"As usual we may make the errors of I) rejecting the null hypothesis when it is true, II) accepting the null hypothesis when it is false. But there is a third kind of error which is of interest because the present test of significance is tied up closely with the idea of making a correct decision about which distribution function has slipped furthest to the right. We may make the error of III) correctly rejecting the null hypothesis for the wrong reason." (Frederick Mosteller, "A k-Sample Slippage Test for an Extreme Population", The Annals of Mathematical Statistics 19, 1948)
"Errors of the third kind happen in conventional tests of differences of means, but they are usually not considered, although their existence is probably recognized. It seems to the author that there may be several reasons for this among which are 1) a preoccupation on the part of mathematical statisticians with the formal questions of acceptance and rejection of null hypotheses without adequate consideration of the implications of the error of the third kind for the practical experimenter, 2) the rarity with which an error of the third kind arises in the usual tests of significance." (Frederick Mosteller, "A k-Sample Slippage Test for an Extreme Population", The Annals of Mathematical Statistics 19, 1948)
"For many purposes graphical accuracy is sufficient. The speed of graphical processes, and more especially the advantages of visual presentation in pointing out facts or clues which might otherwise be overlooked, make graphical analysis very valuable." (Frederick Mosteller & John W Tukey, "The Uses and Usefulness of Binomial Probability Paper?", Journal of the American Statistical Association 44, 1949)
"If significance tests are required for still larger samples, graphical accuracy is insufficient, and arithmetical methods are advised. A word to the wise is in order here, however. Almost never does it make sense to use exact binomial significance tests on such data - for the inevitable small deviations from the mathematical model of independence and constant split have piled up to such an extent that the binomial variability is deeply buried and unnoticeable. Graphical treatment of such large samples may still be worthwhile because it brings the results more vividly to the eye." (Frederick Mosteller & John W Tukey, "The Uses and Usefulness of Binomial Probability Paper?", Journal of the American Statistical Association 44, 1949)
"Scientific and technological advances have made the world we live in complex and hard to understand. […] Science itself shows the same growing complexity. We often hear that 'one man can no longer cover a broad enough field' and that 'there is too much narrow specialization'. And yet these complexities must be met - and resolved. At all levels, decisions must be made which involve consideration of more than a single field." (Frederick Mosteller et al, "The Education of a Scientific Generalist", Science 109,1949)
"Mathematical models for empirical phenomena aid the
development of a science when a sufficient body of quantitative information has
been accumulated. This accumulation can be used to point the direction in which
models should be constructed and to test the adequacy of such models in their
interim states. Models, in turn, frequently are useful in organizing and
interpreting experimental data and in suggesting new directions for
experimental research." (Robert R. Bush & Frederick Mosteller, "A
Mathematical Model for Simple Learning", Psychological Review 58, 1951)
"Almost any sort of inquiry that is general and not
particular involves both sampling and measurement […]. Further, both the
measurement and the sampling will be imperfect in almost every case. We can define
away either imperfection in certain cases. But the resulting appearance of
perfection is usually only an illusion."
"Because representativeness is inherent in the sampling plan and not in the particular sample at hand, we can never make adequate use of sample results without some measure of how well the results of this particular sample are likely to agree with the results of other samples which the same sampling plan might have provided. The ability to assess stability fairly is as important as the ability to represent the population fairly. Modern sampling plans concentrate on both." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)
"By sampling we can learn only about collective properties of populations, not about properties of individuals. We can study the average height, the percentage who wear hats, or the variability in weight of college juniors [...]. The population we study may be small or large, but there must be a population - and what we are studying must be a population characteristic. By sampling, we cannot study individuals as particular entities with unique idiosyncrasies; we can study regularities (including typical variabilities as well as typical levels) in a population as exemplified by the individuals in the sample." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)
"In many cases general probability samples can be thought
of in terms of (1) a subdivision of the population into strata, (2) a
self-weighting probability sample in each stratum, and (3) combination of the
stratum sample means weighted by the size of the stratum."
