Method and system for determining minimum sample size for reliability evaluation of aviation equipment

Abstract

The invention belongs to the technical field of aeronautical equipment quality judgment, and particularly relates to a minimum sample size determination method and system for aeronautical equipment reliability assessment, and the method comprises the steps: S1, carrying out the normality test of the sample size data for aeronautical equipment reliability assessment, and determining the initial minimum sample size; s2, performing skewness detection on the sample size data for reliability evaluation of the aviation equipment; s3, correcting the minimum sample size of reliability evaluation of the aviation equipment according to the sample data skewness judgment parameters; and S4, determining the minimum sample size of aviation equipment reliability evaluation. And S5, carrying out reliability evaluation according to the minimum sample size for reliability evaluation of the aviation equipment finally determined in the step S4. The method provided by the invention has very high engineering value in a small sample scene with high sample acquisition cost and long period in aeronautical equipment development.

Description

Technical Field

[0001] This invention belongs to the field of aviation equipment quality technology, and specifically relates to a method and system for determining the minimum sample size for aviation equipment reliability assessment. Background Technology

[0002] Reliability data of aviation equipment during flight is field data, which needs to reflect the product's condition under actual use and maintenance conditions. However, the sample size is small and the data fluctuates greatly. When processing, it should be classified according to different situations and processing requirements. Often, the reliability assessment cannot be completed due to insufficient sample size. Therefore, the number of tests and the sample size of the tests should be clearly defined to ensure the accuracy and rationality of the field assessment.

[0003] Current methods for calculating the minimum sample size all rely on the empirical value of 30. Especially when other methods yield a minimum sample size less than 30, it is still recommended to wait until the sample size reaches 30 before evaluating reliability parameters. This approach is somewhat conservative, and in flight environments with small sample sizes, it may be difficult to reach the recommended empirical value of 30. Furthermore, another major method requires the sample to follow a specific distribution. In most cases, it is difficult to determine which distribution the data follows, and in flight environments, the limited amount of actual data further increases the difficulty of determining the distribution. Therefore, traditional methods often fail to provide statistically significant and valuable results when assessing reliability in near-flight environments with small sample sizes and large data fluctuations. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides a method and system for determining the minimum sample size for reliability assessment of aviation equipment. It uses a uniform skewness measure to measure data from different distributions and uses the central limit theorem to approximate the sample size of a normal distribution. When calculating the sample size, it is not necessary to distinguish the specific distribution from which the sample size comes, making it more intuitive. Especially in the case of small samples, it can more effectively provide statistically significant and valuable calculation results, thereby expanding the data for reliability assessment and completing the final reliability assessment.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a method for determining the minimum sample size for reliability assessment of aviation equipment, comprising: S1. Perform a normality test on the sample data for the reliability assessment of aviation equipment to determine the initial minimum failure sample size required for the reliability assessment of aviation equipment. Obtain sample size data for reliability assessment of aviation equipment Perform a normality test to determine the initial minimum failure sample size required for reliability assessment of aviation equipment. ; in, The initial minimum number of fault samples required for reliability assessment of aviation equipment; Confidence level The corresponding critical value of the standard normal distribution; The degree of dispersion of the time interval between any failure events of aviation equipment; The set error range; The set confidence level; S2. Skewness detection of sample data for reliability assessment of aviation equipment; Skewness is used to assess the symmetry of the distribution of sample data in the reliability assessment of aviation equipment, and to determine whether there is skewness in the sample data, specifically the skewness of the interval between failures (IBC) data. for: ; in, The skewness of the fault interval time data reflects whether the temporal pattern of fault events triggered by the criterion is symmetrical. This indicates that aviation equipment experienced a concentrated number of malfunctions within a short period of time. Let be the interval between single faults observed for the i-th time; The observed mean time between failures; This represents the total number of fault events observed so far. S3. Adjust the minimum sample size based on skewness. : Calculated using step S2 Number of idealized fault observations Make the first correction: ; in, The number of fault event observations after skewness correction; S4. Determine the final minimum sample size. Specifically, it includes the following sub-steps. S41. Constructing a principle for determining the effectiveness of the central limit theorem under small skewness to establish a general sample size for reliability assessment of aerospace equipment. And generate a database; S42. Determine the final minimum number of fault event observations, using the following formula: ; in, The minimum number of failure event observations required for the final determination of the reliability assessment of aviation equipment; Pass rate of the target normality test The significance level; The database generated in step S41; S5, based on Decision-making based on reliability assessment of aviation equipment: Calculated from the data of the secondary fault event If the reliability is deemed satisfactory, then the airborne avionics system is considered reliable; among which, This represents the one-sided confidence lower bound of the MTBF. The required MTBF threshold is the minimum standard that an aircraft equipment must achieve between critical failure events.

[0006] Preferably, the normality test in step S1 is to ensure that the sample size data used for reliability assessment of aviation equipment meets statistical reasonableness, and to test the statistical indicators of the sample size dataset. for: ; in, To test statistical indicators of the sample size dataset; For the first Each sample size data parameter; For the sorted number Individual sample data values; The mean of the sample size data; This represents the total amount of sample data.

[0007] Preferably, the sample data skewness determination parameter in step S2 is... The value of S can be used to determine the distribution of sample data in the reliability assessment of aviation equipment. Specifically, when S=0, the sample data of the reliability assessment of aviation equipment is symmetrically distributed; when S≠0, the sample data of the reliability assessment of aviation equipment is skewed.

[0008] Preferably, step S41 specifically involves: acquiring aviation equipment reliability data, estimating and calculating using a normal distribution based on the central limit theorem; and constructing a judgment sample size to determine the general sample size for aviation equipment reliability assessment based on the principle that the central limit theorem is effective under small skewness. ; in, The expected distribution of reliability data for aviation equipment; Sample size data for reliability assessment of aviation equipment; This represents the sample mean of reliability data for aviation equipment. The skewness of the sample data is set; Confidence level The expected pass rate is as follows; Confidence level The pass rate is below; The set sample data volume requirement.

