This chapter begins by defining an averaging procedure for random variables, known as the mean. We show that the mean is linear, and also that the mean of the product of independent variables equals the product of their means. Then, we derive the mean of popular parametric distributions. Next, we caution that the mean can be severely distorted by extreme values, as illustrated by an analysis of NBA salaries. In addition, we define the mean square, which is the average squared value of a random variable, and the variance, which is the mean square deviation from the mean. We explain how to estimate the variance from data and use it to describe temperature variability at different geographic locations. Then, we define the conditional mean, a quantity that represents the average of a variable when other variables are fixed. We prove that the conditional mean is an optimal solution to the problem of regression, where the goal is to estimate a quantity of interest as a function of other variables. We end the chapter by studying how to estimate average causal effects.

We introduce the expectation of a random variable on a probability space, which is its best guess in the least-squares sense. The variance is an operator quantifying the dispersion of its distribution. We recall the expression of the expectation and variance of a weighted sum of variables. The moment generating function is a characterization of a distribution which is powerful for finding the distribution of a linear combination of variables. Those concepts are then extended to pairs of variables, for which covariance and correlation can be defined. The Central Limit Theorem gives another interpretation to the expectation, which is also the asymptotic value taken by the average of variables having the same distribution. Finally, we introduce a special class of random variables called Radon–Nikodym derivatives, which are nonnegative and display unit expectation. This family of variables can be used to build new probability measures starting from a reference probability space. Switching probability measures triggers a modification of the distribution of the random variables at hand. Those concepts are illustrated on various examples including coins, dice, and stock price models.

Chapter 4 examines measures of variability. Variability is the degree to which scores in a distribution are spread out or clustered together. Scores in a distribution not only focus around certain reference points or central tendency but also focus in particular ways around these central points. To understand the significance of any distribution of scores, we must know about both its central tendency and its variability. The index of qualitative variation (IQV), range, standard deviation, mean absolute deviation, and variance are all important measures of variability. Variability allows researchers to provide a full picture of a distribution of scores.

Dispersion describes the spread of the data or how it varies from its mean.The chapter begins with the calculation of the variance and then the more important standard deviation, along with their interpretation.Students learn how to calculate these measures by hand and in the R Commander.Other measures of dispersion like skewness and kurtosis are described.The range and interquartile range are also calculated using the R Commander software for ordinal variables.

Chapter 4 explores methods for turning model variance inside out. We discuss the logic and limitations of decomposing variance to the level of cases and introduce two distinct approaches to obtaining case-level contributions to the model variance: the squared residuals approach and the leave-one-out approach.

● Darwin’s theory of evolution by natural selection was influenced by his reading Thomas Malthus’s Essay on Population. The status of Malthusian ideas in evolutionary theory is discussed. ● An organism’s fitness is its ability to survive and reproduce. ● Natural selection influences the evolution of height (for example) precisely when individuals with different heights differ in their fitnesses. ● Fitness can be represented as a mathematical expectation, but mathematical variance matters too. ● It is almost always impossible to estimate the fitness of a single organism, but the fitnesses of traits that are possessed by multiple organisms can often be estimated. ● Two coextensive traits must have the same fitness, even if one of them helps organisms to survive and reproduce while the other does not. ● Natural selection does not inevitably lead the average fitness of organisms in a population to improve, even when the external environment is stable.

How can we use cognitive approaches to embed the dynamic and often variant outcomes of ritual experiences? Key themes that have emerged in both individual and communal rituals are the subjectivity and variation in these experiences: the role of physical and emotional interaction in shaping memory. The concluding chapter begins with a vivid discussion of cognition, sensation and experience, exploring how these elements function together, creating a spectrum of variable experiences and outcomes in modern and ancient ritual contexts. This section is used to further develop ideas, common themes and issues connecting the different chapters – the versatility of ritual experience(s), the role of embodied cognition in constructing ritual experience(s), the importance of the relationship between distributed cognition and ritual experience(s) - and how these themes help to expand disciplinary boundaries in the study of ancient religions and religious rituals. The concluding chapter also situates the themes of the volume within current cognitive science of religion research as well as broader disciplines such as art, heritage and museum studies. These discussions address how the study of ancient religions from a cognitive perspective can contribute to a number of disciplines, opening up new venues for research and interdisciplinary collaboration.

Chapter 3 covers measures of location, spread and skewness and includes the following specific topics, among others: mode, median, mean, weighted mean, range, interquartile range, variance, standard deviation, and skewness.

This chapter covers the two topics of descriptive statistics and the normal distribution. We first discuss the role of descriptive statistics and the measures of central tendency, variance, and standard deviation. We also provide examples of the kinds of graphs often used in descriptive statistics. We next discuss the normal distribution, its properties and its role in descriptive and inferential statistical analysis.

