiado, y si se asume que no hay correlación entre las cantidades V , I y φ , entonces los coeficientes de correlación observados no tienen significado y deben ser iguales a cero. Si esto es hecho en la Tabla H.2, la ecuación (H.9) se reduce a una ecuación equivalente a (F.2) en el numeral F.1.2.3, es decir, N ∂ y1 u(y1 , y m ) = ∑ i =1 ∂ x i
∂ ym ∂ xi
u 3 ( x i )
(H.11)
y su aplicación a los datos de la Tabla H.2 da lugar a los cambios en la Tabla H.3 que se muestran en la Tabla H.5. Tabla H.5. Cambios en la Tabla H.3 suponiendo que los coeficientes de correlación de la Tabla H.2 son cero. Incertidumbre estándar combinada u c( y1) del resultado de la medición uc( R) = 0,195 Ω uc(( R) /R = 0,15 x 10-2 uc( X ) = 0,201 Ω uc( X ) /X = 0,09 x 10-2 uc( Z ) = 0,204 Ω
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GUÍA TÉCNICA COLOMBIANA
GTC 51 (H.12)
b(t ) = y 1 + y 2 ( t - t o)
es ajustada para las correcciones y temperaturas medidas, utilizando el método de los mínimos cuadrados. Los parámetros y1, y y2. los cuales son, respectivamente, la ordenada al origen y la pendiente de la curva de calibración, son los dos mensurandos o resultados a determinar. La temperatura t 0. es una temperatura de referencia exacta elegida convenientemente; no es un parámetro independiente que vaya a ser determinado mediante el ajuste por mínimos cuadrados. Una vez que y1 y y2 son determinados, junto con sus varianzas y covarianzas estimadas, la ecuación (H.12) puede ser usada para predecir el valor y la incertidumbre estándar de la corrección que se aplicará al termómetro para cualquier valor t de la temperatura. H.3.2 Ajuste Ajuste por mínimos mínimos cuadrado cuadradoss Basándose en el método de los mínimos cuadrados, y bajo las consideraciones hechas en el y2 y sus varianzas y covarianzas estimadas se obtienen numeral H.3.1, los mensurandos y1 y y al minimizar la suma n
S
∑=
= [ bk − y1 − y2 (tk − to )]
2
k i
y2, sus varianzas experimentales Esto da como resultado las siguientes ecuaciones para y1 y y experimentales 2 2 s ( y1) y s ( y2) , y sus coeficientes de correlación estimado r ( y1 ,y2) = s( y1 , y2) / s( y1) s( y2), donde
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GUÍA TÉCNICA COLOMBIANA
GTC 51
D
= n ∑ θ 2k (∑ θ k )2
(H.13g)
− 2 − 2 D =n ∑ θ k − θ =n ∑ t k − t / n; y t = (Σt k k ) / / n; [bk Donde todas las sumas van desde k = 1 hasta n, θk = t k k - t 0, θ = (Σθk ) / b(t k k )] es la diferencia entre la corrección medida u observada bk a la temperatura t k k y la corrección b(t k k ) predicha por la curva ajustada b(t)=y1+y2(t - t 0) a tk . La varianza s2 es una medida de la incertidumbre total del ajuste, donde el factor n - 2 refleja el hecho de que, debido a que dos parámetros, y1 y y2, se determinan mediante las n observaciones, los grados de libertad de 2 (véase el numeral G.3.3).
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GUÍA TÉCNICA COLOMBIANA = s ( y1) y u ( y2) = s ( y2),
GTC 51
se puede obtener
uc2 [b(t )] )]= u ( y1) + ( t- t 0) u ( y2) + 2 (t - t 0) u( y1) u( y2) r ( y1 ,y2) 2
2
2
La varianza estimada uc2 [b(t) tiene un mínimo en presente caso, es t min min = 24,0085 °C.
