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Search SpringerLink Search. Abstract A number of criteria of sample data concordance with the normal probability distribution law has been classified, and some recommendations for using these criteria in practice of engineering statistical analysis have been given.
Para la MFU nicamente debe aplicarse un medio de ensayo que haya aprobado el departamento responsable del proceso de ensayo previsto.
Una MFU se refiere exclusivamente a una caracterstica de fabricacin o un parmetro de mquina. Por regla general para la evaluacin es preciso registrar los distintos valores de medicin del muestreo. En el caso de valores de medicin registrados manualmente en forma de rayas en una divisin de clases de la gama de valores nomenclatura , en su lugar tambin puede registrarse la distribucin de frecuencia de los valores de medicin clasificados9 Para registrar ante todo nicamente la influencia de mquina en el marco de una comprobacin de capacidad de mquina MFU es preciso cumplir las siguientes condiciones en la fabricacin de las piezas del muestreo:.
Durante la MFU es preciso que sea el mismo operario quien maneje la mquina o la instalacin. La calidad del mecanizado previo de las caractersticas pendientes de valoracin debe corresponder a las instrucciones de fabricacin requeridas. En caso de que este volumen de muestreo sea difcil de conseguir debido a razones econmicas o tcnicas tambin se admite un volumen inferior.
En su caso deben considerarse unos valores lmite convenientemente superiores segn la tabla 3 o las frmulas 3. Sin embargo, el volumen efectivo del muestreo es decir, sin valores extraos debe ser de 20 como mnimo. Las piezas deben fabricarse inmediatamente una detrs de otra y es preciso numerar el orden de fabricacin de forma correspondiente. En cada pieza deben comprobarse todas las caractersticas especificadas.
Las piezas de ensayo deben fabricarse con las condiciones de serie exigidas para la mquina es decir, con el tiempo del ciclo y los valores de los parmetros de ajuste de mquina de la fabricacin en serie. Las especificaciones especiales deben acordarse en correspondencia con el proyecto para. Los cambios de herramienta, ajustes de herramienta manuales u otras modificaciones de parmetros de mquina no se admiten durante el tiempo de la MFU.
Esto no afecta a las correcciones automticas de la herramienta mediante mandos por medicin integrados. En caso de anomalas de mquina durante la MFU que repercutan en la caracterstica pendiente de comprobacin es preciso reiniciar la MFU. Es preciso que entre el proveedor y el cliente acuerden el mtodo de medicin que debe definirse antes de la comprobacin.
Dado que esta forma de registro de datos manual se contina aplicando en la prctica, se tiene en cuenta en esta norma aunque debido a esta circunstancia las posibilidades para la MFU se vean algo limitadas. Si las condiciones para la toma del muestreo mencionadas en el apartado 4.
Los modelos de distribucin asignados a los tipos de caractersticas ms importantes vase tambin VW deben extraerse de la tabla 2. Para los tipos de caracterstica no listados en la mayora de los casos puede realizarse una asignacin de una distribucin segn la siguiente regla: para caractersticas con tolerancias por ambos lados o con tolerancias unilaterales hacia abajo una distribucin normal.
Los valores extraos son valores de medicin que varan tanto respecto a los otros valores de medicin que es muy probable que no procedan de la misma totalidad que los valores restantes, como por ejemplo, las mediciones incorrectas.
Sin embargo, es preciso no borrar los valores extraos. Por el contrario, deben identificarse de forma correspondiente en la representacin grfica del desarrollo de los valores individuales y la cantidad debe indicarse en la documentacin.
Una variacin significativa de la posicin de la fabricacin puede originarse, por ejemplo, debido a la influencia trmica o a un desgaste de herramienta desarrollo de tendencia. En caso de que se haya registrado nicamente la distribucin de frecuencia de los valores de medicin clasificados, es imposible aplicar este ensayo.
Una desviacin del modelo de distribucin determinado puede producirse, por ejemplo, debido a diferentes lotes de material en la toma de muestreo distribucin mixta, vase tambin el ejemplo 3 en el apartado 5.
