Editing Inductive reasoning (section)

=== Inductive generalization ===
A generalization (more accurately, an ''inductive generalization'') proceeds from premises about a [[Sample (statistics)|sample]] to a conclusion about the [[statistical population|population]].<ref name=":0">{{Cite book|last=Govier|first=Trudy|title=A Practical Study of Argument, Enhanced Seventh Edition|publisher=Cengage Learning|year=2013|isbn=978-1-133-93464-6|location=Boston, MA|pages=283}}</ref> The observation obtained from this sample is projected onto the broader population.<ref name=":0" />

: The proportion Q of the sample has attribute A.
: Therefore, the proportion Q of the population has attribute A.

For example, if there are 20 balls—either black or white—in an urn: to estimate their respective numbers, a ''sample'' of four balls is drawn, three are black and one is white. An inductive generalization may be that there are 15 black and five white balls in the urn. However this is only one of 17 possibilities as to the ''actual'' number of each color of balls in the urn (the ''population)'' -- there may, of course, have been 19 black and just 1 white ball, or only 3 black balls and 17 white, or any mix in between. The probability of each possible distribution being the actual numbers of black and white balls can be estimated using techniques such as [[Bayesian inference]], where prior assumptions about the distribution are updated with the observed sample, or [[maximum likelihood estimation]] (MLE), which identifies the distribution most likely given the observed sample.

How much the premises support the conclusion depends upon the number in the sample group, the number in the population, and the degree to which the sample represents the population (which, for a static population, may be achieved by taking a random sample). The greater the sample size relative to the population and the more closely the sample represents the population, the stronger the generalization is. The [[hasty generalization]] and the [[biased sample]] are generalization fallacies.

==== Statistical generalization ====
A statistical generalization is a type of inductive argument in which a conclusion about a population is inferred using a [[Sample (statistics)|statistically representative sample]]. For example:

:Of a sizeable random sample of voters surveyed, 66% support Measure Z.
:Therefore, approximately 66% of voters support Measure Z.

The measure is highly reliable within a well-defined margin of error provided that the selection process was genuinely random and that the numbers of items in the sample having the properties considered are large. It is readily quantifiable. Compare the preceding argument with the following. "Six of the ten people in my book club are Libertarians. Therefore, about 60% of people are Libertarians." The argument is weak because the sample is non-random and the sample size is very small.

Statistical generalizations are also called ''statistical projections''<ref>Schaum's Outlines, Logic, Second Edition. John Nolt, Dennis Rohatyn, Archille Varzi. McGraw-Hill, 1998. p. 223</ref> and ''sample projections''.<ref>Schaum's Outlines, Logic, p. 230</ref>

==== Anecdotal generalization ====
An anecdotal generalization is a type of inductive argument in which a conclusion about a population is inferred using a non-statistical sample.<ref>{{Cite book|last1=Johnson|first1=Dale D.|url=https://books.google.com/books?id=cMOPtcgQfT8C|title=Trivializing Teacher Education: The Accreditation Squeeze|last2=Johnson|first2=Bonnie|last3=Ness|first3=Daniel|last4=Farenga|first4=Stephen J.|publisher=Rowman & Littlefield|year=2005|isbn=9780742535367|pages=182–83}}</ref> In other words, the generalization is based on [[anecdotal evidence]]. For example:

:So far, this year his son's Little League team has won 6 of 10 games.
:Therefore, by season's end, they will have won about 60% of the games.

This inference is less reliable (and thus more likely to commit the fallacy of hasty generalization) than a statistical generalization, first, because the sample events are non-random, and second because it is not reducible to a mathematical expression. Statistically speaking, there is simply no way to know, measure and calculate the circumstances affecting performance that will occur in the future. On a philosophical level, the argument relies on the presupposition that the operation of future events will mirror the past. In other words, it takes for granted a uniformity of nature, an unproven principle that cannot be derived from the empirical data itself. Arguments that tacitly presuppose this uniformity are sometimes called ''Humean'' after the philosopher who was first to subject them to philosophical scrutiny.<ref>Introduction to Logic. Gensler p. 280</ref>