Issue 
Int. J. Metrol. Qual. Eng.
Volume 13, 2022



Article Number  16  
Number of page(s)  12  
DOI  https://doi.org/10.1051/ijmqe/2022015  
Published online  29 November 2022 
Research Article
A double integration method for generating exact tolerance limit factors for normal populations
Nuclear Technology Expert Committee, Korean Agency for Technology and Standards, 93, Isuro, Maengdongmyeon, Eumseonggun, Chungcheongbukdo 27737, Korea
^{*} Corresponding author: pilsangkang@gmail.com
Received:
2
February
2022
Accepted:
13
October
2022
This article introduces a new method for generating the exact onesided and twosided tolerance limit factors for normal populations. This method does not need to handle the noncentral tdistribution at all, but only needs to do a double integration of a joint probability density function with respect to the two independent variables “s” (standard deviation) and “x” (sample mean). The factors generated by this method are investigated through Monte Carlo simulations and compared with the existing factors. As a result, it is identified that the twosided tolerance limit factors being currently used in practical applications are inaccurate. For the right understanding, some factors generated by this method are presented in Tables along with a guidance for correct use of them. The AQL (Acceptable Quality Level) is a good, common measure about quality of a product lot which was already produced or will be produced. Therefore, when performing sampling inspection on a given lot using a tolerance limit factor, there is a necessity to know the AQL assigned to the factor. This new double integration method even makes it possible to generate the AQLs corresponding to the onesided and twosided tolerance limit factors.
Key words: AQL / acceptable quality level / sampling inspection / sample size / tolerance limit factor / tolerance interval
© P. Kang, Published by EDP Sciences, 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The onesided and twosided tolerance limit factors are needed for quality control by sampling inspection. In response to this necessity, for the past tens of years, a lot of methodologies for generating the factors have been developed. Wald and Wolfowitz [1] developed an approximate formula for setting the tolerance limits in 1946. Even since then, other approximate formulas were developed. Representatively, Lieberman [2] developed a formula for onesided tolerance limit factors and it was cited widely in other popular references (e.g., Natrella [3]). Unfortunately, Lieberman's formula tends to underestimate the factors; as the sample size n decreases, the factor gets much more underestimated. For this reason, it could be used only for sample sizes larger than 50. In 1970, Guttman [4] developed an approximate formula for onesided tolerance limit factors using the statistical properties of the noncentral tdistribution. For reference, tens of years ago, it was not easy to use directly the noncentral tdistribution. Johnson and Welch [5] encouraged applications of the noncentral tdistribution through their research results. Owen and Amos [6] suggested some computational programs needed to use the noncentral tdistribution in order to set the tolerance limits. In 1972, Abramowitz and Stegun [7] proved that the noncentral tdistribution can be approximated by the normal distribution. With the aid of Abramowitz and Stegun's theory, Link [8] also suggested an approximate formula, and then Link compared the three formulas for onesided tolerance limit factors (i.e., Lieberman [2], Guttman [4], and Link [8]). In comparatively recent years, Janiga and Garaj [9], and Young [10] presented their respective computational methodologies. In this way, various research activities relating to the onesided tolerance limit factors have been done for a long time. It seems that most of recent studies are focused mainly on improvement or simplification of computational procedures by statistical software rather than exact calculation of the factors.
Although related studies are still ongoing, fortunately, the sufficiently accurate onesided tolerance limit factors which were generated by Owen [11] in 1963 are being currently used in industry. Owen created each factor by an iterative method based on the noncentral tdistribution with degrees of freedom ν and noncentrality parameter δ. Later in 1993, Wheeler [12] reviewed and compared two methods for generating tolerance limit factors which were called “Method A” and “Method B” in his paper. “Method A” was a method based on the noncentral tdistribution, and “Method B” was a unique method which does not need to utilize the noncentral tdistribution. The calculation results by the two methods are the same. The onesided tolerance limit factors generated by Wheeler coincide with the ones by Owen [11]. From this, in the case of onesided, Wheeler's two methods are all correct and exact as well as Owen's method.
In parallel with the efforts to develop the methodologies for generating the onesided tolerance limit factors, a lot of effort to generate exactly the twosided tolerance limit factors has also been made till now. In 1964, Owen [13] used the term “equaltailed tolerance intervals” for the first time and developed methodologies relating to the twosided tolerance limit factors. Howe [14] also improved relevant methodologies. Odeh et al. [15] wrote a pocket book containing the diverse statistical constants needed for computing normal tolerance intervals. Odeh [16], and Odeh and Owen [17] created tables of the twosided tolerance limit factors. In 1993, Wheeler [12] tried to calculate the twosided tolerance limit factors using the two methods (i.e., “Method A” and “Method B”) in the same manner as used in the case of onesided. In comparatively recent years, Garaj and Janiga [18], Jensen [19], Krishnamoorthy and Mathew [20], Krishnamoorthy and Xie [21], Witkovsky [22], and Young [23] presented their respective methodologies or computational algorithms for calculating the twosided tolerance limit factors.
