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The normal distribution
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
A normal distribution is a very important statistical data distribution pattern occurring in many natural phenomena, such as height, blood pressure, lengths of objects produced by machines, marks in a test, errors in measurements and so on. In general, when data is gathered, we expect to see a particular pattern to the data, called a normal distribution. This is a distribution where the data is evenly distributed around the mean in a very regular way, which when plotted as a histogram will result in a bell curve. The normal distribution is the most important of all probability distributions; it is applied directly to many practical problems in every engineering discipline. There are two principal applications of the normal distribution to engineering and reliability. One application deals with the analysis of items which exhibit failure to wear, such as mechanical devices – frequently the wear-out failure distribution is sufficiently close to normal that the use of this distribution for predicting or assessing reliability is valid. Another application is in the analysis of manufactured items and their ability to meet specifications. No two parts made to the same specification are exactly alike; the variability of parts leads to a variability in systems composed of those parts. The design must take this variability into account, otherwise the system may not meet the specification requirement due to the combined effect of part variability.
The normal distribution
Published in John Bird, Bird's Engineering Mathematics, 2021
A normal distribution is a very important statistical data distribution pattern occurring in many natural phenomena, such as height, blood pressure, lengths of objects produced by machines, marks in a test, errors in measurements and so on. In general, when data is gathered, we expect to see a particular pattern to the data, called a normal distribution. This is a distribution where the data is evenly distributed around the mean in a very regular way, which when plotted as a histogram will result in a bell curve. The normal distribution is the most important of all probability distributions; it is applied directly to many practical problems in every engineering discipline. There are two principal applications of the normal distribution to engineering and reliability. One application deals with the analysis of items which exhibit failure to wear, such as mechanical devices – frequently the wear-out failure distribution is sufficiently close to normal that the use of this distribution for predicting or assessing reliability is valid. Another application is in the analysis of manufactured items and their ability to meet specifications. No two parts made to the same specification are exactly alike; the variability of parts leads to a variability in systems composed of those parts. The design must take this variability into account, otherwise the system may not meet the specification requirement due to the combined effect of part variability.
Groundwater Hydrology
Published in Mohammad Karamouz, Azadeh Ahmadi, Masih Akhbari, Groundwater Hydrology, 2020
Mohammad Karamouz, Azadeh Ahmadi, Masih Akhbari
The normal distribution is a bell-shaped, symmetric distribution. It is described mathematically by its probability density function: () f(x)=1σ2πexp[−(x−μ)22σ2],−∞<x<∞,−∞<μ〈∞,σ〉0
Efficient Machine Learning-based Approach for Brain Tumor Detection Using the CAD System
Published in IETE Journal of Research, 2023
Mohamed Amine Guerroudji, Zineb Hadjadj, Mohamed Lichouri, Kahina Amara, Nadia Zenati
Classification is the last step in a computer-aided diagnosis (CAD) system. The CAD system has three main steps: segmentation, description, and classification. Segmentation step detects the region of interest in an image, description step determines the characteristics of the segmented region of interest, and classification step exploits the description result to be able to decide on the pathological nature of the region of interest (tumor). The classifier used in this system is the Bayes classifier, which is a probabilistic machine-learning algorithm. The training of this classifier was carried out using 36 images, which is two-thirds of the database. The testing was carried out using the remaining one-third of the database. In Figure 11, the frequencies of the different attributes (characteristics) of the tumors were calculated and plotted as normal probability graphs. The purpose of these plots is to determine if the data is normally distributed. A normal distribution is a type of statistical distribution where most of the data is centered around the mean and the data becomes less frequent as you move away from the mean. If the data is normally distributed, the plot will be linear, as demonstrated in Figure 11. The paragraph states that all normal probability graphs for the nine attributes were found to be linear, indicating that the data is normally distributed.
Semi-automatic road extraction from high resolution satellite images by template matching using Kullback–Leibler divergence as a similarity measure
Published in International Journal of Image and Data Fusion, 2022
Xiangguo Lin, Wenhan Xie, Libo Zhang, Huiyong Sang, Jing Shen, Shiyong Cui
in which, is the grey value of some a pixel, is the mean value of the grey values, and is the standard deviation of the grey values. Moreover, according to the 68–95-99.7 (empirical) rule or the 3-sigma rule (Pukelsheim 1994), about 68% of values drawn from a normal distribution are within one standard deviation away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. Thus, we just focus on the region with grey values ranging from to in the histogram in this paper.
A system optimisation design approach to vehicle structure under frontal impact based on SVR of optimised hybrid kernel function
Published in International Journal of Crashworthiness, 2021
Xianguang Gu, Wei Wang, Liang Xia, Ping Jiang
Through the analysis of the load-carrying path, the load-carrying capacity of path C is of great significance to the energy absorption and collision acceleration of vehicles. It is the main load-bearing component of vehicle body structure when the front collision occurs. Therefore, the thickness of eight components is chosen as design variables in this study, as shown in Figure 8 (taking into account the symmetry of the structure). Table 2 displays the probabilistic distribution and variation of the variables, and these variables are assumed to be continuous. Table 2 shows the probability distribution and variation range of design parameters of variables, assuming that the fluctuations of these variables are continuous. The normal distribution is a very important probability distribution in statistics and it is also the widely used continuous distribution in engineering practice [28–30]. As long as the sample data is sufficient, the normal distribution can be adopted to describe its probability characteristics. Thus the design variables are treated as normal distributions in RBDO and robust optimisation in this study, whose coefficient of variation (COV()) is set as 5% from typical manufacturing and assembly tolerance [29]. The variations of design variables are chosen according to possible design range.