Abstract:
The current attempt is aimed to extend previous results, concerning the explicit expression of the arithmetic mean standard deviation distribution, to the general case of the weighted mean standard deviation distribution. To this respect, the integration domain is expressed in canonical form after a change of reference frame in the n-space, which is recognized as an infinitely thin n-cylindrical corona where the axis coincides with a coordinate axis and the orthogonal section is an infinitely thin, homotetic (n-1)-elliptical corona. The semiaxes are formulated in two different ways, namely in terms of (1) eigenvalues, via the eigenvalue equation, and (2) leading principal minors of the matrix of a quadratic form, via the Jacobi formulae. The distribution and related parameters have the same formal expression with respect to their counterparts in the special case where the weighted mean coincides with the arithmetic mean. The reduction of some results to ordinary geometry is also considered.

Abstract:
Let and be the mean and the standard deviation of the components of the vector , where with a positive integer. Here, we prove that if , then for . The equality holds if and only if the largest components of are equal. It follows that is a strictly increasing sequence converging to , the largest component of , except if the largest components of are equal. In this case, for all .

Abstract:
In systematic reviews and meta-analysis, researchers often pool the results of the sample mean and standard deviation from a set of similar clinical trials. A number of the trials, however, reported the study using the median, the minimum and maximum values, and/or the first and third quartiles. Hence, in order to combine results, one may have to estimate the sample mean and standard deviation for such trials. In this paper, we propose to improve the existing literature in several directions. First, we show that the sample standard deviation estimation in Hozo et al. (2005) has some serious limitations and is always less satisfactory in practice. Inspired by this, we propose a new estimation method by incorporating the sample size. Second, we systematically study the sample mean and standard deviation estimation problem under more general settings where the first and third quartiles are also available for the trials. Through simulation studies, we demonstrate that the proposed methods greatly improve the existing methods and enrich the literature. We conclude our work with a summary table that serves as a comprehensive guidance for performing meta-analysis in different situations.

Abstract:
In the classical Newsboy problem, we provide a new proof for the tight range of optimal order quantities for the newsboy problem when only the mean and standard deviation of demand are available. The new proof is only based on the definition of the optimal solution therefore it is the most straightforward method. It is also shown that the classical Scarf’s rule is the mid-point of the range of optimal order quantities. This provides an additional understanding of Scarf’s order rule as a distribution free decision.

Abstract:
We ask the following question: what are the relative contributions of the ensemble mean and the ensemble standard deviation to the skill of a site-specific probabilistic temperature forecast? Is it the case that most of the benefit of using an ensemble forecast to predict temperatures comes from the ensemble mean, or from the ensemble spread, or is the benefit derived equally from the two? The answer is that one of the two is much more useful than the other.

Abstract:
Background: When conducting a meta-analysis of a continuous outcome, estimated means and standard deviations from the selected studies are required in order to obtain an overall estimate of the mean effect and its confidence interval. If these quantities are not directly reported in the publications, they need to must be estimated from other reported summary statistics, such as the median, the minimum, the maximum, and quartiles. Methods: We propose a simulation-based estimation approach using the Approximate Bayesian Computation (ABC) technique for estimating mean and standard deviation based on various sets of summary statistics found in published studies. We conduct a simulation study to compare the proposed ABC method with the existing methods of Hozo et al. (2005), Bland (2015), and Wan et al. (2014). Results: In the estimation of the standard deviation, our ABC method performs best in skewed or heavy-tailed distributions. The average relative error (ARE) approaches zero as sample size increases. In the normal distribution, our ABC performs well. However, the Wan et al. method is best since it is based on the normal distribution assumption. When the distribution is skewed or heavy-tailed, the ARE of Wan et al. moves away from zero even as sample size increases. In the estimation of the mean, our ABC method is best since the AREs converge to zero. Conclusion: ABC is a flexible method for estimating the study-specific mean and standard deviation for meta-analysis, especially with underlying skewed or heavy-tailed distributions. The ABC method can be applied using other reported summary statistics such as the posterior mean and 95% credible interval when Bayesian analysis has been employed.

Abstract:
This quasi-comparative-experimental study explained student achievements scores based on the level of disparity in the means within the control and experimental groups. Four secondary schools (two-government and two-private owned) were studied in Enugu State. The study was comprised of 182 Agricultural Science students in 4 intact classes drawn from four secondary schools in Enugu state. A Student Achievement Test (SAT) was used for data collection. The instrument was faced and content validated. Kuder-Richardson formula (K-R21) was used to determine the internal consistency of the instrument after trial testing, which yielded a reliability index of 0.79. Mean and standard deviation were used for data analysis. Findings of the study strongly supported the use of instructional materials in content delivery but cautions on the diverging effects that could arise among student as revealed in the widening gap between the highest and lowest mean achievement scores.

Abstract:
Prior knowledge of the autocorrelation function (ACF) enables an application of analytical formalism for the unbiased estimators of variance s2a and variance of the mean s2a(xmacr;). Both can be expressed with the use of so-called effective number of observations neff. We show how to adopt this formalism if only an estimate {rk} of the ACF derived from a sample is available. A novel method is introduced based on truncation of the {rk} function at the point of its first transit through zero (FTZ). It can be applied to non-negative ACFs with a correlation range smaller than the sample size. Contrary to the other methods described in literature, the FTZ method assures the finite range 1 < neff ≤ n for any data. The effect of replacement of the standard estimator of the ACF by three alternative estimators is also investigated. Monte Carlo simulations, concerning the bias and dispersion of resulting estimators sa and sa(×), suggest that the presented formalism can be effectively used to determine a measurement uncertainty. The described method is illustrated with the exemplary analysis of autocorrelated variations of the intensity of an X-ray beam diffracted from a powder sample, known as the particle statistics effect.

Abstract:
The mean absolute deviation about the mean is an alternative to the standard deviation for measuring dispersion in a sample or in a population. For stationary, ergodic time series with a finite first moment, an asymptotic expansion for the sample mean absolute deviation is proposed. The expansion yields the asymptotic distribution of the sample mean absolute deviation under a wide range of settings, allowing for serial dependence or an infinite second moment.