Unfortunately, despite such efforts, in the case of twosided, the exact factors have not been established yet. Although both of Wheeler's two methods [12] can exactly generate the onesided tolerance limit factors, either of the two methods does not exactly generate the twosided tolerance limit factors. It is difficult to reflect correctly the concept of the twosided tolerance limit factors in the calculation of the factors. In addition, the calculation procedures are more complicated than onesided. These are presumably the reason that any correct methodology has not been established yet. Nevertheless, the twosided tolerance limit factors generated inaccurately are being used in industry without any criticism or comment and it is a problem.
In this article, a new method for generating correctly the onesided and twosided tolerance limit factors is introduced as the solution for the problem. This method does not need to handle the noncentral tdistribution at all, but only needs to utilize the statistical properties of normal distribution and chisquare distribution, which makes the calculation process much easier. To generate the factors, this method needs a double integration of a joint probability density function with respect to the two independent variables “s” (standard deviation) and “” (sample mean). The related definite integrals are evaluated numerically by a statistical software. Notably, after a onesided or twosided tolerance limit factor has been generated, the AQL (Acceptable Quality Level) corresponding to that factor can also be derived by the same double integration method. For demonstration purposes, some onesided and twosided tolerance limit factors, including some AQLs, obtained by the new method are presented in tables. Through Monte Carlo simulations, it is identified that this new method is correct and exact and that the existing twosided tolerance limit factors which were generated by other methods are inaccurate. In addition, a guidance for correct use of the tolerance limit factors is presented.
2 Method and determination of factors
2.1 Definitions of k_{1}, k_{2}, k_{1,practical} and k_{2,practical}
The four tolerance limit factors k_{1}, k_{2}, k_{1,practical} and k_{2,practical} are defined. (The practical tolerance limit factors k_{1,practical} and k_{2,practical} are determined by the factors k_{1} and k_{2} in a proper manner.) First, imagine that an inspector needs to perform a sampling inspection on a given normal population according to the inspection parameters of sample size n, confidence level (1 − α) and proportion p. Here, 0 < p < 1 and 0 < α < 1.
Let's assume that not “at least”, but “exactly” the proportion p of the given normal population lies below the upper specification limit U (or above the lower specification limit L) and let's assume that the inspector tests only this population. The factor k_{1} is defined as the onesided tolerance limit factor which will be used by the inspector to ensure with confidence level (1 − α) that the proportion p of the normal population is below U (or above L). When the inspector performs the onesided test using the factor k_{1}, the probability that the population does not pass the test is “exactly (1 − α)”.
Let's assume that not “at least”, but “exactly” the proportion p of the given normal population lies between L and U, and the population mean is equal to (U + L)/2. In addition, let's assume that the inspector tests only this population. The factor k_{2} is defined as the twosided tolerance limit factor which will be used by the inspector to ensure with confidence level (1 − α) that the proportion p of the normal population lies between L and U. When the inspector performs the twosided test using the factor k_{2}, the probability that the population does not pass the test is “exactly (1 − α)”.
The factor k_{1,practical} is defined as the onesided tolerance limit factor which can be directly used in all practical applications to ensure with confidence level (1 − α) that at least the proportion p of any given normal population is below U (or above L). In case that more than the proportion p of the normal population is below U (or above L), although k_{1} is used instead of k_{1,practical} as the onesided tolerance limit factor, the inspector can conservatively ensure with confidence level (1 − α) that at least the proportion p of the normal population is below U (or above L). Therefore, k_{1} can be selected as substitute for k_{1,practical} in all practical applications.
The factor k_{2,practical} is defined as the twosided tolerance limit factor which can be directly used in all practical applications to ensure with confidence level (1 − α) that at least the proportion p of any given normal population lies between L and U. Here, unlike the case of k_{1} and k_{1,practical}, k_{2} cannot always be selected as substitute for k_{2,practical}. Whether or not k_{2} can be used instead of k_{2,practical} as the twosided tolerance limit factor in a practical application depends on the situation. More specifically, if k_{2} is larger than its corresponding factor k_{1}, k_{2} can be selected as substitute for k_{2,practical}. However, if k_{2} is smaller than k_{1}, k_{2} cannot be selected as substitute for k_{2,practical}. Instead k_{1} should be selected; the detailed reason is explained in Section 3.