[0009] A second aspect of the present invention provides a sample size determination system for aviation equipment reliability assessment, which is used in the above-mentioned method for determining the minimum sample size for aviation equipment reliability assessment. The system is characterized in that it includes: a distribution state applicable sample size simulation analysis determination module, an applicable sample size module for aviation equipment reliability assessment, and a recommended sample size module for aviation equipment reliability assessment. The simulation analysis module for applicable sample size of the distribution state calculates the minimum applicable sample size for data that follows a set distribution and whose mean satisfies the central limit theorem and follows a normal distribution under different skewnesses through simulation. It generates a standard comparison chart of applicable sample size for the distribution state, obtains the general sample size for data with different distributions and skewnesses, and generates a database. The applicable sample size module for the reliability assessment of aviation equipment can analyze the standard comparison chart of the applicable sample size of the distribution state generated by the simulation analysis module of the applicable sample size of the distribution state. Based on the change of the applicable sample size with skewness in the distribution simulation calculation results, the applicable sample size of aviation equipment reliability assessment with the central limit theorem is summarized. The module for recommending sample size for reliability assessment of aviation equipment is used to provide sample size recommendations for reliability assessment of aviation equipment under the condition of small skewness data. Specifically, it includes: a data processing submodule, a sample size calculation submodule, a normality test submodule, a sample size recommendation submodule, a visualization submodule, and a user interface submodule.

[0010] Preferably, the data processing submodule is capable of data import, data preprocessing, skewness correction, cleaning, skewness correction, and normality testing of the original data.

[0011] Preferably, the sample size calculation submodule can combine the skewness-corrected data, use the coefficient of variation and precision parameters to automatically calculate the recommended sample size, and output a calculation report.

[0012] Preferably, the normality test submodule provides a test function for the reliability sample data of aviation equipment and automatically generates a test report.

[0013] Preferably, the sample size recommendation submodule dynamically adjusts the recommended sample size value according to the accuracy requirements and data skewness set by the user, so as to ensure that the recommended sample size can meet the evaluation accuracy and take into account the data distribution characteristics.

[0014] Preferably, the visualization submodule can provide sample distribution maps, comparison maps before and after skewness correction, and normality test result maps, intuitively displaying data characteristics and sample size recommendation results.

[0015] Compared with the prior art, the present invention has the following beneficial effects: (1) This invention uses skewness to uniformly measure the reliability data of aviation equipment with different distributions. It uses simulation method with skewness as the measurement standard to uniformly approximate the sample size of the normal distribution using the central limit theorem. This measurement method does not need to distinguish the specific distribution from which the sample size comes when calculating the sample size. It is not only universal but also more intuitive to use.

[0016] (2) Compared with traditional methods, the present invention is more flexible in application. Given a specific skewness or distribution of data, the simulation results are given more accurate simulation results with a determined skewness or distribution based on the specific data skewness or distribution. Especially in the case of small samples, it can more effectively give calculation results with statistical significance and reference value.

[0017] (3) The method of the present invention provides a clear minimum sample size target that takes into account the data distribution characteristics. The evaluation conclusions obtained after collecting data based on this target can accurately assess the reliability, and its statistical significance and engineering reference value are far higher than traditional methods that rely on empirical values ​​or ignore data skewness. The effect is particularly significant in small sample scenarios such as aerospace equipment development, where sample acquisition costs are high and cycles are long. Attached Figure Description

[0018] Figure 1 A flowchart illustrating the method for determining the minimum sample size in the reliability assessment of aviation equipment according to the present invention. Figure 2 This is a schematic diagram of the standard normal distribution of the present invention; Figure 3 This is a distribution diagram of sample data obtained by random sampling in this invention; Figure 4 This is the corrected sample data distribution diagram of the present invention; Figure 5 This is a schematic diagram illustrating the principle framework for calculating the minimum sample size of this invention. Figure 6 The graph shows the simulation results of each distribution when the normality test pass rate p≥0.9 of this invention. Figure 7 This is an enlarged view of the simulation results of each distribution when the normality test pass rate p≥0.9 of this invention; Figure 8 The graph shows the simulation results of each distribution when the normality test pass rate p≥0.8 of this invention. Figure 9 This is an enlarged view of the simulation results of each distribution when the normality test pass rate p≥0.8 of this invention; Figure 10 The graph shows the simulation results of each distribution when the normality test pass rate p≥0.7 of this invention. Figure 11This is an enlarged view of the simulation results of each distribution when the normality test pass rate p≥0.7 of the present invention. Detailed Implementation

[0019] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that the specific embodiments described here, and the sample size data (such as failure interval time data) for reliability assessment of aviation equipment (such as airborne avionics systems), are used only to explain the invention and are not intended to limit the scope of the invention.

[0020] This invention provides a reliability assessment method for aviation equipment, specifically an airborne avionics system navigation module. The method includes the following steps: S1. Perform a normality test on the sample data for the reliability assessment of aviation equipment to determine the initial minimum failure sample size required for the reliability assessment of aviation equipment.

[0021] Obtain sample size data for reliability assessment of aviation equipment Perform a normality test to determine the initial minimum failure sample size required for reliability assessment of aviation equipment. ; in, The initial minimum number of fault samples required for reliability assessment of aviation equipment; Confidence level The corresponding critical value of the standard normal distribution; This refers to the dispersion of time intervals between any malfunction events occurring in the navigation module; that is, the dispersion of time intervals between any of the aforementioned malfunction events (such as positioning errors, signal loss, etc.) occurring in the navigation module. The unit is flight hours (FH). The set error range; The set confidence level.

[0022] This step aims to use standard statistical formulas to calculate, under ideal conditions, how many "failure events" defined by the above criteria need to be observed to achieve the target accuracy for the assessment.

