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Concept# Standard deviation

Summary

In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.
Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lower case Greek letter σ (sigma), for the population standard deviation, or the Latin letter s, for the sample standard deviation.
The standard deviation of a random variable, sample, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust, than the average absolute deviation. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same unit as the data.
The standard deviation of a population or sam

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Anaerobic digestion of organic waste to methane gas (CH4) is an attractive method to produce renewable energy. Due to the trend towards the reuse of waste and away from fossil energy sources, this technology saw a worldwide development in the recent years. A fundamental parameter in this domain is the Biochemical Methane Potential (BMP), which defines the amount of CH4 which can be produced out of a certain organic substrate. Such information is essential in order to plan and optimise anaerobic digestion plants, and to evaluate the reactor feed with a new substrate or a new substrate combination (co-digestion). BMPs are assessed in batch test, which can be regarded as the simulation of a full-scale biogas plant. Two experiments are required, the test in which the test substrate is digested on the inoculum, and the blank, in which the inoculum is digested alone. The part of the inoculum which can be digested is called endogenous substrate. Since the test experiment produced not only on the test substrate but only on the endogenous substrate, the CH4 production on the test substrate alone is obtained by the subtraction of the test production minus the blank production. The units of a BMP are generally indicated in litre of CH4 per gram of volatile solids (VS) test substrate [L/gVS]. Despite the importance and the wide application of this parameter, its exact determination remains a challenging issue and results are often not consistent in between (inter-) and within (intra-) laboratories. Experimental protocols exist but do not lead to a satisfactory BMP tests consistency neither. The aim of this study was to identify parameters affecting the outcome of BMP tests, based on the investigation of two different data sets. An inter-laboratory study of BMP tests providing the final BMP values of 327 experiments and information about 40 related experimental parameters, and a second data set containing the complete CH4 production curves of 136 BMP experiments provided by one single laboratory. The method consisted in graphical and statistical analysis (Mixed Effect Modelling), using [R] programming language and software environment. This study found out, that a significant part of the inter-laboratory BMP inconsistency can be explained by an imprecise assessment of the VS, which was not expected. As the VS of the substrate are directly implied in the computation of the BMP, the impact was significant and therefore the BMP were corrected regarding the VS imprecision. With these data, up to 70% of the inconsistency of BMPs could be explained by inter-laboratory effects. The statistical analysis led to the conclusion that the concentration of the endogenous substrate and the moisture content in the digestate would be the principal factors affecting the outcome of BMP tests. An effort was made to identify and correct errors contained in CH4 production curves, which turned out to be delicate for certain cases. Also indications regarding the precision and the reliability of BMPs were formulated. Further, the investigation of the CH4 production curves led to the development of a new method in order to compute the BMP result. This method was based on the fact, that a certain inoculum reaches always the same slope toward the end of the experiment, no matter what was digested before. The advantage of this method would be that the experiment end-point could be set clearly and that the concentration of the endogenous substrate would not have an impact on the outcome of a BMP test. According to the findings of this report, it was proposed to add the following requirements into experimental protocols for BMP tests: Tests, blanks and the analysis of VS should be carried out in triplicates and their standard deviation should be indicated for each of them, together with the BMP result If triplicates contain set-up errors, these experiments must be repeated, including the corresponding blank/test In the annex: An experimental protocol for the TS and VS analyses An indication of a required moisture content of the digestate (mechanism to investigate in detail first) An indication of the range of required VS concentrations or a maximal production rate for blank tests according to certain experimental conditions (mechanism to investigate in detail first). This report identified several parameters, which contribute to the inter-laboratory inconsistency of BMPs. These findings should be investigated further in order to prove and quantify their impact. An effort should be made to demonstrate the newly developed BMP-computation-method, which could eventually lead to more consistent results. The limitation of this study was, that a relatively low amount of data was available compared to their characteristics. Consequently, the findings could only be proven on a few examples and should therefore only be seen as evidences. Further, this led also to the risk of overfitting, since a relative high amount of parameters needed to be included into the statistical model.

2012Although the procedure of total knee replacement (TKR) has been improved for the last decades, patients may still face postoperative complications. Specially, causes of complications related to patellar resurfacing, such as osteonecrosis, patellar bone fracture, anterior knee pain and implant failure, are still unclear. Predicting strain state of the patellar bone after TKR by means of numerical modeling may help in understanding and possibly preventing such complications. First TKR model has been already developed within LBO (built in Abaqus/Standard). The model replicates loaded squat movement (Oxford rig). As the model is planned to be applied to the real TKA patients and compared with their real kinematics, squat movement could be problematic to accomplish for these patients. Thus we want also ask them to perform chair rising movement. In the project we propose to extend existing model of squat to “chair rising” movement. First, the student should study existing in vivo studies of chair rising to estimate kinematics during the movement. And then to implement this movement in Abaqus. The aim is to estimate patellar bone strain during this movement.

2014Safety assessments of road bridges to braking events combine the braking force, acting along the longitudinal axis of the deck, with a vertical load that accounts for the vertical component of the traffic action. In modern design standards the vertical load models result from probabilistic calibration procedures targeting predefined return periods. On the contrary, the braking force was derived from a deterministic characterization of the vehicle configurations and of the braking process. Therefore, the return period of the braking force is unclear and may not be consistent with that of the vertical load model. Significant deviations from the target return period might lead to either uneconomical decisions, e.g. uncalled-for retrofitting interventions, or to inaccurate structural safety verifications. This thesis presents an original stochastic model to compute site-specific values of the braking force as a function of the return period. The developed stochastic model takes into account the length of the bridge deck and its dynamic properties for vibrations in the longitudinal direction, as well as different sources of randomness related to braking events, all of which comply with real-world measurements, including: - vehicle configurations, resorting to a time-history of crossing vehicles; - driver response times, randomly generated from probability distributions defined in the scope of this project; - deceleration profiles of the vehicles, resampled from catalogues of realistic deceleration profiles. The stochastic model uses Monte Carlo simulation of braking events and computes the maximum of the dynamic response of the bridge to each event. The computed maxima are collected in an empirical distribution function of the braking force. In the end, the model returns the quantile of this distribution that is suitable for safety assessments. This value of braking force is specific to the bridge given properties, to the traffic characteristics, and to the target return period. An additional novelty of this research work is the estimation of a rate of occurrence on motorways of braking events per vehicle-distance travelled. This parameter enables the estimation of the period of time covered by the simulations of braking events as a function of traffic flow and of the total number of braking events simulated. This step is fundamental to determine the value of the braking force that has a given return period. The braking forces returned by the stochastic model show significant dependence on the bridge length, the natural vibration period of the deck in the longitudinal direction, and the number of directions of traffic on the deck. On the contrary, damping ratio, traffic on the fast-lane or on weekends, and an augmentation of traffic in 20% show no substantial influence on the braking force. Moreover, the two motorway locations considered as sources of traffic data, Denges and Monte Ceneri, both in Switzerland, yielded braking forces with similar magnitudes, despite the significant differences in traffic characteristics. Finally, the results compiled served to calibrate an updated braking force that depends explicitly on the parameters found relevant, as well as on the return period so that it can be adopted by different standards even if they enforce different safety targets. This updated expression evidences that the braking forces of current codes tend to be conservative and, hence, can be improved based on the findings of this project.