Throughout this article, when referring to the term “two corresponding factors k_{1} and k_{2}”, it includes the meaning that the inspection parameters (i.e., sample size, confidence level, and proportion) given to the factor k_{1} are identical to those given to the factor k_{2}. Similarly, when referring to the term “AQL corresponding to factor k_{1} (or k_{2})”, it includes the meaning that the inspection parameters given to the AQL are identical to those given to the factor k_{1} (or k_{2}). These remarks are also effective for k_{1,practical} and k_{2,practical}. For example, in this Section 2.1, the factors k_{1}, k_{2}, k_{1,practical} and k_{2,practical} have been defined based on the same inspection parameters (i.e., the same sample size n, the same confidence level (1 − α), and the same proportion p). Therefore, in this case, it can be said that the four factors correspond to one another.
2.2 Properties of sdistribution
A distribution named sdistribution is defined. Let's assume that a normal population of the random variable x is given and n samples (i.e., sample size is n) are randomly selected from it. If the same sampling process is repeated numerously, we can envision the probability distribution of the standard devizzation s (s = {)^{2}/(n–1)}^{1/2} and = ). Here, such probability distribution is defined as sdistribution. To proceed further, some basic information is reminded. If x is the normal random variable and its variance is σ^{2}, then the sample mean is normally distributed independently of s and the statistic (n − 1)s^{2}/σ^{2} follows the chisquare distribution with degrees of freedom ν (= n − 1). Based on such definition of sdistribution, we can proceed as follows.
Let s^{′} = { (n − 1)^{1/2}/σ}s = as, a = (n − 1)^{1/2}/σ, and h = s^{′}^{2}. In addition, let φ (s) be the probability density function of the continuous variable s, let ψ (s^{′})be the probability density function of s^{′}, and let ϕ (h) be the probability density function of h (= s^{′}^{2}). Then the following can be found.
This equality holds for all real numbers t (t > 0).
Similarly,
2.3 Determination of k_{1} by double integration
According to the definition of k_{1} in Section 2.1, let's assume that not “at least”, but “exactly” the proportion p of the given normal population lies below the upper specification limit U and let's assume that an inspector tests only this population. In addition, suppose that the inspector needs to ensure with confidence level (1 − α) by a sampling inspection that the proportion p of the normal population is below U, where 0 < p < 1 and 0 < α < 1. If the mean and the variance of the population are μ and σ^{2} respectively and the sample size is n, then the onesided tolerance limit factor k_{1} to be used for the sampling inspection should satisfy the following equation.
The factor K_{p} is the critical value corresponding to the proportion p in onetailed test of normal population, and is the probability density function of the normal variable with the mean μ and the variance σ^{2}/n. Using equations (1) and (2), equation (3) can be rewritten in the following form.
It should be reminded that ϕ (s^{'2}) is the probability density function of the variable s^{′}^{2} with degrees of freedom ν (= n − 1). Equation (4) does not need to handle the noncentral tdistribution. Only the probability density functions of normal distribution and chisquare distribution are needed to solve it. However, this equation can be solved numerically, not analytically.
In real situation, to determine numerically the factor k_{1}, the probability density function in equation (4) should be integrated with respect to s '. In principle, its integration range should be from 0 to ∞. However, the range of 0 to 10 (or the range of 0 to 20 for large sample size, i.e., n > 40) is sufficient, because ϕ(s^{'2}) and/or the definite integral of is nearly 0 for s^{′} greater than 10 (or 20) and hence, does not affect the integration results. The definite integral with respect to s^{′} is evaluated numerically by “division quadrature” with the aid of a statistical software (e.g., program “MINITAB 15”). Each unit interval of s ' from 0 to 10 (or 20) is divided into 100,000 equal parts for the division quadrature.
As an example of determination of k_{1}, let μ = 500 and σ^{2} = 5^{2}. (Although other μ and σ^{2} are selected, the same result is obtained.) In addition, let p = 0.95, (1 − α) = 0.95, and n = 15, then equation (4) is rewritten to determine its corresponding k_{1} as follows.
K_{p} = K_{0.95} = 1.64485 and a = (15−1)^{1/2}/5 = 0.74833 have been substituted into equation (4). Both by “division quadrature” and by “trial and error”, we find that k_{1} is 2.566. (It will take a few minutes for an expert accustomed to this process to generate a factor needed.) In this way, all k_{1} factors can be numerically determined. Table 1 shows some k_{1} factors thus obtained. For reference, the onesided tolerance limit factors being currently used in industry (Owen [11], Wheeler [12] and ISO [24]) coincide with the k_{1} factors which are determined by the above method.
Numerically determined factor k_{1}.
