小偏度数据下通用质量特性评估最小样本量确定方法

doi:10.12305/j.issn.1001-506X.2025.10.22

Abstract

Abstract:

In response to the complex and diverse combat tasks faced by aviation equipment, as well as the dynamic and changing battlefield environment, the evaluation of general quality characteristics such as state identification and combat testing often faces problems such as small sample sizes and large fluctuations in data. Starting from the skewness and central limit theorems, a method for determining the minimum sample size requirement for general quality characteristic parameters is proposed with small skewness data. Firstly, determine the target distribution of a certain parameters and calculate the skewness of the distribution with that parameters. Secondly, take a certain initial sample size, generate multiple random numbers for the distribution, and calculate their mean. Thirdly, perform a normality test on its mean to determine whether it passes the normality test with the given sample. Finally, a case analysis is conducted using the average time between failures of the equipment in the experiment, which proves that the calculation results of small skewness samples still have reference value even when they are less than the recommended value. This provides technical guidance for the accurate evaluation of general quality characteristics of aviation equipment in small sample sizes.

Key words: minimum sample size, skewness, central limit theorem, universal quality characteristic

CLC Number:

U 462.3+5

Jiao LI, Wensheng PENG, Zhaoyang ZENG, Yinghua SHAO, Baoliang ZHANG. Method for determining minimum sample size for general quality characteristics evaluation with small skewness data[J]. Systems Engineering and Electronics, 2025, 47(10): 3378-3388.