[0023] The normality test performed in step S1 is to ensure that the sample size data used for reliability assessment of aviation equipment meets statistical reasonableness, and to test the statistical indicators of the sample size dataset. for: ; in, To test statistical indicators of the sample size dataset; For the first Each sample size data parameter; For the sorted number Individual sample data values; The mean of the sample size data; This refers to the total amount of sample data. Preferably, the sample data skewness determination parameter in step S2... The value of S can be used to determine the distribution of sample data in the reliability assessment of aviation equipment. Specifically, when S=0, the sample data of the reliability assessment of aviation equipment is symmetrically distributed; when S≠0, the sample data of the reliability assessment of aviation equipment is skewed.

[0024] S2. Skewness detection of sample data for reliability assessment of aviation equipment; Skewness is used to assess the symmetry of the distribution of sample data in the reliability assessment of aviation equipment, and to determine whether there is skewness in the sample data, specifically the skewness of the interval between failures (IBC) data. for: ; in, The skewness of the fault interval time data reflects whether the temporal patterns of fault events triggered by the criteria are symmetrical. Physically, it reflects whether the temporal patterns of fault events triggered by criteria such as "location error" and "signal-to-noise ratio" are symmetrical. If... This means that the navigation module tends to experience a concentrated period of failures after a long period of stable operation. Let be the interval between single faults observed for the i-th time; The observed mean time between failures; This represents the total number of fault events observed so far.

[0025] S3. Adjust the minimum sample size based on skewness. : Calculated using step S2 Number of idealized fault observations Make the first correction: in, The number of fault event observations after skewness correction; The skewness value reflects the symmetry of the time pattern of navigation module failures; A higher value indicates that more failure events need to be observed in order to accurately assess the overall MTBF level.

[0026] S4. Determine the final minimum sample size. Specifically, it includes the following sub-steps: S41. Constructing a principle for determining the effectiveness of the central limit theorem under small skewness to establish a general sample size for reliability assessment of aerospace equipment. And generate a database; Reliability data for aviation equipment is obtained, and estimation calculations are performed using a normal distribution based on the Central Limit Theorem. Based on the principle that the Central Limit Theorem is effective under small skewness, a minimum sample size is constructed to determine the reliability assessment of aviation equipment. ; in, The expected distribution of reliability data for aviation equipment; Sample size data for reliability assessment of aviation equipment; This represents the sample mean of reliability data for aviation equipment. The skewness of the sample data is set; Confidence level The expected pass rate is as follows; Confidence level The pass rate is below; The set sample data volume requirement; S42. Determine the final minimum number of fault event observations, using the following formula: ; in, The minimum number of failure event observations required for the final determination of the reliability assessment of the aircraft equipment; this is the final instruction for subsequent testing or field data collection, meaning that at least [number missing] failure events must be observed. Only after a fault event is defined by the fault criteria can the final evaluation be carried out; the unit is the number of fault events. The pass rate of the target normality test is a robustness requirement set by an engineering project, representing the desired evaluation process to have... The probability is statistically valid. The significance level; This refers to the database generated in step S41. The representative looks up the table.

[0027] S5, based on Decision-making based on reliability assessment of aviation equipment: Calculated from the data of the secondary fault event If the reliability is deemed satisfactory, then the airborne avionics system navigation module is considered reliable; among which, This is the one-sided confidence lower limit of MTBF. Physically, it represents the value we are 95% confident will be higher than the actual MTBF of the navigation module after collecting N_min fault events defined by criteria such as "positioning error" and "signal-to-noise ratio". The unit is flight hours (FH). The required MTBF threshold is the minimum standard that the navigation module must achieve between critical failure events. Physically, it represents the minimum standard that the navigation module must achieve between critical failure events, as explicitly required by design specifications, such as 1500 flight hours (FH).

[0028] A second aspect of the present invention provides a sample size determination system for aviation equipment reliability assessment, which is used in the above-mentioned method for determining the minimum sample size for aviation equipment reliability assessment. The system is characterized in that it includes: a distribution state applicable sample size simulation analysis determination module, an applicable sample size module for aviation equipment reliability assessment, and a recommended sample size module for aviation equipment reliability assessment.

[0029] The simulation analysis module for applicable sample size of the distribution state calculates the minimum applicable sample size for data that follows a set distribution and whose mean satisfies the central limit theorem and follows a normal distribution under different skewnesses through simulation. It generates a standard comparison chart of applicable sample size for the distribution state, obtains the general sample size for data with different distributions and skewnesses, and generates a database.

[0030] The applicable sample size module for aviation equipment reliability assessment can analyze the standard comparison chart of applicable sample size for distribution state generated by the distribution state applicable sample size simulation analysis module. Based on the change of applicable sample size with skewness in the distribution simulation calculation results, the applicable sample size for aviation equipment reliability assessment with the central limit theorem in effect is summarized.

[0031] The module for recommending sample size for reliability assessment of aviation equipment is used to provide sample size recommendations for reliability assessment of aviation equipment under the condition of small skewness data. Specifically, it includes: a data processing submodule, a sample size calculation submodule, a normality test submodule, a sample size recommendation submodule, a visualization submodule, and a user interface submodule.