Una desviacin del modelo de distribucin determinado puede producirse, por ejemplo, debido a la toma de muestreo de diferentes herramientas distribucin mixta, vase tambin el ejemplo 3, en el apartado 5. Pgina 27 VW que se debe adaptar segn las frmulas 2.
La documentacin de una comprobacin de capacidad de mquina MFU relativa a una caracterstica debe contener como mnimo las siguientes informaciones y representaciones: Datos de cabecera:.
Departamento, operario y fecha de elaboracin Especificaciones sobre la pieza Denominacin, medida nominal y tolerancia de la caracterstica Especificaciones de mquina Especificaciones de medios de ensayo Perodo de fabricacin.
Representacin grfica del desarrollo de los valores individuales con los valores medios del muestreo con lneas lmite del intervalo de tolerancia en caso de valores individuales registrados. Histograma con el modelo de distribucin adaptado, lneas lmite del intervalo de tolerancia y del margen de dispersin, as como lnea del valor medio o bien del valor mediano.
Representacin en la red de probabilidad con el modelo de distribucin adaptado, lneas lmite del intervalo de tolerancia y del margen de dispersin, as como lnea del valor medio o del valor mediano vase [2].
Cantidad de valores medidos Cantidad de valores de medicin evaluados o valores extraos encontrados Valor estimado de la posicin de fabricacin Valores estimados de los lmites de dispersin o valor estimado del margen de dispersin Modelo de distribucin aplicado Resultado del ensayo de variacin de la posicin de fabricacin Resultado del ensayo de desviacin del modelo de distribucin especificado Valores caractersticos de capacidad calculados Cm y Cmk hasta la segunda decimal Valores lmite requeridos para Cm y Cmk.
En su caso indicacin de MFU limitada En su caso acuerdos especiales entre proveedor y cliente En su caso incidentes especiales durante la toma de muestreo. El hecho de que una mquina pueda evaluarse como idnea en lo que se refiere a la elaboracin de una caracterstica analizada depende de la siguiente valoracin de resultados: En el caso de que en una evaluacin resulten valores extraos, es preciso analizar sus causas.
En caso contrario, la mquina debe valorarse como no idnea. En su caso es preciso repetir la MFU. Cuando se presenta una variacin significativa de la posicin de fabricacin durante la toma de muestreo, por regla general es preciso conocer sus causas y aceptar sus efectos con el fin de cumplir la condicin previa de la capacidad de mquina como excepcin vase el ltimo prrafo de este apartado. En el caso de una evaluacin no distribuida debido a una desviacin significativa del modelo de distribucin especificado y a la caracterstica analizada no se le pueda asignar sin contradiccin ningn otro modelo de distribucin, es preciso conocer las causas y aceptar los efectos 10 con el fin de cumplir la condicin previa de la capacidad de mquina como excepcin vase el ltimo prrafo de este apartado.
Si no se ha acordado de otra forma, los valores caractersticos de capacidad determinados en el caso de un volumen de muestreo efectivo de n e 50 es decir, sin valores extraos deben cumplir el requisito. Para algunos volmenes de muestreo los valores lmite adaptados estn indicados en la tabla 3. En el caso de haber acordado otros valores lmite sobre la base de n e 50 es preciso determinar los correspondientes valores lmite adaptados segn las frmulas 3.
Tabla 3 - Valores lmite de la capacidad de mquina para 20 ne Pgina 29 VW En el caso de que resulte un valor caracterstico de capacidad inferior al valor lmite correspondiente, la mquina debe valorarse como no idnea. En el caso de un valor caracterstico de capacidad determinado c mk 2 ,33 corresponde a 14 y para caractersticas con tolerancias por ambos lados adems en caso de c m 2 ,67 corresponde a 16 , la mquina puede valorarse como idnea con respecto a la caracterstica analizada independientemente de si existe una variacin significativa de la posicin o una desviacin del modelo de distribucin especificado.