Figures/Tables 21

Fig.1

Fig.2

Fig.3

Fig.4

Fig.5

Fig.6

Table 1

Fig.7

Fig.8

Fig.9

Table 2

Fig.10

Fig.11

Fig.12

Table 3

Table 4

Table 5

Fig.13

Table 6

Fig.14

Table 7

References 34

1	YAZDI M. Progressive decision-making tools and applications in project and operation management: approaches, case studies, multi-criteria decision-making, multi-objective decision-making, decision under uncertainty[M]. Switzerland: Springer Cham, 2024.
2	AGUILAR A. Lowering mean time to recovery (MTTR) in responding to system downtime or outages: an application of lean six sigma methodology[C]//Proc. of the 13th Annual International Conference on Industrial Engineering and Operations Management, 2023.
3	崔镇韬, 李健, 宋月, 等. 故障率对装备通用质量特性的作用分析[J]. 环境技术, 2023, 41 (12): 78- 81.
	CUI Z T, LI J, SONG Y, et al. Analysis of the effect of failure rate on general quality characteristics of equipment[J]. Environmental Technology, 2023, 41 (12): 78- 81.
4	LI X Q, GAO J, LI W. Reliability testing technology of digital diagnosis and treatment equipment[M]//Reliability Engineering Technology of Digital Diagnosis and Treatment Equipment. Singapore: Springer, 2024: 361−485.
5	SONG M, SHI Q, DONG E, et al. A small sample reliability assessment method based on Bayesian theory[C]//Proc. of the 13th International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering, 2023.
6	XU A C, WANG B B, ZHU D, et al. Bayesian reliability assessment of permanent magnet brake under small sample size[J]. IEEE Trans. on Reliability, 2024, 74 (1): 2017- 2117.
7	FANG G Q, PAN R, HONG Y L. Copula-based reliability analysis of degrading systems with dependent failures[J]. Reliability Engineering & System Safety, 2020, 193, 106618.
8	CAO J Z, WANG T, SHENG M, et al. Assessment of multi-dimensional joint probability distribution for uncertain mechanical strength parameters under small sample test data[J]. Probabilistic Engineering Mechanics, 2023, 74, 103511. doi: 10.1016/j.probengmech.2023.103511
9	LI Y, ZHENG S L, JIANG H X. Research on reliability assessment method based on Bootstrap method[C]//Proc. of the International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering, 2012: 871−874.
10	MA W T, MU H N, LIU W, et al. Reliability evaluation method for initiating explosive device output performance based on SMOTE-Bootstrap method in small sample sizes[C]//Proc. of the 5th International Conference on System Reliability and Safety Engineering, 2023: 439−444.
11	YILMAZ R, BUZLUCA F. A fuzzy logic-based quality model for identifying microservices with low maintainability[J]. Journal of Systems and Software, 2024, 216, 112143. doi: 10.1016/j.jss.2024.112143
12	MIRZAEI D, BEHBAHANINIA A, ABDALISOUSAN A, et al. A novel approach to repair time prediction and availability assessment of the equipment in power generation systems using fuzzy logic and Monte Carlo simulation[J]. Energy, 2023, 282, 128842. doi: 10.1016/j.energy.2023.128842
13	LI M, MEEKER W Q. Application of Bayesian methods in reliability data analyses[J]. Journal of Quality Technology, 2014, 46 (1): 1- 23. doi: 10.1080/00224065.2014.11917951
14	杨凡, 何晋, 周石金, 等. 采用中心极限定理改进可拓灰云聚类的断路器状态评估方法[J]. 现代电子技术, 2021, 44 (21): 124- 128.
	YANG F, HE J, ZHOU S J, et al. Circuit breaker state assessment method based on extended gray cloud clustering improved by central limit theorem[J]. Modern Electronic Technology, 2021, 44 (21): 124- 128.
15	ZHAO Y G, ZHANG X Y, LU Z H. A flexible distribution and its application in reliability engineering[J]. Reliability Engineering & System Safety, 2018, 176, 1- 12.
16	李娇, 蒋觉义. 多源数据驱动下的航空装备状态鉴定通用质量特性评估方法研究[J]. 航空标准化与质量, 2023, (1): 39- 44.
	LI J, JIANG J Y. Research on the general quality characteristics evaluation method of aviation equipment condition identification driven by multi-source data[J]. Aviation Standardization and Quality, 2023, (1): 39- 44.
17	ALTHUBAITI A. Sample size determination: a practical guide for health researchers[J]. Journal of General and Family Medicine, 2023, 24 (2): 72- 78. doi: 10.1002/jgf2.600
18	LIU S, YANG D Z, REN Y, et al. A general quality characteristic configuration management method for equipment[C]//Proc. of the International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering, 2019: 285−298.
19	EASTERLING R G. Approximate confidence limits for system reliability[J]. Journal of the American Statistical Association, 1972, 67 (337): 220- 222. doi: 10.1080/01621459.1972.10481234
20	DU S Y, LI Z H, YU D, et al. Exact confidence limit for complex system reliability based on component test data[J]. Quality Technology & Quantitative Management, 2020, 17 (1): 75- 88.
21	LI Z A, FU H M, WU Q. Phased system reliability modeling and assessment for construction of lunar scientific base[J]. Applied Sciences, 2023, 13 (17): 9823. doi: 10.3390/app13179823
22	LIU Y, LIU B D. A modified uncertain maximum likelihood estimation with applications in uncertain statistics[J]. Communications in Statistics-Theory and Methods, 2024, 53 (18): 6649- 6670. doi: 10.1080/03610926.2023.2248534
23	尹子盟, 陈志会. 基于维修性试验的平均修复时间与最大修复时间验证与评估技术研究[J]. 国际航空航天科学, 2020, 8 (4): 102- 107.
	YIN Z M, CHEN Z H. Research on verification and evaluation technology of mean repair time and maximum repair time based on maintainability test[J]. International Journal of Aeronautics and Astronautics, 2020, 8 (4): 102- 107.
24	SONG X. Optimal maintenance strategy for large-scale production systems under maintenance time uncertainty[J]. Reliability Engineering & System Safety, 2023, 240, 109594.
25	ALJEDDANI S M A, MOHAMMED M A. Parameter estimation of a model using maximum likelihood function and Bayesian analysis through moment of order statistics[J]. Alexandria Engineering Journal, 2023, 75, 221- 232. doi: 10.1016/j.aej.2023.05.079
26	GUO J, KONG X W, WU N X, et al. Weibull parameter estimation and reliability analysis with small samples based on successive approximation method[J]. Journal of Mechanical Science and Technology, 2023, 37 (11): 5797- 5811. doi: 10.1007/s12206-023-1019-z
27	ZHANG J S, JIANG Y C, LI X, et al. An adaptive remaining useful life prediction approach for single battery with unlabeled small sample data and parameter uncertainty[J]. Reliability Engineering & System Safety, 2022, 222, 108357.
28	ISLAM M R. Sample size and its role in central limit theorem （CLT）[J]. Computational and Applied Mathematics Journal, 2018, 4 (1): 37- 47.
29	HATEM G, ZEIDAN J, GOOSSENS M, et al. Normality testing methods and the importance of skewness and kurtosis in statistical analysis[J]. BAU Journal-Science and Technology, 2022, 3 (2): 7.
30	姜伟杰, 权冀川, 刘勇, 等. 一种指控装备试验数据的检验与统计分析框架[J]. 兵工自动化, 2022, 41 (9): 41- 45.
	JIANG W J, QUAN J C, LIU Y, et al. A test and statistical analysis framework for command and control equipment test data[J]. Ordnance Industry Automation, 2022, 41 (9): 41- 45.
31	王维, 向瀚淋, 龚雯丽, 等. 常见分布中心极限定理适用样本量研究[J]. 高师理科学刊, 2021, 41 (7): 20- 25.
	WANG W, XIANG H L, GONG W L, et al. Study on the applicable sample size of the central limit theorem of common distributions[J]. Journal of Science of Teachers Colleges and Universities, 2021, 41 (7): 20- 25.
32	茆诗松, 程依明, 濮晓龙. 概率论与数理统计教程[M]. 北京: 高等教育出版社, 2011.
	MAO S S, CHENG Y M, PU X L. A course in probability theory and mathematical statistics[M]. Beijing: Higher Education Press, 2011.
33	LEI J Z, XIE M, KUO W. Design for solar floor tiles systems under competing risks: a case study[J]. Journal of Reliability Science and Engineering, 2025, 1, 015006. doi: 10.1088/3050-2454/adbaf6
34	HUANG H Z, LI H, SHI Y, et al. Theory and application of possibility and evidence in reliability analysis and design optimization[J]. Journal of Reliability Science and Engineering, 2025, 1, 015007. doi: 10.1088/3050-2454/adbaf7