[0032] The data processing submodule can import data, preprocess data, and correct skewness, cleaning, correcting skewness, and performing normality tests on the raw data. The sample size calculation submodule can automatically calculate the recommended sample size using the coefficient of variation and precision parameters, based on the skewness-corrected data, and output a calculation report. The normality test submodule provides a function for verifying the reliability sample data of aerospace equipment and automatically generates a test report. The sample size recommendation submodule dynamically adjusts the recommended sample size value according to the user-defined precision requirements and data skewness, ensuring that the recommended sample size meets both the evaluation accuracy and the data distribution characteristics. The visualization submodule can provide sample distribution maps, comparison maps before and after skewness correction, and normality test result maps, intuitively displaying data characteristics and sample size recommendation results. Specific Implementation Example 1 This invention provides a method for determining the minimum sample size for evaluating the quality characteristics of aviation equipment. The method involves performing a normality test on the sample data for evaluating the quality characteristics of aviation equipment to determine the minimum sample size; performing a skewness test on the sample data for evaluating the quality characteristics of aviation equipment; correcting the minimum sample size for evaluating the quality characteristics of aviation equipment based on the skewness judgment parameters of the sample data; and constructing a principle for determining the effectiveness of the central limit theorem under small skewness to determine the minimum sample size used for evaluating the quality characteristics of aviation equipment. The method includes: Step S1: Perform a normality test on the sample data for the quality characteristic assessment of aviation equipment to determine the minimum sample size.

[0034] Obtain sample size data for the quality characteristic assessment of aviation equipment To ensure the statistical reasonableness of the sample data used for evaluating the quality characteristics of aviation equipment, a normality test was conducted, examining the statistical indicators of the sample data set. for: ; in, To test statistical indicators of the sample size dataset; For the first Each sample size data parameter; For the sorted number Individual sample data values; The mean of the sample size data; This represents the total amount of sample data.

[0035] Determine the minimum sample size for evaluating the quality characteristics of aviation equipment. for: ; in, The minimum sample size for evaluating the quality characteristics of aviation equipment; Confidence level The corresponding critical value of the standard normal distribution; For sample size data Standard deviation; The set error range; The set confidence level.

[0036] S2: Skewness detection of sample data for the quality characteristic assessment of aviation equipment.

[0037] The skewness of the sample data distribution used in the assessment of the quality characteristics of aerospace equipment is evaluated to determine whether the sample data is skewed. The skewness determination parameters for the sample data are as follows. for: ; in, This is a parameter for determining the skewness of the sample data; For the first The sorted sample data values; The mean of the sample data. ; The sample size represents the total amount of data. Number the sample size data.

[0038] Sample data skewness determination parameters The value can be used to determine the distribution of sample data for the quality characteristic assessment of aviation equipment. Specifically, when S=0, the sample data for the quality characteristic assessment of aviation equipment is symmetrically distributed; when S≠0, the sample data for the quality characteristic assessment of aviation equipment is skewed.

[0039] S3: Determine the skewness parameters based on the sample data skewness from step S2. The minimum sample size for the assessment of the quality characteristics of aerospace equipment in step S1 is corrected. .

[0040] To reduce the impact of skewness on the distribution of the sample mean, the sample size is increased to improve the applicability of the central limit theorem. To ensure the reasonableness of the sample size, the expression for the sample size after skewness correction is as follows: ; in, This is the minimum sample size for the revised assessment of the quality characteristics of aviation equipment.

[0041] like Figure 2 The diagram shown is a schematic representation of a standard normal distribution according to an embodiment of the present invention; the skewness of the distribution often reflects the sample size required for the central limit theorem to take effect when samples following the distribution are valid. Figure 3 The diagram shows the distribution of sample data obtained by random sampling according to the present invention, with a sample size of 1000. The data was randomly sampled from samples with an actual skewness of 60 and an actual skewness of 2.5. It can be seen that when the sample size is simultaneously 1000, as... Figure 4 The diagram shows the modified sample data distribution of this invention. The small skewness in the diagram is quite close to a normal distribution, confirming that it follows a normal distribution. However, even with a sample size of 1000, a large skewness still results in significant skewness, meaning that a noticeable skewed distribution greatly increases the required sample size. If we can simulate the central limit theorem for some common distributions and verify that a definite value exists that makes the central limit theorem effective for these common distributions within a certain skewness range, then we can guide the selection of sample size when the historical data skewness is close to 0. Because in some cases, data may still have significant kurtosis even with skewness close to 0, affecting the effectiveness of the central limit theorem, the range of kurtosis should be limited when using simulation results.

[0042] S4: Constructing the principle of the central limit theorem's effectiveness under small skewness to determine the minimum sample size for evaluating the quality characteristics of aerospace equipment. .

[0043] Data on the quality characteristics of aviation equipment are obtained, and estimation calculations are performed using a normal distribution based on the Central Limit Theorem. Based on the principle that the Central Limit Theorem is effective under small skewness, the minimum sample size for evaluating the quality characteristics of aviation equipment is determined as follows: ; in, The expected distribution state of the quality characteristic data of aviation equipment; Sample size data for evaluating the quality characteristics of aviation equipment; This represents the sample mean of the quality characteristic data for aviation equipment. The skewness of the sample data is set; Confidence level The expected pass rate is as follows; Confidence level The pass rate is below; The set sample data volume requirement.

[0044] The second aspect of this invention proposes a sample size determination system for evaluating the quality characteristics of aviation equipment, which includes: a simulation analysis module for determining the applicable sample size for distribution status, an applicable sample size module for evaluating the quality characteristics of aviation equipment, and a recommended sample size module for evaluating the quality characteristics of aviation equipment.

[0045] The distribution state applicable sample size simulation analysis module calculates the minimum applicable sample size for data that follows a set distribution and whose mean satisfies the central limit theorem and follows a normal distribution under different skewnesses through simulation. It generates a standard comparison chart of applicable sample sizes for distribution states, obtains the general sample size for data with different distributions and skewnesses, and uses the general sample size as the minimum sample size.

[0046] like Figure 5 The diagram shows the framework of the minimum sample size calculation principle of this invention. This embodiment simulates and calculates the applicable sample size for common distributions such as Weibull distribution, exponential distribution, chi-square distribution, F distribution, gamma distribution, log-normal distribution, t distribution, and uniform distribution. For highly skewed distributions such as the exponential distribution, logarithmic processing is performed first, followed by simulation calculation. By simulating and calculating the minimum applicable sample size for data following different distributions with means satisfying the central limit theorem and following a normal distribution under different skewnesses, the results are plotted to obtain the general sample size for data with different distributions and skewnesses. The general sample size is used as the minimum sample size for the skewed data. The specific steps are as follows: Step S51: Determine the target distribution with certain parameters and calculate the skewness of the distribution under those parameters.

