This paper highlights the critical role of pH or proton activity measurements in environmental studies and emphasises the importance of applying proper statistical approaches when handling pH data. This allows for more informed decisions to effectively manage environmental data such as from mining influenced water. Both the pH and {H+} of the same system display different distributions, with pH mostly displaying a normal or bimodal distribution and {H+} showing a lognormal distribution. It is therefore a challenge of whether to use pH or {H+} to compute the mean or measures of central tendency for further environmental statistical analyses. In this study, different statistical techniques were applied to understand the distribution of pH and {H+} from four different mine sites, Metsämonttu in Finland, Felsendome Rabenstein in Germany, Eastrand and Westrand mine water treatment plants in South Africa. Based on the statistical results, the geometric mean can be used to calculate the average of pH if the distribution is unimodal. For a multimodal pH data distribution, peak identifying methods can be applied to extract the mean for each data population and use them for further statistical analyses.

Introduction: With the widespread use of wearable sensors, various methods to evaluate external physical loads using acceleration signals measured by inertial sensors in sporting activities have been proposed. Acceleration-derived external physical loads have been evaluated as a simple indicator, such as the mean or cumulative values of the target interval. However, such a conventional simplified indicator may not adequately represent the features of the external physical load in sporting activities involving various movement intensities. Therefore, we propose a method to evaluate the external physical load of tennis player based on the histogram of acceleration-derived signal obtained from wearable inertial sensors. Methods: Twenty-eight matches of 14 male collegiate players and 55 matches of 55 male middle-aged players wore sportswear-type wearable sensors during official tennis matches. The norm of the three-dimensional acceleration signal measured using the wearable sensor was smoothed, and the rest period (less than 0.3 G of at least 5 s) was excluded. Because the histogram of the processed acceleration signal showed a bimodal distribution, for example, high- and low-intensity peaks, a Gaussian mixture model was fitted to the histogram, and the model parameters were obtained to characterize the bimodal distribution of the acceleration signal for each player. Results: Among the obtained Gaussian mixture model parameters, the linear discrimination analysis revealed that the mean and standard deviation of the high-intensity side acceleration value accurately classified collegiate and middle-aged players with 93% accuracy; however, the conventional method (only the overall mean) showed less accurate classification results (63%). Conclusion: The mean and standard deviation of the high-intensity side extracted by the Gaussian mixture modeling is found to be the effective parameter representing the external physical load of tennis players. The histogram-based feature extraction of the acceleration-derived signal that exhibit multimodal distribution may provide a novel insight into monitoring external physical load in other sporting activities.

In this paper, we propose two multimodal circular distributions which are suitable for modeling circular data sets with two or more modes. Both distributions belong to the regular exponential family of distributions and are considered as extensions of the von Mises distribution. Hence, they possess the highly desirable properties, such as the existence of non-trivial sufficient statistics and optimal inferences for their parameters. Fine particulates (PM2.5) are generally emitted from activities such as industrial and residential combustion and from vehicle exhaust. We illustrate the utility of our proposed models using a real data set consisting of fine particulates (PM2.5) pollutant levels in Houston region during Fall season in 2019. Our results provide a strong evidence that its diurnal pattern exhibits four modes; two peaks during morning and evening rush hours and two peaks in between.

Traditional lecture and active learning methods of teaching a university course are compared. The particular course is university calculus. The lecture method was applied to two sections of calculus. The active learning method was applied to two other sections. In all cases students were given an examination near the beginning of the course and a final examination at the end of the course. The score averages for the active learning method were higher than for the lecture method. The distribution of scores for the lecture method were non-normal multimodal in the first and final examinations. The distribution for the active learning method went from non-normal multimodal in the first examination to unimodal normal in the final examination. A new undeceivable nature evidence-based method is presented for measuring teaching efficacy by probability distribution.
BACKGROUND: The online version contains supplementary material available at 10.1007/s43545-021-00154-1.

Flow cytometry (FC) analysis of erythrocyte shape and related biomechanical properties, such as osmotic fragility, have not moved from a research tool to regular clinical testing. The main reason is existing evidence that various pre-analytical factors influence the mathematical interpretation of the data obtained. With an aim to contribute to the standardization and broaden the use of FC for human erythrocyte shape assessment, freshly prepared peripheral blood erythrocytes isolated from healthy donors were incubated in iso and hypo-osmotic solutions (pure saline, saline with potassium and calcium, and phosphate buffered saline) and examined by FC using values of forward scatter (FSC) and side scatter (SSC). Kurtosis, skewness, Pearson\'s second skewness coefficient of dissymmetry (PCD), and spherical index, calculated from FSC distributions, were used for the erythrocyte shape evaluation. In all isotonic media FSC distribution and FSC-based morphology parameters showed huge inter-individual and inter-medium variation. With decreasing osmolality, in all media and samples, the size of the erythrocytes increased, and swelling index and kurtosis decreased. However, changes in skewness and PCD were influenced by the medium used and the sample tested. Compared to FSC, SSC signal in isotonic and its change in hypotonic media showed lower inter-individual variation and was not influenced by the type of medium. We propose a spherical index and kurtosis as FSC-based indicators of erythrocyte shape. As more resistant to the influence of the preanalytical treatment, SSC data appeared to be unfairly neglected for the assessment of erythrocyte shape, in comparison to the usually employed FSC data.