2.4 Determination of k_{2} by double integration
According to the definition of k_{2} in Section 2.1, let's assume that not “at least”, but “exactly” the proportion p of the given normal population lies between the lower specification limit L and the upper specification limit U, and the population mean is equal to (U + L)/2. In addition, let's assume that an inspector tests only this population. Also suppose that the inspector needs to ensure with confidence level (1 − α) by a sampling inspection that the proportion p of the normal population lies between L and U, where 0 < p < 1 and 0 < α < 1. If the mean and the variance of the normal population are μ and σ^{2} respectively, and the sample size is n, then the twosided tolerance limit factor k_{2} to be used for the sampling inspection should satisfy the following equation.
The factor K '_{p} is the critical value corresponding to the proportion p in twotailed test of normal population. On the righthand side of equation (5), the first term includes the definite integral with respect to the normal variable greater than μ, and the second term includes the definite integral with respect to smaller than μ. In the same manner as used in the case of onesided, equation (6) is changed into the integral expression with respect to the variable s ' as follows
Only the probability density functions of normal distribution and chisquare distribution are needed to solve equation (7). However, this equation cannot be solved analytically, but can be solved numerically like equation (4).
Similar to the case of k_{1}, let's determine numerically a factor k_{2} by equation (7). First, let μ = 500 and σ^{2} = 5^{2}, and let p = 0.95, (1 − α) = 0.95, and n = 15. Then equation (7) is rewritten in the following form to determine its corresponding k_{2}.
See equation below.
Both by “division quadrature” and by “trial and error”, we obtain k_{2} = 2.616. At this time, K ' _{p} = K ' _{0.95}= 1.95996 has been used as the critical value corresponding to the proportion 0.95 in twotailed test of normal population. In this way, all k_{2} factors can be numerically determined. Table 2 shows some k_{2} factors thus obtained. From Tables 1 and 2, when the sample size n is small, some k_{2} factors are smaller than their respective corresponding k_{1} factors. As the proportion p and the confidence level (1 − α) come closer to 1, this phenomenon becomes more apparent. For reference, the twosided tolerance limit factors being currently used in industry (Wheeler [12] and ISO [24]) are different from the k_{2} factors which are determined by the above method.
Numerically determined factor k_{2}.
2.5 Simulation study
The factors k_{1} and k_{2} presented in Tables 1 and 2 can be investigated by Monte Carlo simulations. As an example, one twosided tolerance limit factor is investigated to identify its correctness. In the case of n = 10, p = 0.90, and (1 – α) = 0.90, the twosided tolerance limit factor k_{2} numerically determined is 2.112 from Table 2. Before simulating with this factor, suppose that the mean μ and the variance σ^{2} of the normal population to be twosided tested are 500 and 5^{2} respectively. After then, 491.77575 (= = 500–1.64485 × 5) and 508.22425 (= μ + K ' _{p}σ = 500 + 1.64485 × 5) are assigned as the lower specification limit L and the upper specification limit U respectively so that exactly the proportion 0.90 of the population lies between L and U. As the first stage of the simulation, the sample mean and the standard deviation s of the ten independent random numbers generated from the given normal population (μ = 500, σ^{2} = 5^{2}) by a statistical software (e.g., program “MINITAB 15”) are calculated. Then it is observed whether or not the lower tolerance limit − k_{2}s) is larger than L, and whether or not the upper tolerance limit + k_{2}s) is smaller than U. This observation is repeated 1,000,000 times. Only in cases where ( − k_{2}s) is larger than L and simultaneously + k_{2}s) is smaller than U, the twosided test confirms that the population is acceptable.
In the simulation, only 99,959 observations showed that − k_{2}s) was larger than L (= 491.77575) and simultaneously + k_{2}s) was smaller than U (= 508.22425). From this, the observed α value is 99,959/1,000,000 = 0.099959; in other words, the confidence level identified by the simulation is (1 − α) = 0.900041. Therefore, it can be said that the twosided tolerance limit factor k_{2} = 2.112 determined numerically by the new integration method is correct. In this way, the correctness of all the onesided and twosided tolerance limit factors can be investigated by Monte Carlo simulations. Table 3 shows the results from the simulations that were conducted to investigate some k_{1} and k_{2} factors.
In Table 4, for n = 10, p = 0.90, and (1–α) = 0.90, the twosided tolerance limit factor being used in industry is 2.546. This factor is significantly different from the above factor 2.112; the factor 2.546 is the one that was unduly overestimated. Table 4 additionally shows that regardless of sample size, confidence level, and proportion, the k_{2,existing} factors are always larger than the k_{2} factors. Therefore, from Tables 3 and 4, it can be said that the twosided tolerance limit factors being used in industry are inaccurate. However, we can observe that as the sample size n increases, the difference between k_{2} and k_{2,existing} gets smaller.
Simulation result.