[0047] Step S52: Take an initial sample size of n=2, generate n random numbers for this distribution, and calculate the mean.

[0048] Step S53: Generate a total of m=1000 sets of such random numbers, and perform a normality test on the m means (take the default significance level of 5%).

[0049] Step S54: Repeat steps S52~53, set k=200 times, calculate the pass rate p of the normality test. If the pass rate p≥0.9, then n is the applicable sample size. If the pass rate p<0.9, then the sample size n is increased by 1, and steps S52~S54 are repeated.

[0050] The applicable sample size module for evaluating the quality characteristics of aviation equipment can analyze the standard comparison chart of the applicable sample size generated by the distribution state applicable sample size simulation analysis module. Based on the change of the applicable sample size with skewness in the distribution simulation calculation results, the applicable sample size for evaluating the quality characteristics of aviation equipment under the central limit theorem is summarized.

[0051] A module for recommending sample size for evaluating the quality characteristics of aviation equipment is constructed to provide sample size recommendations for the evaluation of aviation equipment quality characteristics under conditions of low skewness data. Specifically, it includes: a data processing submodule, a sample size calculation submodule, a normality test submodule, a sample size recommendation submodule, a visualization submodule, and a user interface submodule. The data processing submodule can import data, preprocess data, and perform skewness correction, cleaning, and normality testing on the raw data. The sample size calculation submodule can automatically calculate the recommended sample size using the coefficient of variation and precision parameters, based on the skewness-corrected data, and output a calculation report. The normality test submodule provides the function of verifying the sample data of aviation equipment quality characteristics and automatically generates a test report. The sample size recommendation submodule dynamically adjusts the recommended sample size value according to the precision requirements and data skewness set by the user, ensuring that the recommended sample size meets both the evaluation accuracy and the data distribution characteristics. The visualization submodule can provide sample distribution maps, comparison maps before and after skewness correction, and normality test result maps, intuitively displaying data characteristics and sample size recommendation results. The user interface submodule provides users with a user-friendly interactive interface, supporting parameter input, data import, and result output.

[0052] The simulation results of various distributions in this invention are as follows. The figure shows the variation of applicable sample size with skewness under different parameters. Figure 6 The figure shows the simulation results of each distribution when the normality test pass rate p≥0.9 according to the present invention; the results show that when the data sample skewness is within ±0.15, a sample size greater than 5 can ensure that the sample mean passes the normality test at a significance level of 5% with a 90% probability, thus satisfying the central limit theorem. Figure 7 This is an enlarged view of the simulation results of each distribution when the normality test pass rate p≥0.9 according to the present invention. When the data sample skewness is within ±0.5, the sample size is greater than 30, which allows the sample mean to pass the normality test at a significance level of 5% with a 90% probability, thus satisfying the central limit theorem. Furthermore, this criterion does not apply to highly skewed distributions such as the exponential distribution.

[0053] In this embodiment of the invention, considering that p ≥ 0.9 may be too high in some practical applications, the commonly used value of 0.8 is also used to simulate the case where p ≥ 0.8 in the simulation steps. For example... Figure 8 The figure shows the simulation results of each distribution when the normality test pass rate p≥0.8 according to the present invention; the results show that when the data sample skewness is within ±0.2, the sample size is greater than 5, which allows the sample mean to pass the normality test at a significance level of 5% with an 80% probability, thus satisfying the central limit theorem; as shown Figure 9 This is an enlarged view of the simulation calculation results of each distribution when the pass rate of the normality test p≥0.8 of the present invention; when the skewness of the data sample is in the range of ±0.5, the sample size is greater than 20, which enables the sample mean to pass the normality test at a significance level of 5% with a probability of 80%, thereby satisfying the central limit theorem.

[0054] In this embodiment of the invention, the same method was used to calculate the result p≥0.7; for example... Figure 10 The figure shows the simulation results of each distribution when the normality test pass rate p≥0.7 according to the present invention; the results show that when the data sample skewness is within ±0.3, a sample size greater than 5 can make the sample mean pass the normality test at a significance level of 5% with a 70% probability, thus satisfying the central limit theorem; Figure 11 This is an enlarged view of the simulation calculation results of each distribution when the normality test pass rate p≥0.7 of the present invention; when the data sample skewness is in the range of ±0.5, the sample size is greater than 15, which enables the sample mean to pass the normality test at a significance level of 5% with a probability of 70%, thereby satisfying the central limit theorem.

[0055] The above conclusions show that the applicable sample size increases with increasing skewness. A summary of the applicable sample sizes for the Central Limit Theorem is shown in Table 1. Table 1. Reference Table of Applicable Sample Size Models for Quality Characteristic Assessment of Aviation Equipment Normality test significance level Normality test pass rate p Skewness range Applicable sample size 2.5% 0.9 ±0.5 25 ±0.15 5 0.8 ±0.5 17 ±0.3 5 0.7 ±0.5 10 ±0.36 5 5% (default) 0.9 ±0.5 30 ±0.15 5 0.8 ±0.5 20 ±0.2 5 0.7 ±0.5 15 ±0.3 5 10% 0.9 ±0.35 50 ±0.15 20 0.8 ±0.5 28 ±0.2 5 0.7 ±0.66 27 ±0.25 5 The simulation results with a significance level of 5% show that taking the minimum sample size of 30 is reasonable. At the default significance level, taking the minimum sample size of 30 ensures that when the data skewness is kept within ±0.5, the sample mean can pass the normality test at a significance level of 5% with a 90% probability, thus satisfying the central limit theorem. That is, taking a sample size of 30 has high statistical significance and can also be used directly to replace the population for relevant calculations.