t min min = t o - u( y1) r ( y1 ,y 2) /u(y 2 ), que,
en el
Tabla H.6. Datos usados para obtener una curva de calibración lineal para un termómetro mediante el método de mínimos cuadrados. Lectura número K
Lectura del termómetro t k (°C)
Corrección observada b k = t R.k -t k (°C)
Corrección predicha b (t k k ) (°C)
Diferencia entre la observación y la corrección predicha b k - b(t k ) (°C)
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GUÍA TÉCNICA COLOMBIANA
GTC 51
uc2 [b(30 ° C)] = (0,0029 °C)2 + (10 °C) (0,000 67) + 2 (10 °C ) (0,0029 °C) (0,00067) (0,00067) (-0,930) (-0,930) 2
2
2
17,1x10 6 °C
o uc[b(30 °C)] = 0,0041 °C
Así, la corrección a 30 °C es -0,1 -0,149 4944 °C, con una incertidumbre estándar combinada de uc=0,0041 °C; uc, tiene v = n -2= 9 grados de libertad. H.3.5 Eliminación de de las correlaciones entre la pendiente pendiente y la ordenada ordenada al origen La ecuación (H. 13e) para el coeficiente de correlación n
n
r ( y1 , y2),
implica que si to es elegido
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GUÍA TÉCNICA COLOMBIANA
GTC 51
La aplicación de estas relaciones a los resultados dados en H.3.3. proporciona proporciona b(t ) = -0,1625 (11) + 0,002 18 (67) ( t-24,0085 °C) uc2 [b(t )] )] = (0,0011) 2 + ( t - 24,0085°C) 2 (0,000 67) 2
(H.17a) (H.17b)
El hecho de que estas expresiones dan el mismo resultado que las ecuaciones (H.14) y (H. 15) puede ser verificado repitiendo el cálculo de b(30 °C) y uc[b(30 °C)]. La substitución de t = 30 °C en las ecuaciones (H.17a) y (H. 17b) da como resultado: b(30 ° C) = - 0,149 0,1494 4 °C uc[b(30 ° C)] = 0,0041 °C
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GUÍA TÉCNICA COLOMBIANA
GTC 51
e s la varianza experimental de la í-ésima serie de ni observaciones Donde si2 , es repetidas independientes [ecuación (4) en el numeral 4.2.2] y tienen vi = ni - 1
grados de libertad. Los grados de libertad de
s p2
son v = ∑
n k
=1
vi,.
La varianza
experimental s p2 / m (y la desviación es sp / m ) de la la media media aritmética aritmética de m observaciones independientes caracterizadas por la estimación ponderada de la varianza s p2 tiene también v grados de libertad. 2)
Suponga que cada una de las t k k tienen incertidumbre despreciable, que una corrección εk , se aplica a cada uno de los n valores de t ,k y que cada r r corrección tiene la misma incertidumbre estándar ua. Entonces la incertidumbre y’ 1) es reempla y1) + estándar de cada bk = t R,K - t k k es también ua, y s2( y’ reemplazada zada por por s2( y 2 2 2 2 uo y s ( y y’ 1 ) es reemplazada por s ( y y’ 1) + ua
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GUÍA TÉCNICA COLOMBIANA t s ,t B ,t x
GTC 51
son los lapsos desde el tiempo de referencia t = 0 hasta el punto medio de los intervalos de conteo corregidos por tiempo muerto T 0 = 60 min para el patrón, el testigo, y los frascos de muestra respectivamente; aunque t B, se da por completez, no es necesario para el análisis;
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Las ecuaciones (H.18a) y (H. 18b) indican que no pueden promediarse directamente ni los seis valores individuales de C s s ni los de C x x dados en la Tabla H.7 debido al decaimiento exponencial de la actividad del patrón y de la muestra, y a las ligeras variaciones en el conteo de fondo de un ciclo al otro. En lugar de eso, se debe tratar con el decaimiento y el conteo de fondo corregidos (o razones de conteo definidos como el número de cuentas dividido por T = 60 min). Estos sugiere combinar las ecuaciones (H.18a) y (H.18b) para
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Las medias aritméticas de R s, R x y R , y sus desviaciones estándar experimentales s( R s ), s( R x) y s( R ), se calculan de la manera usual [ecuaciones (3) y (5) en el numeral 4.2]. El R x, R ) coeficiente de correlación r( R s se calcula a partir de la ecuación (17) en el numeral 5.2.3 y la ecuación (14) en el numeral 5.2.2.