Comparison of k_{2} and k_{2,existing.}
3 Practical factors and guidance for correct use
According to the definition of k_{1,practical} in Section 2.1, even in case that more than the proportion p of any given normal population is below U (or above L), the use of k_{1,practical} should enable inspectors to ensure with confidence level (1 − α) that at least the proportion p of the normal population is below U (or above L). Fortunately, although its corresponding factor k_{1} is used as the onesided tolerance limit factor instead of k_{1,practical}, the inspectors can conservatively ensure the requirement with confidence level (1 − α). Therefore, k_{1} can be selected as substitute for k_{1,practical} in all practical applications. Thus, we can say
According to the definition of k_{2,practical}, even in case that more than the proportion p of any given normal population lies between L and U, and moreover, the population mean is not equal to (U + L)/2, the use of k_{2,practical} should enable inspectors to ensure with confidence level (1 − α) that at least the proportion p of the normal population lies between L and U. Here, unlike the case of k_{1} and k_{1,practical}, k_{2} cannot always be selected as substitute for k_{2,practical}. Whether or not k_{2} can be used instead of k_{2,practical} depends on the situation. Imagine the two corresponding factors k_{1} and k_{2} which were determined according to the same inspection parameters (i.e., the same sample size n, the same confidence level (1 − α), and the same proportion p). Here, if k_{2} is larger than k_{1}, k_{2} can be selected as substitute for k_{2,practical}. On the contrary, if k_{2} is smaller than k_{1} (see the k_{2} factors marked with superscript “a” in Table 2), k_{2} cannot be selected as substitute for k_{2,practical}. Instead k_{1} should be selected. To summarize,
In case that the population mean μ is not equal to (U + L)/2, i.e., the population mean does not coincide with the center of the specification range, the factor k_{2,practical} will be affected by the deviation magnitude of μ from the center. If the population mean comes closer to U (or L) and the variance gets smaller, the sampling inspection will become more similar to the onesided test. Moreover, the inspector performs the test, not knowing the mean and variance of the given normal population. Therefore, if k_{2} is not larger than its corresponding factor k_{1}, k_{1} (not k_{2}) should be selected as substitute for k_{2,practical} for the twosided test. This is the reason that the determination of k_{2,practical} depends on whether k_{2} is larger than k_{1} or not. For reference, the twosided tolerance limit factors being currently used in industry (Wheeler [12] and ISO [24]) are always larger than their respective corresponding onesided tolerance limit factors.
Although this assumption is not realistic, let's assume that the inspector knows the mean of a given normal population to be tested. In this case, k_{2,practical} needed for the twosided test can be directly and numerically determined, not relying on k_{1} and k_{2}. As an example, suppose that the inspector needs to perform a twosided test on the normal population according to the inspection parameters of n = 5, (1 − α) = 0.95, p = 0.95, L = 490, and U = 510. In addition, let's assume that the population mean μ is known to be 503.4206. Then, k_{2,practical} (here, denoted by k'_{2} for convenience during calculation) can be determined by the following equation. (Before calculating, we assign 4^{2} as the variance σ^{2} of the population so that nearly the proportion p = 0.95 of the population lies between L and U.) (503.4206 = 510–1.64485 × 4)
Notably, in the last term on the righthand side of equation (10), U − k ^{′} _{2}s^{′}/a is smaller than μ. If we substitute related parameters, i.e., n = 5, α = 0.05, L = 490, U = 510, μ = 503.4206, σ^{2}= 4^{2} and a = (n − 1)^{1/2}/σ = 0.5, into equation (10) and then solve it using both “division quadrature” and “trial and error” in a similar manner to determination of k_{1} and k_{2}, we obtain k ^{′} _{2} (= k_{2,practical}) = 4.156, which is needed for the twosided test. However, if n is 15, a different result is obtained. At this time, if we substitute n =15, α = 0.05, L = 490, U = 510, μ = 503.4206, σ^{2}= 4^{2} and a = (n − 1)^{1/2}/σ = 0.93541 into equation (10) and then solve it, we obtain k ^{′} _{2}(= k_{2,practical}) ≒ 2.566. The above two calculation results are summarized as follows.
when n = 5; k_{2} = 3.917 < k ^{′} _{2}(= k_{2,practical}) = 4.156 < k_{1} = 4.203
when n = 15; k_{1} = 2.566 ≒ k ^{′} _{2}(= k_{2,practical}) ≒ 2.566 < k_{2} = 2.616
Thus, irrespective of sample size n, the factor k_{2,practical} calculated directly is always placed somewhere between the two corresponding factors k_{1} and k_{2}. However, as previously mentioned, inspectors usually perform twosided tests (including onesided tests), not knowing the mean and variance of normal populations, and hence cannot calculate the k_{2,practical} factors. Therefore, k_{2,practical} should be selected from the two corresponding factors k_{1} and k_{2} by equation (9). Tables 5 and 6 show some k_{1,practical} and k_{2,practical} factors that should be used in practical applications.
