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Basic Univariate Statistics
Published in Jhareswar Maiti, Multivariate Statistical Modeling in Engineering and Management, 2023
If we denote the first, second and third quartiles as Q1, Q2, and Q3, then the inter-quartile range (IQR) is Q3–Q1. But the measure that is widely used to define variability is standard deviation. Standard deviation is the square root of variance of a given data set where variance is the average squared deviation of the data points from their mean. Consider a sample of size n with mean x¯. The variance of the sample, denoted by s2, is computed using the following formula:s2=1n−1∑i=1n(xi−x¯)2
Statistics for Quality
Published in K. S. Krishnamoorthi, V. Ram Krishnamoorthi, Arunkumar Pennathur, A First Course in Quality Engineering, 2018
K. S. Krishnamoorthi, V. Ram Krishnamoorthi, Arunkumar Pennathur
The average X¯ is simply the arithmetic average or sum of all observations in the data divided by the number of observations. For the data shown in Table 2.1, X¯=264.05. The standard deviation is the square root of the “average” of the squared deviations of the individual values from the average of the data. Note that the formula uses (n − 1) rather than n in the denominator for finding the “average” of the squared deviations. This is done for a good reason, which will be explained later when discussing the unbiasedness as a desirable property of a good estimator. The standard deviation can also be calculated by using another formula, which is mathematically equivalent to the one given above: S=nΣXi2−(ΣXi)2n(n−1)
Design considerations for additive manufacturing of feed channel spacers for spiral wound membrane modules
Published in Fernando Moreira da Silva, Helena Bártolo, Paulo Bártolo, Rita Almendra, Filipa Roseta, Henrique Amorim Almeida, Ana Cristina Lemos, Challenges for Technology Innovation: An Agenda for the Future, 2017
J. An, W.S. Tan, C.K. Chua, T.H. Chong, A.G. Fane
Figure 3 below shows the benchmark study results. Comparison is made among three 3D printing technologies: SLS, FDM and Polyjet. All features are designed less than 2 mm to match the size of a real spacer. Positive deviation means expansion or larger than designed value, and negative deviation means shrinkage or smaller than designed value. In general, most features with protrusion (e.g. pins and walls) become smaller after printing, perhaps due to the volumetric contraction of polymeric materials when solidifying from the liquid or semi-liquid state (Gurrala & Regalla, 2014; Senthilkumaran et al., 2009). Features with depression (e.g. gaps and holes) tend to become larger if the feature involves a curvature (Figure 3(a) & (b)). This could also be due to the contraction of polymers however in outward radial direction. AM techniques have limited effects on the angled features. Among the examined features, circular walls and gaps are the most difficult to print with accuracy because the data points are the most scattered. In fact, this is one of the limitations when using line by line strategy to print curves. The direct implication of these results to spacer feature design is that, to ensure a better part to model accuracy, features with curvatures should be avoided and features with protrusion should be designed slightly larger to overcome the shrinkage effect.
Performance Evaluation Towards Automatic Building and Road Detection Technique for High-Resolution Remote Sensing Images
Published in IETE Journal of Research, 2023
The IGV is computed by first computing the average gray value within the window w × w (w = 5). This is the sum of the gray values of all the pixels in the window divided by the number of pixels. The average gray value for each pixel (x, y) in the enhanced image (h) within the window can be calculated as Then, the candidate pixel value is generated by computing the standard deviation which is the difference between the gray values of each pixel in the enhanced image (h) to the average gray value. The standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. Compute the IGV at (x, y) within the window as The sum of deviations at the boundary of man-made objects is significantly more than that of points inside the man-made objects. In contrast, the sums of deviations at the boundary of non-man made (natural) objects are not as high as man-made objects.
Predicting loads and dynamic responses of an offshore wind turbine in a nonlinear mixed sea
Published in Ships and Offshore Structures, 2021
We can notice from Figure 5 that the three wave time series have approximately the same mean value. However, the standard deviation values of the red curve and green curve wave time series in Figure 5 are quite different. In statistics, the standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values. The standard deviation values in Table 2 quantitatively show that in Figure 5 the red curve from the nonlinear simulation with bottom effects fits perfectly with the blue curve from the transformed linear simulation with bottom effects. These standard deviation values in Table 2 also show quantitatively that in Figure 5 the two curves from nonlinear simulations with and without bottom effects deviate quite substantially from each other.
Evaluation of Measured Digital Output of Gas Sensors During Spontaneous Heating of Coal
Published in IETE Technical Review, 2019
Subhash Kumar, P. K. Mishra, Jitendra Kumar
The derivation of the mathematical formula for gas sensors for measurement of the concentration level of gases in the atmosphere in terms of standard units requires accuracy, numerical stability and reliability [15,16]. For this purpose, a number of gas canisters for each gas have been used with different concentration levels for identifying the respective digital value (DL) of the output of the sensor. Those DLs were plotted with respect to the respective concentration of gases during various laboratory analyses as shown in Figure 3. With the help of the laboratory analysis and the graph, the mathematical functions have been deduced. Although a good approximation to the underlying situation was performed, one cannot assume that it is exactly correct. Therefore, the mathematical derivation should possess statistical procedure [17]. For this purpose, mean of the observations have been taken to eliminate the measuring deviation.