Statistics histograms

Longer datasets give possibility to compare distributions of concentrations (distribution statistics, histograms), which give better view of concentrations, processes, and variability than short-term comparisons. 

Figure 9 Subduction zone statistics histograms of depths of (a) the slab surface below the volcanic front and (b) the width of volcanic arcs. The vertical axis denotes arc lengths in km measured at the trench. This is our own compilation (unpublished) based on locations of quaternary volcanoes and slab surfaces from tomography...

Figure 7. Univariate statistics histogram of the C-0 bond length used to assess suitable geometric restriction embodied in Figure 5.

Tests for normality Tests for normality are statistical methods (Descriptive Statistics, Histograms) used to determine if the data collected is normal or abnormal so as to be properly analyzed by other tools. Correlation/regression analysis These tools help to identify the relationship between inputs and outputs or the correlation between two different sets of variables. 

Ferrenberg A M, Landau D P and Swendsen R H 1995 Statistical errors in histogram reweighting Phys. Rev. E 51 5092-100... 

One therefore needs a smooth density estimation techniques that is more reliable than the histogram estimates. The automatic estimation poses additional problems in that the traditional statistical techniques for estimating densities usually require the interactive selection of some smoothing parameter (such as the bin size). Some publicly available density estimators are available, but these tended to oversmooth the densities. So we tried a number of ideas based on numerical differentiation of the empirical cdf to devise a better density estimator. 

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

Mixmre models have come up frequently in Bayesian statistical analysis in molecular and structural biology [16,28] as described below, so a description is useful here. Mixture models can be used when simple forms such as the exponential or Dirichlet function alone do not describe the data well. This is usually the case for a multimodal data distribution (as might be evident from a histogram of the data), when clearly a single Gaussian function will not suffice. A mixture is a sum of simple forms for the likelihood ... 

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a basic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency histogram of a repeated measurement and the normal distribution f(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity. 

When the underlying distribution is not known, tools such as histograms, probability curves, piecewise polynomial approximations, and general techniques are available to fit distributions to data. It may be necessary to assume an appropriate distribution in order to obtain the relevant parameters. Any assumptions made should be supported by manufacturer s data or data from the literature on similar items working in similar environments. Experience indicates that some probability distributions are more appropriate in certain situations than others. What follows is a brief overview on their applications in different environments. A more rigorous discussion of the statistics involved is provided in the CPQRA Guidelines. ... 

Process capability indices [49], histograms, statistical analysis of system variation, legislation [4]... 

MINITAB readily produces many useful manipulations of data such as were obtained in this experiment. Figure 2 shows histograms of the responses, indicating that, for the limited number of data points, the experimental values for each response approach a normal distribution. Thus, the statistical analysis was considered valid. Table III shows a copy of the computer printout of a correlation table with all the responses. Clearly, Property A and Property B are negatively correlated, as predicted, but Property B and Property E are not well correlated. 

Precise thickness measurements by TEM require sections transverse to the basal lamellar surface. Conversely, only lamellae that can be identified as untilted "edge-on" or "flat-on" in AFM images are suitable for thickness analysis. The average thickness obtained by these techniques is based on sampling microscopic areas and will only be correct if the morphology is uniform in the sample. Micrographs taken from different areas of the specimen are usually studied, and statistical analysis of histograms used for quantitative analysis [255,256]. 

Transition-matrix estimators are typically more accurate than their histogram counterparts [25,26,46], and they offer greater flexibility in accumulating simulation data from multiple state conditions. This statistical improvement over histograms is likely due to the local nature of transition probabilities, which are more readily equilibrated than global measures such as histograms [25], Fenwick and Escobedo... 

Kobrak, M. N., Systematic and statistical error in histogram-based free energy calculations, J. Comput. Chem. 2003, 24, 1437-1446... 

Wang, F. Landau, D. P., Determining the density of states for classical statistical models a random walk algorithm to produce a flat histogram, Phys. Rev. E 2001, <54, 05 6101... 

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