En el caso de no haber podido demostrar la capacidad de mquina respecto a la caracterstica analizada es preciso iniciar medidas para la optimizacin de la mquina. Para ello deben identificarse las influencias correspondientes por ejemplo, mediante la metdica estadstica de ensayo DOE y eliminarlas. En caso de no alcanzar la capacidad de mquina mediante optimizaciones de mquina econmicamente viables, en primer lugar debera comprobarse con ayuda del clculo estadstico de tolerancias segn la norma VW , si existe la posibilidad de una ampliacin de tolerancias para conseguir la capacidad de mquina.
Si mediante esta medida tampoco es posible alcanzar la capacidad de mquina es preciso tomar una decisin sobre si la aceptacin de la mquina es posible o no despus de haber acordado regulaciones especiales.
Estas regulaciones especiales deberan contener los siguientes puntos:. Se han determinado los siguientes valores caractersticos de muestreo:. La figura 12 ilustra el resultado de evaluacin. Dado que este valor es inferior al valor lmite 1, debido a la dispersin aleatoria de los valores caractersticos de muestreo segn la condicin 2.
Por consiguiente, el segundo valor de parmetro de la distribucin de valor tipo 2 que se debe adaptar se calcula segn el caso especial 2. Para el valor caracterstico Cm en el caso de una tolerancia unilateral hacia arriba no est definido ningn. Pgina 32 VW valor lmite, pero en comparacin con el valor Cmk se obtiene informacin sobe la posicin de fabricacin, con lo cual el valor Cm ms pequeo indica que ste est ms cerca del lmite cero natural que de la medida mxima.
Se realiza por tanto una evaluacin no distribuida segn el apartado 3. Envergadura segn la frmula 2. Los valores caractersticos de capacidad determinados indican que la mquina no cumple el requisito de capacidad respecto a la caracterstica analizada.
Una indicacin interesante en este contexto es la desviacin significativa de una desviacin normal esperada. De este modo se detecta un potencial de optimizacin, como aqu en el caso de una desviacin mixta. The discussion below is therefore confined to providing examples in the case of two quality.
For brevity, all the. In all cases, it is assumed that a single acceptance criterion is stipulated for each class of nonconformity, and. With two quality characteristics, some new notation is necessary. The two quality characteristics are denoted. The lower and upper specification limits on x are denoted by L x and U x respectively, and on y.
The process fraction nonconforming beyond each of these four limits is denoted by. Due to the independence of x and y, the total process fraction nonconforming in a class containing. The class of nonconformity is indicated by the appropriate subscript from A, B, etc. Expressions 2. The following examples. This PDF file may contain embedded typefaces.
This document establishes single sampling plans for conformance testing, i. Sampling plans are provided corresponding to four levels of discriminatory ability.
The limiting quality ratio LQR see Clause 4 of each sampling plan is given for reference. This document specifies an acceptance sampling system for inspection by attributes indexed by limiting quality LQ.
In many industrial situations, in which switchin This International Standard specifies sequential sampling plans and procedures for inspection by variables of discrete items.
The plans are indexed in terms of producer's risk point and the consumer's risk point. Therefore, they are suitable not only for the purposes of acceptance sampling, but for the more general purpose of the testing of simple statistical hypotheses for proportions. The purpose of this International Standard is to provide procedures for the sequential assessment of inspectio ISO specifies a system of single sampling schemes for lot-by-lot inspection by attributes.
All the sampling plans of the present system are of accept-zero form, i. The schemes depend on a suitably-defined average outgoing quality limit AOQL , the value of which is chosen by the user; no restrictions are placed on the choice of the value of the AOQL or on the sizes of successive lots in the series.
It also provides guidance on the selection of the appropriate inspection system for use in a particular situation. ISO provides double sampling plans by attributes for the acceptance inspection of lots of discrete items. Plans are provided for inspection for percent nonconforming and for inspection for nonconformities per items.
The lot is accepted if there are no nonconforming items ISO specifies sequential sampling plans and procedures for inspection by attributes of discrete items. The plans are indexed in terms of the producer's risk point and the consumer's risk point. Therefore, they can be used not only for the purposes of acceptance sampling, but for a more general purpose of the verification of simple statistical hypotheses for proportions.
The purpose of this International Standard is to provide procedures for sequential assessment of inspection results In addition, this International Standard provides requirements for alternative acceptance methods proposed by the supplier.