[0056] This invention applies the calculation results obtained from Table 1 to analyze actual aviation equipment quality characteristic data. The experimental data processing procedure is as follows: Step S61: Obtain aviation equipment test data and remove outliers.

[0057] Step S62: Determine the type of aviation equipment test data.

[0058] Step S63: When the experimental data are continuous parameters and involve the central limit theorem, in addition to theoretically calculating the sample size, the minimum sample size for statistical significance should be determined according to step S64.

[0059] Step S64: Determine the minimum sample size.

[0060] Step S641: Calculate the kurtosis and skewness of this type of data by combining historical data. If the kurtosis is large (the kurtosis value of data that completely follows a normal distribution is 3, and the kurtosis of related data should not be much higher than this value) or the skewness value is far from 0 (such as the exponential distribution, this method cannot be used), it is necessary to consider the sample size requirements for special analysis.

[0061] Step S642: When the confidence level of the normality test is 95% and the pass rate of the normality test is p≥0.9, the minimum sample size is 30 when the skewness is in the range of ±0.5; and the minimum sample size is 5 when the skewness is in the range of ±0.15 (the results when p≥0.8 and p≥0.7 can also be referenced in combination with the actual situation).

[0062] Step S643: If the experimental data sample size is not less than the minimum sample size, continue to step S65; if it is less than the minimum sample size, continue the experiment to obtain more data.

[0063] Step S65: Take the larger of the two values.

[0064] The failure interval time of a certain equipment in this case follows a Weibull distribution with a shape parameter of 3 and a scale parameter of 2. There are 7 sets of data, as shown in Table 2. Due to the limitation of the test time, each set of data has only five samples. Try to give the confidence interval of the mean failure interval time of the equipment at a 95% confidence level.

[0065] Table 2. Confidence Intervals for the Mean Equipment Failure Interval at 95% Confidence Level Group 1 2 3 4 5 6 7 2.6505 2.3848 2.1153 2.5304 2.5176 1.0404 1.8749 2.1706 1.9989 1.7555 0.7817 1.0387 2.7042 1.9754 1.6902 1.5539 1.7517 0.7106 1.6339 1.9429 1.2882 0.703 1.2568 1.1722 1.6416 1.6851 2.2091 1.2239 0.6588 2.7187 1.2247 2.8246 2.4907 1.2129 2.3763 Skewness 0.0115 0.0036 0.0243 0.0843 -0.1103 0.0363 0.0518 mean 1.57462 1.98262 1.60388 1.69778 1.8732 1.8219 1.74774 In this embodiment of the invention, the data follows a Weibull distribution with a shape parameter of 3 and a scale parameter of 2, with an expected value E = 2 * gamma(1 + 1 / 3) = 2 * 0.893 = 1.786. According to Table 3 from the simulation, at the default significance level, when the data skewness is maintained at ±0.15, only 5 samples are needed per group to make its mean approximately conform to a normal distribution. The following uses 3 to 7 groups of data sequentially (two groups of data lose practical reference value because the quantiles of the t-distribution are too large). The confidence intervals of the normal distribution mean at a 95% confidence level are recorded in Table 3.

[0066] Table 3. Confidence Intervals for the Mean of Normal Distribution at 95% Confidence Level The calculation results show that the more sample groups used, the more accurate the confidence interval becomes. However, with four or more samples, further increasing the sample size has limited effect on improving the accuracy of the confidence interval. Therefore, the confidence interval obtained using four samples already has some reference value. This method can achieve a reference-value confidence interval for the sample mean using a sample size of 20. Compared with traditional methods that require at least 30 samples to obtain a reference-value result, this method obtains statistically significant and reference-value calculation results with a sample size of less than 30. Specific Implementation Example 2 This invention assesses the reliability of a certain type of aviation equipment using its method. This embodiment aims to address the challenge of scientifically evaluating the reliability of a specific type of aviation equipment using limited failure data during flight testing or initial field use. The evaluation object is the navigation module of a certain type of airborne avionics system. The reliability being evaluated is measured by the Mean Time Between Failures (MTBF), a key metric for reliability. The physical quantities involved are the interval between failures, measured in flight hours (FH). Failure modes and criteria are also discussed. Failure Mode 1 (Signal Loss): Under specified flight conditions, the navigation module continuously loses satellite signals for more than 30 seconds.

[0068] Fault Mode 2 (Positioning Error): The output positioning accuracy exceeds the design requirement of 50 meters for 1 minute continuously.

[0069] Fault Mode 3 (Startup Failure): After the system is powered on, it fails to complete initialization and enter normal working state within 2 minutes.

[0070] Any of the above situations will be recorded as a fault, and the number of flight hours since the last fault will be recorded as a set of "fault interval time" sample data.

[0071] The specific process of applying the method of this invention is as follows: Through field data collection, seven sets of failure interval time (MTBF) data (in flight hours) for this navigation module have been obtained: {1250, 1400, 1100, 1800, 1350, 1950, 1050}. The design requirement is that the module's MTBF should not be less than 1500 hours. We need to determine whether the current seven samples are sufficient to support a statistically significant evaluation conclusion.

[0072] Step S1: Perform a normality test on the sample data to determine the minimum sample size.

[0073] Obtain the above 7 fault interval time data samples X = {1250, 1400, ..., 1050}.

[0074] The sample mean μ ≈ 1414.3 FH and the sample standard deviation σ ≈ 325.7 FH are calculated.

[0075] Set the evaluation requirements: confidence level α = 95% (corresponding to Z_α / 2 = 1.96), acceptable error range e (for example, we want the mean error of the estimate to be within 200 hours), then e = 200.

[0076] The initial minimum sample size n_0 is calculated using the formula: n_0 = (Z_α / 2 * σ / e) 2 = (1.96 * 325.7 / 200) 2 ≈ 10.2. Preliminary calculations indicate that at least approximately 11 samples are needed.