正态性检验通过率p	偏度范围	适用样本量（建议大于此值）
0.9	±0.5	30
0.9	±0.15	5
0.8	±0.5	20
0.8	±0.2	5
0.7	±0.5	15
0.7	±0.3	5

正态性检验通过率p	偏度范围	适用样本量（建议大于此值）
0.9	±0.35	50
0.9	±0.15	20
0.8	±0.5	28
0.8	±0.2	5
0.7	±0.66	27
0.7	±0.25	5

正态性检验通过率p	偏度范围	适用样本量（建议大于此值）
0.9	±0.5	25
0.9	±0.15	5
0.8	±0.5	17
0.8	±0.3	5
0.7	±0.5	10
0.7	±0.36	5

组别	实验数据	试验次数	成功次数
1	2.650 5，2.170 6，1.690 2，0.703 0，0.658 8	5	3
2	2.384 8，1.998 9，1.553 9，1.256 8，2.718 7	5	4
3	2.115 3，1.755 5，1.751 7，1.172 2，1.224 7	5	3
4	2.530 4，0.781 7，0.710 6，1.641 6，2.824 6	5	3
5	2.517 6，1.038 7，1.633 9，1.685 1，2.490 7	5	4
6	1.040 4，2.704 2，1.942 9，2.209 1，1.212 9	5	3
7	1.874 9，1.975 4，1.288 2，1.223 9，2.376 3	5	3

组别	偏度	均值
1	0.011 5	1.574 62
2	0.003 6	1.982 62
3	0.024 3	1.603 88
4	0.084 3	1.697 78
5	−0.110 3	1.873 20
6	0.036 3	1.821 90
7	0.051 8	1.747 74

Method for determining minimum sample size for general quality characteristics evaluation with small skewness data

RichHTML

PDF (PC)

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 21

References 34

Related Articles 1

Recommended Articles

Metrics

Comments

使用的数据	组数	置信下限	置信上限
1~3组	3	1.149	2.284
1~4组	4	1.415	2.01
1~5组	5	1.525	1.963
1~6组	6	1.588	1.925
1~7组	7	1.62	1.892

样本量数	置信区间大小	增加的样本量 $\Delta n$	置信区间减小量$\Delta \theta $	边际效益 $\varphi $
3	1.1350	—	—	—
4	0.5950	1	0.5400	0.5400
5	0.4380	1	0.1570	0.1570
6	0.3370	1	0.1010	0.1010
7	0.2720	1	0.0650	0.0650