These guidelines are designed for inspection of populations of any product supplied or delivered in discrete items in lots. They are applicable to - supplier inspection final inspection, product certification upon supplier ISO specifies, for quality levels expressed as nonconforming items per million items, procedures for estimating the quality level of a single entity e.
Procedures are also specified for using this information when selecting a suitable sampling plan so as to verify that the quality level of a given lot does not exceed a stated limiting q This part of ISO addresses: - supplier inspection final inspection, product certification upon supplier's request ; - customer inspection incoming inspection, surveillance, acceptance sampling ; - third party inspection. This part of ISO may also be applicable when only one inspectio ISO specifies an acceptance sampling system of single sampling plans for inspection by variables.
It is indexed in terms of the acceptance quality limit AQL and is of a technical nature, aimed at users who are already familiar with sampling by variables or who have complicated requirements. A more introductory treatment is given in ISO The objectives of the methods laid down in ISO are to ensure that lots of an acceptable quality have a high probability of acc It is indexed in terms of the acceptance quality limit AQL and is designed for users who have simple requirements. A more comprehensive and technical treatment is given in ISO The objectives of the methods laid down in ISO are to ensure that lots of acceptable quality have a high probability of acceptance and that th ISO establishes sampling plans and procedures by variables that can be used to assess whether the quality level of an entity lot, process, etc.
The sampling plans have been devised so that their operating characteristic curves match those of the corresponding attributes plans in ISO as closely as possible, so that the choice between using sampling by attributes and sampling by variables is not influenced by attempts to increase the chance of acc ISO defines procedures for random sampling and randomization.
Several methods are provided, including approaches based on mechanical devices, tables of random numbers, and portable computer algorithms. ISO is applicable whenever a regulation, contract, or other standard requires random sampling or randomization to be used.
The methods are applicable to such situations as a acceptance sampling of discrete units presented for inspection in lots, b sampling for survey purpos Each item of product is countable and has specific characteristics that are measurable or classifiable as being conforming or nonconforming to a given specificatio ISO TR gives general guidance on the selection of an acceptance sampling system, scheme or plan.
It does this principally in the context of standards that either already exist or are presently under development. The guidance is confined to acceptance sampling of products that are supplied in lots and that can be classified as consisting of discrete items i. It is assumed that each item in a lot can be identified and segregated from the other items in ISO specifies an acceptance sampling system of double sampling schemes for inspection by variables for percent nonconforming.
It is indexed in terms of the acceptance quality limit AQL. The objectives of the methods laid down in ISO are to ensure that lots of acceptable quality have a high probability of acceptance and that the probability of non-accepting inferior lots is as high as practicable. This is achieved by means of the switching rules, which provide automatic ISO specifies a system of sequential sampling plans schemes for lot-by-lot inspection by variables.
The schemes are indexed in terms of a preferred series of acceptance quality limit AQL values, ranging from 0,01 to 10, which are defined in terms of percent nonconforming items.
The schemes are designed to be applied to a continuing series of lots. ISO is designed for use under the following conditions: where the inspection procedure is to be applied to a continuing se Your shopping cart is empty! With this additional information it can be decided whether to carry out a directional test, or to carry out either a regression test or a characteristic function test, or no test at all.
In addition, although such a graphical representation cannot be considered as a rigorous test, the summary information that it provides is an essential supplement to any test for departure from the normal distribution. In the case of rejection of the null hypothesis it is often possible to envisage by this means the type of alternative that might be applicable.
It consists of the calculation of a function T of the observations, which is called the test statistic. The null hypothesis of a normal distribution is then not rejected or rejected depending on whether or not the value of T lies within a set of values near to the expected value that corresponds to the normal distribution.
The significance level of the test is the probability P of obtaining a value of T within the critical region when the null hypothesis is correct. This level gives the probability of erroneously rejecting the null hypothesis error of the first kind.
The boundary of the critical region is or, in the case of a two-sided test, the boundaries of the critical region are the critical value s of the test statistic.
For example, a departure from the normal distribution which would become apparent when using a test for departure from the normal distribution on a large sample might not be detected by the same test if there were fewer observations.