Practical factor k_{1,practical} (= k_{1}).
Practical factor k_{2,practical.}
4 AQLs corresponding to factors
According to Owen [11], AQL is defined as “percentage defective in a lot corresponding to a 0.95 chance of accepting the lot”. The AQL is a good, common measure about quality of a product lot as a whole, which was already produced or will be produced. Therefore, when testing a given lot using a tolerance limit factor, there is a necessity to know the AQL which is assigned to the sampling inspection. In this regard, the method for generating the AQLs corresponding to the tolerance limit factors is introduced.
Suppose that the mean μ and the variance σ^{2} of the normal population to be twosided tested are 500 and 5^{2} respectively and that the inspection parameters are n = 5, (1 − α) = 0.90, and p = 0.90. Also assume that the population mean coincides with the center of the specification range. In this case, the twosided tolerance limit factor is k_{2} = 2.597 from Table 2. To obtain its corresponding AQL, we should first replace the confidence level (1 − α) = 0.90 with (1 − α) = 0.95. After then, we rewrite equation (7) to determine its twotailed critical value K ' _{p} as follows.
When rewriting, a = (n − 1)^{1/2}/σ = (5–1)^{1/2}/5 = 0.4, μ = 500, and k_{2} = 2.597 were substituted into equation (7). Now, both by “division quadrature” and by “trial and error”, we obtain K ' _{p} = 1.38253. This twotailed critical value is matched to the proportion p = 0.8332. Therefore, the AQL corresponding to k_{2} = 2.597 is (1 − p) × 100 = 16.68%. In this way, the AQLs corresponding to the onesided and twosided tolerance limit factors can be obtained. Table 7 shows some AQLs thus obtained. The unit “%” is omitted in practical uses of AQLs, however, it is not omitted for clear discernment of the AQLs from the tolerance limit factors in this article.
In Table 7, although two corresponding factors k_{1} and k_{2} are distinctly different from each other, the difference between the two AQLs matched to the two factors k_{1} and k_{2} is very small. For example, the AQL in the onesided test of n = 5, (1 − α) = p = 0.90, and k_{1} = 2.742 is 16.66% and it is nearly equal to the AQL 16.68% in the twosided test of n = 5, (1 − α) = p = 0.90, and k_{2} = 2.597. As another particular observation, in the case of (1 − α) = p = 0.99, as the sample size n increases, the AQL also increases. In contrast, in the case of (1 − α) = p = 0.90, the opposite phenomenon is seen.
The AQLs shown in Table 7 are based on the factors k_{1} and k_{2}. However, the practical AQLs should be used in practical applications in the same manner as k_{1,practical} and k_{2,practical} are used in practical applications. Table 8 presents some practical AQLs corresponding to k_{1,practical} and k_{2,practical}. It is common that people estimate the quality level of a lot from the AQL assigned to the sampling inspection. When establishing a sampling inspection plan, it is recommended that the fixed confidence level “(1 − α) =0.95” be adopted, because, in this case, simply the value “(1− p)” itself becomes the AQL corresponding to the planned sampling inspection.
AQLs corresponding to k_{1} and k_{2.}
Practical AQLs corresponding to k_{1,practical} and k_{2,practical.}
5 Results and discussion
The factors generated by the new double integration method were investigated through Monte Carlo simulations and it was identified that they are correct and exact. During calculation and simulation, the onesided tolerance limit factors were compared with their respective corresponding twosided tolerance limit factors generated by the new method. As a result, a new fact was found. When the sample size n is small, some k_{2} factors are smaller than their respective corresponding k_{1} factors. As the proportion p and the confidence level (1 − α) come closer to 1, this phenomenon becomes more apparent. Particularly, in Table 4, through direct comparison of the twosided tolerance limit factors determined by the new method with the ones generated by the existing methods, we can see that the existing twosided tolerance limit factors being used in industry are inaccurate. As an example, for n = 10 and (1 – α) = p = 0.90, the twosided tolerance limit factor k_{2} determined by the new method is 2.112. However, according to the existing methods, k_{2} is 2.546. Here, it was already demonstrated by the simulation results in Table 3 that the factor k_{2} (= 2.112) is correct and exact. Therefore, we can say that the factor k_{2} (= 2.546) being currently used is inaccurate.