[0077] S2: Perform skewness detection on the sample data.

[0078] The skewness determination parameter S for these 7 sample data was calculated using the skewness formula. The calculated value is S ≈ 0.62. This value is not 0, indicating that the data is right-skewed (with some relatively long failure intervals) and the distribution is not symmetrical.

[0079] S3: Minimum sample size corrected based on the sample data skewness determination parameter in step S2. .

[0080] To reduce the impact of skewness on the distribution of the sample mean, the sample size is increased to improve the applicability of the central limit theorem. To ensure the reasonableness of the sample size, the expression for the sample size after skewness correction is as follows: ; in, This is the minimum sample size for the revised reliability assessment of aviation equipment.

[0081] S4: Construct a database to determine the sample size based on the principle of the central limit theorem under small skewness, and then look up the table in the database to calculate the minimum required sample size.

[0082] The distribution state applicable sample size simulation analysis module in this invention system is the core of the database generation. It determines the applicable sample size for data with different skewnesses through the following simulation steps: Determine the parameters of a target distribution (such as the Weibull distribution) and calculate its theoretical skewness. Take an initial sample size of n=2, generate n random numbers from this distribution, and calculate their mean. Repeat step S52 for a total of m=1000 times to obtain 1000 means, and perform a normality test on these 1000 means. Repeat the entire process k=200 times and calculate the pass rate p of the normality test. If p meets a preset threshold (e.g., p≥0.9), then the current n is the applicable sample size; otherwise, n is increased by 1, and steps S52 to S54 are repeated. Through the above simulations on various common distributions, the relationship between skewness and applicable sample size is finally established, forming Table 4. Other modules of the system (such as data processing, sample size calculation, visualization, etc.) provide automated tool support for users to perform steps S1-S5.

[0083] Table 4. Reference Table of Applicable Sample Size Models for Reliability Assessment of Aviation Equipment Normality test significance level Normality test pass rate p Skewness range Applicable sample size 2.5% 0.9 ±0.5 25 ±0.15 5 0.8 ±0.5 17 ±0.3 5 0.7 ±0.5 10 ±0.36 5 5% (default) 0.9 ±0.5 30 ±0.15 5 0.8 ±0.5 20 ±0.2 5 0.7 ±0.5 15 ±0.3 5 10% 0.9 ±0.35 50 ±0.15 20 0.8 ±0.5 28 ±0.2 5 0.7 ±0.66 27 ±0.25 5 Step S5: Based on the determined minimum sample size Perform reliability assessment and decision-making. This step is the final application stage of the invention, where the calculated reliability is used to make a decision. Reaction to the evaluation process: S51. Experimental Design and Data Acquisition: This serves as the objective, guiding subsequent experiments or field data collection. If the current sample size is insufficient, additional data must be collected until the requirements are met.

[0084] S52. Reliability Parameter Estimation: Using a sample that meets the quantity requirements, calculate the confidence interval of key reliability indicators, such as the 95% one-sided confidence lower limit of MTBF.

[0085] S53. Evaluation Conclusion Judgment: The calculated confidence interval is compared with the design requirements or acceptance criteria to make a scientific judgment on the reliability of the aviation equipment as "qualified", "unqualified" or "requires further analysis".

[0086] Refer to the database of this invention, “Applicable Sample Size Model Reference Table for Reliability Assessment of Aviation Equipment” (i.e., Table 1).

[0087] Set the test conditions: the default significance level is 5%, and the expected pass rate for the normality test is p ≥ 0.8.

[0088] Referring to the reference table: With p ≥ 0.8, a sample size of 20 is suitable when the skewness S is within ±0.5. The sample skewness S = 0.62 slightly exceeds this range. This means that 20 samples may still be insufficient to achieve a high confidence level.

[0089] Decision: Considering the calculated value of 17 from S3 and the table lookup results, and taking into account that S > 0.5, we should choose a more conservative value. To ensure the reliability of the evaluation, we determine the minimum sample size ultimately used for the MTBF evaluation of this navigation module. = 25. S5: Based on the determined minimum sample size N_min = 25, react on the evaluation.

[0090] Data Acquisition: Currently, there are only 7 samples, which does not meet the requirement of 25. Therefore, the testing team must continue to conduct more rigorous testing or field tracking of the navigation module until at least 18 more fault interval time data points are collected.

[0091] Parameter estimation: Assume the team has successfully collected 25 samples of failure interval time (MTBF). Using this data, they calculate the sample mean (i.e., the point estimate of MTBF) and its 95% one-sided confidence lower bound. For example, the calculated 95% confidence lower bound for MTBF is 1580 hours.

[0092] Conclusion: The calculated results (MTBF confidence lower limit = 1580 hours) were compared with the design requirements (MTBF ≥ 1500 hours). Because 1580 > 1500, we have a 95% confidence level that the reliability of the navigation module meets the design requirements. The final evaluation conclusion is: qualified.

[0093] The simulation results with a significance level of 5% show that taking the minimum sample size of 30 is reasonable. At the default significance level, taking the minimum sample size of 30 ensures that when the data skewness is kept within ±0.5, the sample mean can pass the normality test at a significance level of 5% with a 90% probability, thus satisfying the central limit theorem. That is, taking a sample size of 30 has high statistical significance and can also be used directly to replace the population for relevant calculations.

[0094] The beneficial effects of the embodiments of the present invention are as follows: This invention avoids the erroneous conclusion that "the mean is 1414.3 hours, which does not meet the 1500-hour requirement" when there are only 7 samples. This method scientifically points out that the current sample size is insufficient to support effective evaluation and provides a clear minimum sample size target (25 samples) that considers the data distribution characteristics. Evaluation conclusions drawn from data collected based on this target have far greater statistical significance and engineering reference value than traditional methods that rely on empirical values ​​(such as 30) or ignore data skewness. This is particularly effective in small-sample scenarios such as aerospace equipment development, where sample acquisition is costly and time-consuming.