When the form of departure from the normal distribution is specified in the alternative hypothesis, then the test is a directional test. However, when the form of departure from the normal distribution is not specified in the alternative hypothesis, the test is an omnibus test. In a directional test, the critical region is determined in such a way that the power of the test reaches its maximum value.
In an omnibus test, it is necessary to divide the critical region in such a way that the critical region consists of those values of the test statistic which lie far away from the expected value. In the case of asymmetry, for example, it centres either 4. However, when several alternatives are considered jointly, the test is multidirectional.
This is the case particularly when a non-null asymmetry and a kurtosis different from that of the normal distribution are considered together. Note that a test may result in the rejection of the null hypothesis at the 0,05 level and the non-rejection of this same hypothesis at the 0,Ol level. Subtotals, inter- mediate results and auxiliary quantities shall not be rounded to less than six significant digits.
On this paper, one of the axes in this International Standard it is the vertical axis is non-linearly scaled according to the area under the standardized normal distribution function and is marked with the corresponding values of the cumulative relative frequency.
The cumulative distribution function of the variable X then approximates to a straight line. Sometimes these two axes are interchanged with each other Furthermore, if a normalizing transformation of the variable X is made, the linear scale may be replaced by a logarithmic, quadratic, reciprocal or other scale.
On the vertical axis the values of the cumulative relative frequency are given as percentages, while the horizontal axis has an arbitrary linear scale. If a plot on this paper gives a set of points that appears to be scattered around a straight line, this provides crude support for the assumption that the sample can reasonably be regarded as having come from a normal distribution.
However, if there is a systematic departure from the straight line, the plot often suggests the type of distribution to be taken into consideration. The importance of this approach is that it easily provides visual information on the type of departure from the normal distribution. If the graph indicates that the data do not come from a simple homogeneous distribution, but rather from a mixture of two or more homogeneous subpopulations e.
It should be kept in mind that such a plot is in no way a test for departure from the normal distribution in the strict sense. In the case of small samples, pronounced curves may occur for normal distributions, whilst for large samples slight curves may indicate non-normal distributions.
It is immediately seen from this graph that these points do not form a straight line. Therefore, and since the scale for the cumulative relative frequency widens towards the extremes, a few values at either end of the cumulative distribution which distinctly depart from the straight line defined by the middle values cannot be regarded as indicators of departure from the normal distribution. The upper parts of figures 3 to 7 show the cumulative distribution function in comparison with the corresponding density function shown in the lower part of each figure.
The graphs of the cumulative distribution functions shown in figures 5 and 6 correspond to a density function with positive skewness and negative skewness respectively. Figure 7 shows the cumulative distribution function and the density function of a superposition of two different density functions. Generally, a d istribution wit h posit ive skewness has a h igher dispersion a mongst the high values of the variable than amongst the low ones; the co ntrary is the case for n egative skewness 6.
Compared with the normal distribution, a distribution with kurtosis in excess tends to have a preponderance of values of the variable both close to the average and towards both extremes. The contrary is the case for a kurtosis in default. For example, the fact that a variable is non-negative, with a mean close to zero in comparison with the value of the standard deviation, may be a physical reason for positive skewness of the real distribution.
In any case, the choice of a directional test should be based on general considerations regarding the nature 6. Table 2 gives 50 independent measurements of the depth of the sapwood in pieces of wood intended for use as telegraph poles.
As the depth of sapwood is a characteristic having essentially non-negative values close to zero, positive skewness may be assumed. Table 2 - Depth of sapwood I,25 2,05 2,60 3,lO 4,00 2,60 I,35 2,lO 3,15 4,00 I,40 2,15 2,70 3,15 4,05 I,50 2,15 2,75 3,20 4,05 2,15 2,75 3,30 I,55 4,lO I,60 2,20 2,80 3,45 4,20 2,25 2,95 I,75 3,50 4,45 I,75 2,35 2,95 3,50 4,50 I,85 2,40 3,00 3,80 4,70 2,55 3,05 3,90 5,lO I,95 NOTE - Series arranged according to the non- decreasing values of 50 observations.
Table 3 shows a series of 50 independ
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