On Averages II

"An average value is a single value within the range of the data that is used to represent all of the values in the series. Since an average is somewhere within the range of the data, it is sometimes called a measure of central value." (Frederick E Croxton & Dudley J Cowden, "Practical Business Statistics", 1937)

"An average is a single value selected from a group of values to represent them in some way, a value which is supposed to stand for whole group of which it is part, as typical of all the values in the group." (Albert E Waugh, "Elements of Statistical Methods" 3rd Ed., 1952)

"An average does not tell the full story. It is hardly fully representative of a mass unless we know the manner in which the individual items scatter around it. A further description of the series is necessary if we are to gauge how representative the average is." (George Simpson & Fritz Kafka, "Basic Statistics", 1952)

"An average is sometimes called a 'measure of central tendency' because individual values of the variable usually cluster around it. Averages are useful, however, for certain  types of data in which there is little or no central tendency." (William A Spirr & Charles P Bonini, "Statistical Analysis for Business Decisions" 3rd Ed., 1967)

"The most widely used mathematical tools in the social sciences are statistical, and the prevalence of statistical methods has given rise to theories so abstract and so hugely complicated that they seem a discipline in themselves, divorced from the world outside learned journals. Statistical theories usually assume that the behavior of large numbers of people is a smooth, average 'summing-up' of behavior over a long period of time. It is difficult for them to take into account the sudden, critical points of important qualitative change. The statistical approach leads to models that emphasize the quantitative conditions needed for equilibrium-a balance of wages and prices, say, or of imports and exports. These models are ill suited to describe qualitative change and social discontinuity, and it is here that catastrophe theory may be especially helpful." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"The arithmetic mean has another familiar property that will be useful to remember. The sum of the deviations of the values from their mean is zero, and the sum of the squared deviations of the values about the mean is a minimum. That is to say, the sum of the squared deviations is less than the sum of the squared deviations about any other value." (Charles T Clark & Lawrence L Schkade, "Statistical Analysis for Administrative Decisions", 1979)

"Having NUMBERSENSE means: (•) Not taking published data at face value; (•) Knowing which questions to ask; (•) Having a nose for doctored statistics. [...] NUMBERSENSE is that bit of skepticism, urge to probe, and desire to verify. It’s having the truffle hog’s nose to hunt the delicacies. Developing NUMBERSENSE takes training and patience. It is essential to know a few basic statistical concepts. Understanding the nature of means, medians, and percentile ranks is important. Breaking down ratios into components facilitates clear thinking. Ratios can also be interpreted as weighted averages, with those weights arranged by rules of inclusion and exclusion. Missing data must be carefully vetted, especially when they are substituted with statistical estimates. Blatant fraud, while difficult to detect, is often exposed by inconsistency." (Kaiser Fung, "Numbersense: How To Use Big Data To Your Advantage", 2013)

"What is so unconventional about the statistical way of thinking? First, statisticians do not care much for the popular concept of the statistical average; instead, they fixate on any deviation from the average. They worry about how large these variations are, how frequently they occur, and why they exist. [...] Second, variability does not need to be explained by reasonable causes, despite our natural desire for a rational explanation of everything; statisticians are frequently just as happy to pore over patterns of correlation. [...] Third, statisticians are constantly looking out for missed nuances: a statistical average for all groups may well hide vital differences that exist between these groups. Ignoring group differences when they are present frequently portends inequitable treatment. [...] Fourth, decisions based on statistics can be calibrated to strike a balance between two types of errors. Predictably, decision makers have an incentive to focus exclusively on minimizing any mistake that could bring about public humiliation, but statisticians point out that because of this bias, their decisions will aggravate other errors, which are unnoticed but serious. [...] Finally, statisticians follow a specific protocol known as statistical testing when deciding whether the evidence fits the crime, so to speak. Unlike some of us, they don’t believe in miracles. In other words, if the most unusual coincidence must be contrived to explain the inexplicable, they prefer leaving the crime unsolved." (Kaiser Fung, "Numbers Rule the World", 2010) 

"A very different - and very incorrect - argument is that successes must be balanced by failures (and failures by successes) so that things average out. Every coin flip that lands heads makes tails more likely. Every red at roulette makes black more likely. […] These beliefs are all incorrect. Good luck will certainly not continue indefinitely, but do not assume that good luck makes bad luck more likely, or vice versa." (Gary Smith, "Standard Deviations", 2014)

"Average deviation is the average amount of scatter of the items in a distribution from either the mean or the median, ignoring the signs of the deviations. The average that is taken of the scatter is an arithmetic mean, which accounts for the fact that this measure is often called the mean deviation." (Charles T Clark & Lawrence L Schkade)