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GUÍA TÉCNICA COLOMBIANA
GTC 51
-
ince incert rtid idum umbr bree aso asoci ciad adaa con con la posi posibl blee dep depen ende denc ncia ia de la efic eficie ienc ncia ia de detección del contador de centelleo con la fuente usada (patrón, testigo y muestra),
-
ince incert rtid idum umbr bree en en la la cor corre recci cción ón del del tie tiemp mpoo mue muert rtoo del del cont contad ador or y en la corrección de la dependencia de la eficiencia de conteo con el nivel de
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GUÍA TÉCNICA COLOMBIANA uc2 ( A x ) A x2
GTC 51
=
u2 ( R )
u2 ( As ) As2
+
u (R )
u2 ( m ) s ms2
+
u2 ( m ) x mx2
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GUÍA TÉCNICA COLOMBIANA
GTC 51
me d i c i ó n
3.1, 3.1.1, B.2.5
medición, jerarquía de la
7.1.1
medición, método de
v éa s e m é t od o d e m edi c i ón
medición, modelo matemático de la
3.1.6, 3.4.1, 3.4.2, 4.1, 4.1.1, 4.1.2
medición, papel de NAOVA en la
H . 5 . 3 e t s eq q
medición, exactitud de la
véase exactitud de la medición
medición, principio de
véase principio de medición
medición, procedimiento de
3 . 1 . 1 , 7 . 1 . 2 , B. 2 . 8 , F . 1 . 1 . 2
medición, resultado de la
1 . 3 , 3 . 1 . 2 , B . 2 . 11
medición, resultado de la, y su incertidumbre, disponibilidad disponibilidad de la información describiendo describiendo el 7.1.1, 7.1.3 medición, resultado de la, y su incertidumbre, formatos para reportar el 7.2.2, 7.2.4 medición, resultado de la, y su incertidumbre, reportando en detalle e 7.1.14, 7.2.7 mediciones, espectro de las, al cual se aplican los principios de esta guía 1.1 me n s u r a n d o
1.2, 3.1.1, 3.1.3, B.2.19, D.1, D.1.1, D.1.2, D.3.4
mens mensua uara rand ndo, o, defi defini nici ción ón o espe especi cififica caci ción ón del del
véas véasee mens mensur uran ando do
mensurando, estimación de un
4.1.4, 4.1.5, 7.2.5
mensurando, mensurando, incertidumbre debida a la definición incompleta del véase incertidumbre debida a la definición incompleta del mensurando mensurando, mejor medición posible del
D.3.4
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GUÍA TÉCNICA COLOMBIANA
GTC 51
mensurandos mensurandos, covarianza de relacionados véase correlación de estimaciones de mensuranados método de medición
3.1.1, B.2.7
métdpo de medición, incertidumbre del métod todo de de med medic iciión, ón, un unida idad dep depeendien diente te del
véase incertidumbre del método de medición H.6 H.6
metrología legal
3.4.5
mínima, incertidumbre
véase incertidumbre mínima
mínimos cuadrados, método de los
4,2,5, G.3.3, H.3, H.3.1, H.3.2
modelo matemático de la medición
véase medición, modelo matemático de la
momento central de orden q
C.2.13, C.2.22, nota 1 de E.3.1
muestra, incertidumbre de la
véase incertidumbre de la muestra
N nivel de confianza
