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Basics of probability and statistics
Published in Amit Kumar Gorai, Snehamoy Chatterjee, Optimization Techniques and their Applications to Mine Systems, 2023
Amit Kumar Gorai, Snehamoy Chatterjee
Median is defined as the middle value after sorting the observation either in ascending or decreasing order. If the number of the observation is an odd number, say 2n+1, where n is an integer, the value of the (n+1)th observation gives the median value, while if the number is even, say 2n, then the median is the mean of nth and (n+1)th observation values. The median of a grouped frequency distribution data can be determined using simple interpolation using Eq. 2.24. Md=L+N2−Ff*i
Working with Data
Published in David E. Clough, Steven C. Chapra, Introduction to Engineering and Scientific Computing with Python, 2023
David E. Clough, Steven C. Chapra
When outliers exist, especially in smaller data sets, there is a concern that they may influence the average calculation inordinately. One answer to this is to use an alternate measure of central tendency. The most common of these is the median, designated x˜. The definition of the median is simple: sort the data and pick the one in the middle — if the number of the data is even, pick the two in the middle and compute their average. The calculation of the median for our two data sets in Figure 10.5 is shown in Figure 10.6. You will note that the median values are less than the average values, especially for the second data set. This is because the high values, possibly outliers, in the two data sets influence the average calculation more than the median calculation. The high value in the second data set is more extreme than that of the first data set. We will address diagnosing outliers a bit later.
Statistical Data Analysis
Published in Timothy Bower, ®, 2023
The median of a data set is the number which is greater than half of the numbers and less than the other half of the numbers. Although the mean is a more commonly used metric for the center value of a data set, the median is often a better indicator of the center. This is because values that are outliers to the main body of data can skew a mean, but will not shift the median value.
Mode differentiation in partitioning of mixed bi-modal urban networks
Published in Transportmetrica B: Transport Dynamics, 2023
Mansour Johari, Shang Jiang, Mehdi Keyvan-Ekbatani, Dong Ngoduy
The sample size calculation in the second step of the algorithm is the main assumption considered in the present study. We expect this method to be more successful than partitioning methods with the problem of partitions stability (due to local optimality), like the K-mean algorithm, as discussed in Ji and Geroliminis (2012), or the algorithm developed in Fu et al. (2020). We indeed assumed that the obtained best solutions would not be significantly improved after a certain sample size due to the considered constraints. This means the results will be more stable if one selects a suitable sample size. We now evaluate this assumption by running the algorithm for a big range of sample sizes and studying the obtained best solutions through plotting the corresponding notched box plot and the percentage differences of the medians. In the notched box plot, the minimum and maximum are indicated by the black lines. The bottom and top of the blue box depict the first and third quartile, respectively. The red line indicates the median. The width of the notches around the median, which varies with the interquartile range of data, and inversely with the square root of the data set size, is a useful tool to compare the medians of two box plots.
Data-Driven Selection of Typical Opaque Material Reflectances for Lighting Simulation
Published in LEUKOS, 2023
Table 3 provides descriptive statistics of the data visualized in Fig. 2. The number of measurements in each category, their medians and IQRs are reported. The median is one measure of centrality, a statistical method of identifying the midpoint of a data set. The author chose the median such that a few very low or very high values cannot significantly alter the result – for example, the painted black ceiling measurements apparent in the “All Ceilings” category of Fig. 2. The IQR is the difference between the 25th percentile and 75th percentile values of each category. A low IQR indicates that the spread of reflectances about the median central value is small. A larger IQR indicates that there is higher variability in the measured reflectance data. Categories with a low reflectance IQR will also exhibit narrower boxes in Fig. 2 (left). For example, the “All Ceilings” category has a median reflectance of 85.1% and an IQR of 5%, suggesting that 85.1% is a reasonable value to select for ceiling reflectance during a lighting simulation. On the other hand, “Tile Floors” have a median reflectance of 41.8% and an IQR of 37.5%, indicating that tile finish materials have a large variety of possible photopic reflectances, so the selection of a reasonable value for tile floors depends on the specific product, color, lightness, and surface finish chosen.
Fusing separated representation into an autoencoder for magnetic materials outlier detection
Published in Systems Science & Control Engineering, 2022
The outlier detection algorithm will eventually compute an outlier score for each sample. Other literature directly uses the reconstruction error value as the outlier score after sorting the abnormality scores and sets the cut-off to determine the outliers. Some researchers use the variance of the reconstruction error to determine the outliers. Inspired by (Leys et al., 2013), we define the outlier scoring function by the MAD of the reconstruction error. Absolute deviation from the median was (re)discovered and popularized by Hampel (1974). The median (M) is, similar to the mean, a measure of central tendency but offers the advantage of being very insensitive to the presence of outliers. The MAD is immune to the sample size. Therefore we define the mad score as where is the MAD score of , and b is set to 1.4826, which is a constant related to the assumption of normality of the data, disregarding the abnormality induced by outliers. Since is scalar, eliminating outliers can be transformed into sorting the MAD score and determining the optimal threshold. Samples that have a MAD score larger than the threshold are identified as outlier candidates.