If the population mean μ does not coincide with the center of the specification range, the practical twosided tolerance limit factor k_{2,practical} is affected by the deviation magnitude of μ from the center. Even in this case, if the population mean is known, the factor k_{2,practical} can be calculated by the new method. In Section 3, it was illustrated that the factor k_{2,practical} thus calculated is always placed somewhere between its two corresponding factors k_{1} and k_{2}. As an example, for the inspection parameters of n = 5, (1 − α) = p = 0.95, L = 490, and U = 510, if the population mean is known to be 503.4206, in this case, we can calculate the practical factor k_{2,practical} (= 4.156) by the new method, and hence, we can identify that the practical factor lies between its two corresponding factors k_{1} (= 4.203) and k_{2} (= 3.917). However, inspectors have to conduct the sampling inspection, not knowing the mean and variance of the normal population to be tested, and hence cannot calculate k_{2,practical} directly. Therefore, k_{2,practical} should be selected from the two corresponding factors k_{1} and k_{2} in a proper manner. If k_{2} is smaller than k_{1}, k_{1} (not k_{2}) should be used as substitute for k_{2,practical}. Unlike k_{2}, k_{1} can always be used as substitute for k_{1,practical}.
The AQL is regarded as a common, familiar index representing the expected quality level of a product lot from the aspect of statistical quality control by sampling inspection. Therefore, it is necessary to match the tolerance limit factors to their respective corresponding AQLs. Fortunately, the AQL can also be derived by double integration of a joint probability density function of normal distribution and chisquare distribution in the same manner as used to generate the tolerance limit factor itself. In the AQL Tables produced by the new method, although two corresponding factors k_{1} and k_{2} are distinctly different from each other, the AQL matched to k_{1} is nearly (but not exactly) equal to that matched to k_{2}. As an example, in Table 7, for n = 5 and (1 − α) = p = 0.90, the factor k_{1} and its corresponding AQL determined by the new integration method are 2.742 and 16.66% respectively. In addition, the factor k_{2} and its corresponding AQL determined in the same way are 2.597 and 16.68% respectively. Therefore, we can see that although the two factors k_{1} and k_{2} are distinctly different from each other, the two AQLs are nearly equal. As another interesting observation, in the case of p = (1 − α) = 0.99, as the sample size n increases, the AQL also increases. In contrast, in the case of p = (1 − α) = 0.90, the opposite phenomenon is seen. Especially, in case that (1 − α) is 0.95, the value “(1–p)” itself can be regarded as the AQL corresponding to the tolerance limit factor. Therefore, it is recommended that when establishing a sampling inspection plan, the fixed confidence level “(1 − α) = 0.95” be reflected in the plan.
6 Conclusion
The tolerance limit factors are important to conducting the statistical quality control by sampling inspection. The methodology for generating the onesided tolerance limit factors was established in early 1960s. The factors generated by Owen [11] and Wheeler [12] are correct and exact, and are being fully utilized in industry. However, in the case of twosided, any correct methodology has not been established yet, and hence the inaccurate twosided tolerance limit factors are being currently used. In addition, it seems that conceptual understanding needed for applications of the twosided tolerance limit factors is still insufficient.
To resolve such problems, this article introduced a new double integration method for generating correctly the onesided and twosided tolerance limit factors for normal populations along with a guidance for correct use of them. The new method generates the factors, not relying on the noncentral tdistribution, by double integration of a joint probability density function with respect to “s” (standard deviation) and “” (sample mean). Even AQLs corresponding to the factors can be calculated in the same way. In this article, the factors thus generated were investigated through Monte Carlo simulations and it was identified that they are correct and exact.
In the nuclear fuel fabrication plants, great numbers of fuel pellets are manufactured in the pellet production process every day. Each cylindrical pellet is approximately 5 grams in weight and approximately 1 cm in length. After the production of each lot, from 5 to 50 pellets as per quality inspection item are randomly sampled from the lot, and then pellet's diameter, length, density, etc. are precisely measured for quality control and inspection. However, in order to decide “conformance” or “nonconformance” for each of the pellet's quality inspection items (i.e., diameter, length, density, etc.), the inaccurate twosided tolerance limit factors have been used till now. Now we know that those factors being currently used in the fabrication plants are the ones unduly overestimated by the existing methods.
In addition, it is not easy for the inspectors to use other special onesided and twosided tolerance limit factors than the ones that are frequently used in practical applications and can be obtained directly from the statistical handbooks. As an example, let's assume that the inspectors need the onesided or twosided tolerance limit factor to be used specifically for “n = 53 and p = (1 − α) = 0.88”. In this special case, the inspectors cannot easily obtain the needed factor. The reason is that any statistical handbook cannot contain all the factors, therefore, the books contain only the factors to be frequently used. Moreover, in most cases, the inspectors cannot calculate directly the needed factor, because the existing methods are difficult and complicated to handle. Additionally, even in cases where the needed factors are found in the handbooks, the inspectors should be satisfied with the simple use of the factors only for inspection, not knowing the AQLs corresponding to those factors; because there have been no means of matching the factors to the AQLs till now.