"That which can be and should be representative is the
sampling plan, which includes the manner in which the sample was drawn
(essentially a specification of what other samples might have been drawn and
what the relative chances of selection were for any two possible samples) and
how it is to be analyzed. [...] It is clear that many [...] groups fail to be represented
in any particular sample, yet this is not a criticism of that sample.
Representation is not, and should not be, by groups. It is, and should be, by
individuals as members of the sampled population. Representation is not, and
should not be, in any particular sample. It is, and should be, in the sampling
plan."
"The main purpose of a significance test is to inhibit the natural enthusiasm of the investigator." (Frederick Mosteller, "Selected Quantitative Techniques", 1954)
"We must emphasize that such terms as 'select at random', 'choose at random', and the like, always mean that some mechanical device, such as coins, cards, dice, or tables of random numbers, is used." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)
"We have made the sampling plan representative, not by
giving each individual an equal chance to enter the sample and then weighting
them equally, but by a more noticeable process of compensation, where those
individuals very likely to enter the sample are weighted less, while those
unlikely to enter are weighted more when they do appear. The net result is to
give each individual an equal chance of affecting the (weighted) sample mean."
"We realize that if someone just 'grabs a handful', the individuals in the handful almost always resemble one another (on the average) more than do the members of a simple random sample. Even if the 'grabs' [sampling] are randomly spread around so that every individual has an equal chance of entering the sample, there are difficulties. Since the individuals of grab samples resemble one another more than do individuals of random samples, it follows (by a simple mathematical argument) that the means of grab samples resemble one another less than the means of random samples of the same size. From a grab sample, therefore, we tend to underestimate the variability in the population, although we should have to overestimate it in order to obtain valid estimates of variability of grab sample means by substituting such an estimate into the formula for the variability of means of simple random samples. Thus using simple random sample formulas for grab sample means introduces a double bias, both parts of which lead to an unwarranted appearance of higher stability." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)
"Weighing a sample appropriately is no more fudging the data than is correcting a gas volume for barometric pressure." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)
"A primary goal of any learning model is to predict correctly the learning curve - proportions of correct responses versus trials. Almost any sensible model with two or three free parameters, however, can closely fit the curve, and so other criteria must be invoked when one is comparing several models." (Robert R Bush & Frederick Mosteller, "A Comparison of Eight Models?", Studies in Mathematical Learning Theory, 1959)
"A satisfactory prediction of the sequential properties of learning data from a single experiment is by no means a final test of a model. Numerous other criteria - and some more demanding - can be specified. For example, a model with specific numerical parameter values should be invariant to changes in independent variables that explicitly enter in the model." (Robert R Bush & Frederick Mosteller,"A Comparison of Eight Models?", Studies in Mathematical Learning Theory, 1959)
"In the testing of a scientific model or theory, one rarely has a general measure of goodness-of-fit, a universal yardstick by which one accepts or rejects the model. Indeed, science does not and should not work this way; a theory is kept until a better one is found. One way that science does work is by comparing two or more theories to determine their relative merits in handling relevant data."(Robert R Bush & Frederick Mosteller, "A Comparison of Eight Models?", Studies in Mathematical Learning Theory, 1959)
"In a problem, the great thing is the challenge. A problem
can be challenging for many reasons: because the subject matter is intriguing,
because the answer defies unsophisticated intuition, because it illustrates an
important principle, because of its vast generality, because of its difficulty,
because of a clever solution, or even because of the simplicity or beauty of
the answer." (Frederick Mosteller, "Fifty Challenging Problems in Probability with
Solutions", 1965)
"Using data from the population as it stands is a dangerous substitute for testing." (Frederick Mosteller & Gale Mosteller, "New Statistical Methods in Public Policy. Part I: Experimentation", Journal of Contemporary Business 8, 1979)
"Although we often hear that data speak for themselves,
their voices can be soft and sly." (Frederick Mosteller, "Beginning Statistics
with Data Analysis", 1983)
"The law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen." (Frederick Mosteller, "Methods for Studying Coincidences", Journal of the American Statistical Association Vol. 84, 1989)
"It is easy to lie with statistics, but easier to lie without them [...]" (Frederick Mosteller)
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