0.4, nota 1 de 2.3.3, nota 1 y 2 de 2.3.5, 3.3.7, 3.3.7, 4.3.4, 6.2.2,6.3 6.2.2,6.3.1 .1 - 6.3.3, G, G.1.1 G.1.1 G.1.3, G.2.3, G.3.2, G.3.4, G.4.1, G.6.1, G.6.4, G.6.4, G.6.6
nivel de confianza mínimo
F.2.3.2
no corregido, resultado
B . 2 . 12
no lineal, relación funcional
véase funciona, relación
no lineal normal, distribución
nota 1 de 4.2.3, nota de 4.3.2, 4.3.4 4.3.6, nota 1 de 4.3.9, 4.4.2, 4.4.6, C.2.14, E.3.3, F.2.3.3, G.1.3, G.1.4, G.2.1, - G-2.3, nota 2 de G.5.2
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GUÍA TÉCNICA COLOMBIANA
GTC 51
O observaciones, observaciones, pares independientes independientes de, simultáneas 5.2.3, C.3.4, F.1.2.2, F.1.2.2, H.2.2, H.2.4, H.4.2 observaciones repetidas
3.1.4 - 3.1.6, 3.2.2, 3.3.5, 4.2.1, 4.2.3, 4.3.1, 4.4.1, 4.4.3, 5.2.3, E.4.2, E.4.3, F.1, F.1.1, F.1.1.1, F.1.1.2, G.3.2
OIML
i, ii, v, A.3, B.1
Oficina Internacional de Pesas y medidas
v é as e B I P M
orden mayor, términos de
nota de 5.1.2, E.3.1, H.1.7
Organizac Organización ión Internac Internaciona ionall para la Normaliza Normalización ción véase véase ISO Organización Organización Internacional para la metrología legal véase OIML P parámetros
C . 2. 7
particular, magnitud
3. 1. 1 , n o t a 1 B . 2. 1
p o b l ac i ó n
C . 2 . 16
ponderada, estimación, de la varianza
véase va varianza, es estimación po ponderada de de la
precisión
n o t a 2 d e B. 2 . 1 4
principio de medición
B.2.6
probabilidad
3.3.5,4.3.7 - 4.3.9, C.2.1, E.3.5, E.3.6, F.2.3.3
probabilidad de cobertura
véase cobertura, probabilidad de
probabilidad, probabilidad, convolución de distribuciones distribuciones de
nota 2 de 4.3.9, G.1.4 - G-1.6, G.2.2. G.6.5
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GUÍA TÉCNICA COLOMBIANA
GTC 51
probabilidad, función de, de masa
C.2.6
probabilidad, función de densidad de
3.3.5, no nota 2 de 4. 4.3.8, 4. 4.4.2, 4. 4.4.5, 4. 4.4.6, C.2.5, F.2.4.4
probabilidad subjetiva
3.3.5, D.6.1
promedio
véase media aritmética
propagación del error, ley general de
véase err error, ley ley gen general de pr propagación del
propagac propagación ión de la incertidum incertidumbre, bre, ley general general de
véase incertid incertidumbr umbre, e, ley general general de propagación propagación de la
R realizada, magnitud
D.2, D.2.1, D.3.1 - D.3.3, D.4
Recomendación INC-1 (1980)
i, v, 0.5, 0.7, 3.3.3, 6.1.1, 6.1.2, 6.3.3, A.1, A.3, E, E.2.3, E.3.7
Recomendación 1 (CI-1981),CIPM
i , 0. 5 , 6. 1 . 1 , A . 2 , A . 3
Recomendación 1 (CI-1986),CIPM
0.5, 6.1.1, 6.1.2, A.3
referencia, certificación de materiales de
H.5, H.5.3.2
relativo, error
B . 2 . 20
repetibilidad, repetibilidad, condiciones de
3.1.4, nota 1 de B.2.15
repetibilidad de resultados de mediciones
B . 2 . 15
repeticiones independientes
véase independientes, repeticiones
repetidas, observaciones
véase observaciones repetidas
repr reprod oduc ucib ibililid idad ad de resu resultltad ados os de medi medici cion ones es
B.2. B.2.16 16
resultado corregido
B.2.13, D.3.1, D.3.4, D.4
resultado de la medición