[0095] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made by those skilled in the art to the technical solutions of the present invention without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for determining a minimum sample size for reliability assessment of airborne equipment, characterized in that, It comprises: S1, normality test of sample size data of aviation equipment reliability evaluation, determine the initial minimum failure sample size required; Obtaining sample size data for reliability assessment of aviation equipment , performing normality test to determine the initial minimum failure sample size required for reliability assessment of aviation equipment : ； wherein, is the initial minimum sample size of failures required for reliability assessment of the aviation equipment; is the confidence level is the corresponding standard normal distribution critical value; is the dispersion degree of the time interval between any failure events of the aviation equipment; is the set error range; is the set confidence level; S2, skewness detection of sample size data of aviation equipment reliability evaluation; The skewness is used to evaluate the symmetry of the distribution of the sample data of the reliability evaluation of the aviation equipment, to determine whether the sample data is skewed, and the skewness of the failure interval time data is: ； wherein, is skewness of the failure interval data, reflecting whether the time regularity of the failure events triggered by the criterion is symmetric, if then it represents that the aviation equipment has concentrated failures in a short period of time; is the i th observed single failure interval time; is the observed average failure interval time; is the total number of failure events that have been observed so far; S3、Correct the minimum sample size according to skewness : The minimum sample size required for the initial reliability assessment of aviation equipment is calculated by using the skewness calculated in step S2 First correction:​ ； wherein, is the number of failure events observed after skew correction; S4. Determine the minimum sample size required for final air equipment reliability assessment comprising the following sub-steps: S41, Constructing the principle of central limit theorem with small skewness to determine the general sample size for reliability evaluation of aviation equipment and generate a database; S42, determine the final minimum failure event observation number, the formula is as follows: ； wherein, the minimum number of failure events to be observed for final determination of the reliability evaluation of the aviation equipment; the target normality test pass rate; the significance level; the database generated in step S41; S5、based on Evaluate decision on the reliability of the aviation equipment: if based on the calculated greater than or equal to , the reliability is determined to be qualified; wherein, is the lower one-sided confidence limit of the MTBF; is the required MTBF threshold, i.e. the minimum standard reached by the average time between occurrence of critical failure events of the aviation equipment.

2. The method of claim 1, wherein: The normality test in step S1 is to make the sample data for reliability evaluation of the aviation equipment satisfy the statistical rationality, and to test the statistical quantity index of the sample data set is: ； wherein, is a statistical measure of the sample size data set; is the sample size data parameter; is the sample data value in the sorted order; is the mean of the sample size data; is the total of the sample size data.

3. The method of claim 1, wherein: The sample data skewness determination parameter in step S2 The numerical value of S can be used to determine the sample data distribution of the reliability evaluation of the aviation equipment. Specifically, when S=0, the sample data of the reliability evaluation of the aviation equipment is symmetrically distributed; and when S≠0, the sample data of the reliability evaluation of the aviation equipment is skewly distributed.

4. The method of claim 1, wherein: Step S41 is specifically: obtaining aviation equipment reliability data, using normal distribution for estimation calculation according to central limit theorem; Based on the principle of central limit theorem under small skewness, the general sample size for determining aviation equipment reliability evaluation is constructed as: ； in, The expected distribution of reliability data for aviation equipment; Sample size data for reliability assessment of aviation equipment; This represents the sample mean of reliability data for aviation equipment. The skewness of the sample data is set; Confidence level The expected pass rate is as follows; Confidence level The pass rate is below; The set sample data volume requirement.

5. A system for determining sample size for reliability assessment of aerospace equipment according to the method for determining minimum sample size for reliability assessment of aerospace equipment according to any one of claims 1-4, characterized in that, It comprises: distribution state applicable sample size simulation analysis determination module, aviation equipment reliability evaluation applicable sample size module and aviation equipment reliability evaluation recommended sample size module; The distribution state applicable sample size simulation analysis module calculates the minimum applicable sample size of the mean value of the data subject to the set distribution under different skewness, which satisfies the central limit theorem subject to normal distribution, generates a distribution state applicable sample size standard comparison chart, obtains the general sample size of different distribution and skewness data, and generates a database; The aviation equipment reliability evaluation applicable sample size module can analyze the distribution state applicable sample size standard comparison chart generated by the distribution state applicable sample size simulation analysis module, and according to the applicable sample size change with skewness of the distribution simulation calculation result, the applicable sample size of the central limit theorem is obtained. The construction of aviation equipment reliability evaluation recommended sample size module is used for providing sample size recommendation for aviation equipment reliability evaluation under the condition of small skewness data, specifically including: data processing submodule, sample size calculation submodule, normality test submodule, sample size recommendation submodule, visualization submodule and user interface submodule.

6. The system for sample size determination for reliability assessment of aerospace equipment of claim 5, wherein: The data processing submodule can perform data import, data preprocessing, skewness correction, data cleaning, skewness correction and normality test.

7. The system method of sample size determination for reliability assessment of aerospace equipment as claimed in claim 5, wherein: The sample size calculation submodule can automatically calculate the recommended sample size by using the coefficient of variation and the precision parameter combined with the skewness corrected data, and output the calculation report.

8. The system for sample size determination for reliability assessment of aerospace equipment of claim 5, wherein: The normality test submodule provides the test function of aviation equipment reliability sample data, and automatically generates the test report.

9. The system for sample size determination for reliability assessment of aerospace equipment of claim 5, wherein: The sample size recommendation submodule dynamically adjusts the sample size recommendation value according to the user's set precision requirement and data skewness.

10. The system for sample size determination for reliability assessment of aerospace equipment of claim 5, wherein: The visualization submodule can provide sample distribution chart, comparison chart before and after skewness correction, normality test result chart, and intuitively display data characteristics and sample size recommendation result.

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