Abstract: This chapter presents a third-order predictive modelling methodology which aims at obtaining best-estimate results with reduced uncertainties (acronym: 3rd-BERRU-PM) for applications to large-scale models comprising many parameters. The building blocks of the 3rd-BERRU-PM methodology include quantification of third-order moments of the response distribution in the parameter space using third-order adjoint sensitivity analysis (which overcomes the curse of dimensionality), assimilation of experimental data, model calibration, and posterior prediction of best-estimate model responses and parameters with reduced best-estimate variances/covariances for the predicted responses and parameters. Applications of these concepts to an inverse radiation transmission problem, to an oscillatory dynamical model, and to a large-scale computational model involving 21,976 uncertain parameters, respectively, are also presented, thus illustrating the actual computation and impacts of the first-, second-, and third-order response sensitivities to parameters on the expectation, variance, and skewness of the respective model responses.

This chapter provides an overview of the processes that are commonly used for analyzing data. Our intention is to explain what these processes achieve and why they are done. Analyzing data goes through four stages. For each stage, we explain the most important concept and then explain the practical steps that are involved. This begins with the data themselves as variables. Next, we move on to describing the data, their variance and covariance, with linear models. Next, we cover interpreting effects and focus on effect sizes. We end with a discussion of inferences about the population and how the presence of uncertainty has to be taken into account in reaching conclusions.

A review of basic probability theory – probability density, expectation, mean, variance/covariance, median, median absolute deviation, quantiles, skewness/kurtosis and correlation – is first given. Exploratory data analysis methods (histograms, quantile-quantile plots and boxplots) are then introduced. Finally, topics including Mahalanobis distance, Bayes theorem, classification, clustering and information theory are covered.

This chapter reviews statistics and data-analysis tools. Starting from basic statistical concepts such as mean, variance, and the Gaussian distribution, we introduce the principal tools required for data analysis. We discuss both Bayesian and frequentist statistical approaches, with emphasis on the former. This leads us to describe how to calculate the goodness of fit of data to theory, and how to constrain the parameters of a model. Finally, we introduce and explain, both intuitively and mathematically, two important statistical tools: Markov chain Monte Carlo (MCMC) and the Fisher information matrix.

Quantum mechanics is inherently a probabilistic theory, so we present a brief review of some important concepts in probability theory. We distinguish between discrete probabilities, encountered in spin measurements, and continuous probabilities, encountered in position measurements.

This chapter explores how writing affected the nature of custom itself. The writing of custom did not fix or petrify custom and end its malleability or creativity, as scholars have assumed. Drawing on the work of literary scholars on the nature of medieval vernacular text and manuscript culture, I argue instead that each manuscript provided one authoritative version of custom and constituted one voice in a conversation over custom that continued with written text. This conversation can be seen where differences between manuscripts show diverging opinions about proper custom. This, in turn, meant that custom remained creative and dynamic even in written form. The purpose of the coutumiers was not to copy practised custom and faithfully record it with perfect accuracy in text. Rather, the coutumiers were meant to change the patterns of thought of ‘those who would hear or read the text’ and show them how to think in a legal manner, like a lawyer, judge, or sophisticated litigant. Teaching modes of thought rather than rules permitted those who did not spend years in university to understand the framework and rhetoric of lay courts, and enabled them to navigate these through changing rules, time, and circumstance.

Quantum mechanics is inherently a probabilistic theory, so we present a brief review of some important concepts in probability theory. We distinguish between discrete probabilities, encountered in spin measurements, and continuous probabilities, encountered in position measurements.

This chapter reviews some essential concepts of probability and statistics, including: line plots, histograms, scatter plots, mean, median, quantiles, variance, random variables, probability density function, expectation of a random variable, covariance and correlation, independence the normal distribution (also known as the Gaussian distribution), the chi-square distribution. The above concepts provide the foundation for the statistical methods discussed in the rest of this book.

The Innovation Pyramid, like any tool, comes down to how well we learn to use it to improve our collective judgement. Reviewing innovation projects allows us to better understand the linkages between the specific-level decisions and the overall project outcome. This deeper discernment requires a critical analysis of the entire innovation system – every decision taken between the initial situation and overall desired outcome must be reviewed. Fortunately, the structure of The Innovation Pyramid is equally useful in guiding the identification of the well-rationalized choice made during the design and execution of the innovation that inadvertently caused us to veer away from our overall, macro-level, desired impact of the original situation. This chapter lays out a structure to systematically review the project elements that could be associated with the variance between the actual and forecasted impact of the innovation. The guidance that The Pyramid provides for a project's post-execution review must often be augmented with additional research. Firms should consider this additional research an investment in their emerging core competency of serial innovation.

This chapter discusses two types of descriptive statistics: models of central tendency and models of variability. Models of central tendency describe the location of the middle of the distribution, and models of variability describe the degree that scores are spread out from one another. There are four models of central tendency in this chapter. Listed in ascending order of the complexity of their calculations, these are the mode, median, mean, and trimmed mean. There are also four principal models of variability discussed in this chapter: the range, interquartile range, standard deviation, and variance. For the latter two statistics, students are shown three possible formulas (sample standard deviation and variance, population standard deviation and variance, and population standard deviation and variance estimated from sample data), along with an explanation of when it is appropriate to use each formula. No statistical model of central tendency or variability tells you everything you may need to know about your data. Only by using multiple models in conjunction with each other can you have a thorough understanding of your data.