véase medición, resultado de la
resultado no corregido
véase no corregido, resultado
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GUÍA TÉCNICA COLOMBIANA
GTC 51
S seguridad, límite de
nota de 6.3.1
s e n s i t i v i d a d , c o e f ic i e n t e d e
5.13, 5.14
sensitividad, determinación determinación de los coeficientes experimentales de 5.1.4 sesgo
nota de 3.2.3
Sistema Internacional de Unidades (SI)
0.3, 3.4.6
s i s t em á t i c o
3.3.3, E.1.3, E.3.4 - E.3.7
sistemático, efecto
3.2.3, 3.2.4, 3.3.1,3.3.2, 3.3.3, D.6.1, E.1.1, E.3, E.4.4
sistemático, error
3.2.1, 3.2.3, B.2.22
Student, distribución de
C . 3. 8 , G. 3 . 2
T Taylor, serie de
5.1.2, E.3.1, G.1.5, G.4.2, H.1.7, H.2.4
t, distribución
nota 1 de 4.2.3, C.3.8, G.3, G.3.2, G.3.4, G.4.1, G.4.2, G.5.4, G.6.2
t, distribución, cuantiles de la
nota de G.3.4
Teorema del límite central
G.1.6, G.2, G.2.1 - G.2.3, G.6.2, G.6.5, G.6.6
t, factor
E.3.3, G.3.2, G.3.4, G.4.1, G.5.4, G.6.4 G.6.6
Tipo A, evaluación de la covarianza
5.2.3
Tipo A, evaluación de la incertidumbre
2 . 3. 2 , 3 . 3 . 3 - 3. 3. 5 , 4 . 1. 6 , 4. 2, 4 . 2 . 1 4.2.8, 4.2.8, 4.3.2, 4.3.2, 4.4.1 - 4.4.3, 4.4.3, E.3.7, E.3.7, F.1, F.1, F.1.1.1 - F.1.2.4
Tipo A, incertidumbre estándar
3.3.5, 4.2.3, C.3.3
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GUÍA TÉCNICA COLOMBIANA
GTC 51
Tipo b, evaluación de la incertidumbre
2.3.3, 3.3.3 -3.3.5,4.1.6, 4.3, 4.3.1 4.3.11, 4.4.4 - 4.4.6, E.3.7, F.2 et seqq
Tipo B, incertidumbre estándar
3.3.5, 4.3.1, C.3.3
Tipo ipo B, ince ncertidu tidumb mbrre está stándar dar comb combin inad adaa
7.2 7.2.1, .1, not notaa 3 de G.4 G.4.1
Tipo B, necesidad de evaluaciones
F.2.1
Tipo B, varianza
4.3.1
tolerancia, intervalo de, estadístico
no t a 2 d e C . 2 . 3 0
total, incertidumbre
véase incertidumbre total
U unidad, uso del valor adoptado de un patrón de medición como una 3.4.6, nota de 4.2.8 unilateral, intervalo de confianza
C . 2 . 28
Unión Internacional de Química Pura y Aplicada véase IUPAC Unió Uniónn Inte Intern rnac acio iona nall de de Físi Física ca Pura Pura y Apli Aplica cada da
véas véasee IUPA IUPAP P
V valo valorr con conve venncio cional nal ve verdad dadero de la magnit gnituud
B.2. B.2.44
valor de una magnitud
311B22
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GUÍA TÉCNICA COLOMBIANA
GTC 51
varianza de la media
4.2.3, C.3.2
varianza de la media experimental
4.2.3, C.3.2
varianza, estimación ponderada de la (o desviación estándar experimental ponderada) 4.2.4, nota de 4.2.8, H.1.3.2, nota de H.3.6, H.5.2.2, H.5.2.5,H.6.3.1, nota de H.6.3.2 var varian ianza expe xperime rimenntal tal (o esti stimaci macióón de la)
4.2 4.2.2, .2, no nota de H.3. H.3.66
varianza relativa
5.1.6
varianza relativa combinada
5.1.6
VIM
2.1, 2.2.3, 2.2.4, B.1
vocabulario internacional internacional de términos básicos y generales en metrología véase VIM W Welch - Satterhwaite, fórmula de
G.4.1, G.4.2, G.6.2, G.6.4