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Statistical Process and Quality Control
Published in William M. Mendenhall, Terry L. Sincich, Statistics for Engineering and the Sciences, 2016
William M. Mendenhall, Terry L. Sincich
The information provided by tolerance intervals is often used to determine whether product specifications are being satisfied. Specification limits, unlike tolerance or control limits, are not determined by sampling the process. Rather, they define acceptable values of the quality variable that are set by customers, management, and/or product designers. To determine whether the specifications are realistic, the specification limits are compared to the “natural” tolerance limits of the process, that is, the tolerance limits obtained from sampling. If the tolerance limits do not fall within the specification limits, a review of the production process is strongly recommended. An investigation may reveal that the specifications are tighter than necessary for the functioning of the product, and, consequently, should be widened. Or, if the specifications cannot be changed, a fundamental change in the production process may be necessary to reduce product variability.
Statistical Analysis of Fatigue Test Data according to Eurocode 3
Published in Nigel Powers, Dan M. Frangopol, Riadh Al-Mahaidi, Colin Caprani, Maintenance, Safety, Risk, Management and Life-Cycle Performance of Bridges, 2018
The tolerance interval describes an interval that is expected to enclose the intercept of a certain proportion, for example 95%, of the whole population. In (Holický 2005, p. A-23) the tolerance interval approach is also referred to as coverage method of fractile estimation. The lower tolerance bound is computed by (Hahn & Meeker 1991, p. 60): () loga^−k1−α,p,n⋅s
Basic Mathematics
Published in M. Modarres, What Every Engineer Should Know About Reliability and Risk Analysis, 2018
It is important not to confuse confidence interval with other types of intervals in statistics. (e.g., tolerance interval or prediction interval). The confidence interval displays limits with some confidence (e.g., 95%) within which a population parameter (e.g., mean or standard deviation) exists. On the other hand, tolerance interval displays limits (with some confidence) within which some prespecified percentage of the population is contained. Finally, prediction interval displays limits within which some future observations are expected to occur. For further discussion in different types of intervals refer to Hill and Prane (1984).
Integrated parameter and tolerance optimization of a centrifugal compressor based on a complex simulator
Published in Journal of Quality Technology, 2020
Mei Han, Xuejun Liu, Min Huang, Matthias H. Y. Tan
In this section, we formulate the parameter and tolerance design problem for the centrifugal compressor as an optimization problem. Let the nominal values and tolerances of the design variables be denoted by and respectively. Then, each design variable has a tolerance interval given by and a manufactured compressor is accepted for delivery to the customer if each design variable lies within its tolerance interval. Note that the term “tolerance interval” in the article does not refer to a statistical tolerance interval; rather, it refers to what is commonly called a specification interval. It is also noted that each compressor delivered can be assumed to have a randomly distributed with support equal to the hypercube bounded by and This is more reasonable than assuming has a Gaussian distribution, since the geometric design variables represented by are bounded.
Approximate two-sided tolerance interval for sample variances
Published in Quality Engineering, 2020
Yuhui Yao, Martin G. C. Sarmiento, Subha Chakraborti, Eugenio K. Epprecht
Tolerance intervals are useful in several fields, including manufacturing and industrial statistics (statistical quality control), engineering (reliability), pharmacy, environmental science, occupational and industrial hygiene, hydrology, medicine, meteorology, and economics (see, e.g., Fernandez 2010; Gibbons, Bhaumik, and Aryal 2009; Gibbons, Coleman, and Coleman 2001; Lee and Liao 2012; Millard and Neerchal 2000). Statistical tolerance intervals, which are different from confidence intervals and prediction intervals, provide a range of values in which a specified proportion of a population of interest lie with a specified high confidence level. For details about tolerance intervals and their usage see, e.g., Hahn (1970a, 1970b), Krishnamoorthy and Mathew (2009), Meeker, Hahn, and Escobar (2017), Ryan (2007), and Vardeman (1992). Tolerance intervals were initially introduced by Wilks (1941, 1942) for the normal distribution. Since then, several authors have studied tolerance intervals for a given continuous or a discrete distribution, called parametric intervals, as well as nonparametric tolerance intervals that apply to any continuous distribution. Some tolerance intervals have also been considered for regression and multivariate normal settings. For comprehensive literature reviews up to 1989, readers are referred to Patel (1986) and Jílek and Ackermann (1989). For more recent developments and extensive theoretical treatment of tolerance intervals, the books by Krishnamoorthy and Mathew (2009) and Meeker, Hahn, and Escobar (2017) are highly recommended.
A Probability-based Approach for the Definition of the Expected Seismic Damage Evaluated with Non-linear Time-History Analyses
Published in Journal of Earthquake Engineering, 2019
Cristina Cantagallo, Guido Camata, Enrico Spacone
This paper proposes a new approach to compute the expected seismic damage and demand which accounts for the number of the ground motion records used in the NLTHAs, the variability of the seismic demand, and its probability distribution. More specifically, the proposed procedure uses a new response quantity measure that stems from the statistical concepts of tolerance region and upper tolerance limit. Tolerance regions are widely used in economics and industrial applications to consider the uncertainty propagation of data. The first works on tolerance regions are attributed to Wilks [1941, 1942], Wald [1943] and Wald and Wolfowitz [1946], but the definition of this statistical measure has a long development history [Owen, 1962; Guttman, 1970; Aitchison and Dunsmore, 1975]. In general, a tolerance interval is a statistical interval within which a specified proportion of results falls, with a given confidence level. From an engineering viewpoint, this interval defines a range of allowable variation for the structural response variable. The endpoints of a tolerance interval are also called tolerance limits; the tolerance limits provide the upper and lower limits between which a prescribed proportion of the structural response is confidently expected to be found [Natrella, 1963]. When applied to NLTHAs, the upper tolerance limit of the results is shown to account for the uncertainty inherent to the seismic input and the variability of the seismic response.