However, from now on, if necessary, the inspectors or engineers in industry can directly generate the exact onesided and twosided tolerance limit factors, including the AQLs corresponding to the factors, by using the new double integration method introduced in this article. If anyone becomes accustomed to the use of this new method, it will presumably take a few minutes to generate a factor needed. Finally, we hope that this new method is widely disseminated and hence numerous inspectors performing the sampling inspection practically use the tolerance limit factors and AQLs generated by the new method.
References
 A. Wald, J. Wolfowitz, Tolerance limits for a normal distribution, Ann. Math. Statist. 17, 208–215 (1946) [CrossRef] [Google Scholar]
 G.J. Lieberman, Tables for onesided statistical tolerance limits, Ind. Qual. Control 14, 7–9 (1958) [Google Scholar]
 M.G. Natrella, Experimental Statistics, Handbook 91 (National Bureau of Standards, United States Government Printing Office, Washington, 1963) [CrossRef] [Google Scholar]
 I. Guttman, Statistical Tolerance Regions (Hafner Publishing Company, Darien CT, 1970) [Google Scholar]
 N.L. Johnson, B.L. Welch, Applications of the noncentral tdistribution, Biometrika 31, 362–389 (1940) [CrossRef] [Google Scholar]
 D.B. Owen, D.E. Amos, Programs for computing percentage points of the noncentral tdistribution, Sandia Corporation Monograph No. SCR551 (Sandia Corporation, Washington, 1963) [Google Scholar]
 M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1972) [Google Scholar]
 C.L. Link, An Equation for OneSided Tolerance Limits for Normal Distributions, Research Paper FPL F 458 (Forest Products Laboratory, Madison WI, 1985) [Google Scholar]
 I. Janiga, I. Garaj, Onesided tolerance factors of normal distributions with unknown mean and variability, Measur. Sci. Rev. 6, 2 (2006) [Google Scholar]
 D.S. Young, Normal tolerance interval procedures in the tolerance package, R Journal 8/2 (2016) [Google Scholar]
 D.B. Owen, Factors for OneSided Tolerance Limits and for Variables Sampling Plans, Sandia Corporation Monograph No. SCR607 (Sandia Corporation, Washington, 1963) [CrossRef] [Google Scholar]
 J.T. Wheeler, Statistical Computation of Tolerance Limits, NASA TM108424 (George C. Marshall Space Flight Center, Huntsville, 1993) [Google Scholar]
 D.B. Owen, Control of percentages in both tails of the normal distribution, Technometrics 6, 377–387 (1964) [CrossRef] [Google Scholar]
 W.G. Howe, Twosided tolerance limits for normal populations, some improvements, J. Am. Stat. Assoc. 64, 610–620 (1969) [Google Scholar]
 R.E. Odeh, D.B. Owen, Z.W. Birnbaum, L. Fisher, Pocket Book of Statistical Tables (Marcel Dekker Inc., New York, 1977) [Google Scholar]
 R.E. Odeh, Tables of twosided tolerance factors for a normal distribution, Commun. Stat. Simul. Comput. 7, (1978) [Google Scholar]
 R.E. Odeh, D.B. Owen, Tables for Normal Tolerance Limits, Sampling Plans, and Screening (Marcel Dekker Inc., New York, 1980) [Google Scholar]
 I. Garaj, I. Janiga, Twosided tolerance limits of normal distributions for unknown mean and variability, Bratislava: Vyd. STU (2002) [Google Scholar]
 W.A. Jensen, Approximations of tolerance intervals for normally distributed data, Qual. Reliab. Eng. Int. 25, 571–580 (2009) [CrossRef] [Google Scholar]
 K. Krishnamoorthy, T. Mathew, Statistical Tolerance Regions: Theory, Applications, and Computation (John Wiley & Sons Inc., Hoboken, 2009) [Google Scholar]
 K. Krishnamoorthy, F. Xie, Tolerance intervals for symmetric location − scale families based on uncensored or censored samples, J. Stat. Plan. Inference 141, 1170–1182 (2011) [CrossRef] [Google Scholar]
 V. Witkovsky, On the exact twosided tolerance intervals for univariate normal distribution and linear regression, Aust. J. Stat. 43/34, 279–292 (2014) [CrossRef] [Google Scholar]
 D.S. Young, An R package for estimating tolerance intervals, J. Stat. Softw. 36, 1–39 (2010) [CrossRef] [Google Scholar]
 ISO 162696:2014, Statistical interpretation of data − Part 6: Determination of statistical tolerance intervals (ISO, Geneva, 2014) [Google Scholar]
Cite this article as: Pilsang Kang, A double integration method for generating exact tolerance limit factors for normal populations, Int. J. Metrol. Qual. Eng. 13, 16 (2022)
All Tables
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.