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104-459: By reducing detailed measures to a sequence of ordinal numbers , rankings make it possible to evaluate complex information according to certain criteria. Thus, for example, an Internet search engine may rank the pages it finds according to an estimation of their relevance , making it possible for the user quickly to select the pages they are likely to want to see. Analysis of data obtained by ranking commonly requires non-parametric statistics . It

208-496: A ↦ T < a {\displaystyle a\mapsto T_{<a}} defines an order isomorphism between T and the set of all subsets of T having the form T < a := { x ∈ T ∣ x < a } {\displaystyle T_{<a}:=\{x\in T\mid x<a\}} ordered by inclusion. This motivates the standard definition, suggested by John von Neumann at

312-425: A color wheel —there is no mean to the set of all colors. In these situations, you must decide which mean is most useful. You can do this by adjusting the values before averaging, or by using a specialized approach for the mean of circular quantities . The Fréchet mean gives a manner for determining the "center" of a mass distribution on a surface or, more generally, Riemannian manifold . Unlike many other means,

416-421: A probability distribution is the long-run arithmetic average value of a random variable having that distribution. If the random variable is denoted by X {\displaystyle X} , then the mean is also known as the expected value of X {\displaystyle X} (denoted E ( X ) {\displaystyle E(X)} ). For a discrete probability distribution ,

520-462: A supremum , the ordinal obtained by taking the union of all the ordinals in the set. This union exists regardless of the set's size, by the axiom of union . The class of all ordinals is not a set. If it were a set, one could show that it was an ordinal and thus a member of itself, which would contradict its strict ordering by membership. This is the Burali-Forti paradox . The class of all ordinals

624-400: A truncated mean . It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of the total number of values. The interquartile mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing

728-401: A "lower" step—then the computation will terminate. It is inappropriate to distinguish between two well-ordered sets if they only differ in the "labeling of their elements", or more formally: if the elements of the first set can be paired off with the elements of the second set such that if one element is smaller than another in the first set, then the partner of the first element is smaller than

832-401: A canonical labeling of the elements of any well-ordered set. Every well-ordered set ( S ,<) is order-isomorphic to the set of ordinals less than one specific ordinal number under their natural ordering. This canonical set is the order type of ( S ,<). Essentially, an ordinal is intended to be defined as an isomorphism class of well-ordered sets: that is, as an equivalence class for

936-452: A definition is normally said to be by transfinite recursion – the proof that the result is well-defined uses transfinite induction. Let F denote a (class) function F to be defined on the ordinals. The idea now is that, in defining F (α) for an unspecified ordinal α, one may assume that F (β) is already defined for all β < α and thus give a formula for F (α) in terms of these F (β). It then follows by transfinite induction that there

1040-417: A function f ( x ) {\displaystyle f(x)} . Intuitively, a mean of a function can be thought of as calculating the area under a section of a curve, and then dividing by the length of that section. This can be done crudely by counting squares on graph paper, or more precisely by integration . The integration formula is written as: In this case, care must be taken to make sure that

1144-683: A gap. This method is called "High" by IBM SPSS and "max" by the R programming language in their methods to handle ties. In dense ranking, items that compare equally receive the same ranking number, and the next items receive the immediately following ranking number. Equivalently, each item's ranking number is 1 plus the number of items ranked above it that are distinct with respect to the ranking order. Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B gets ranking number 2 ("joint second"), C also gets ranking number 2 ("joint second") and D gets ranking number 3 ("Third"). This method

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1248-518: A given group of data , illustrating the magnitude and sign of the data set . Which of these measures is most illuminating depends on what is being measured, and on context and purpose. The arithmetic mean , also known as "arithmetic average", is the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x 1 , x 2 , ..., x n is typically denoted using an overhead bar , x ¯ {\displaystyle {\bar {x}}} . If

1352-483: A larger ordinal). The smallest uncountable ordinal is the set of all countable ordinals, expressed as ω 1 or ⁠ Ω {\displaystyle \Omega } ⁠ . In a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice , this is equivalent to saying that the set is totally ordered and there is no infinite decreasing sequence (the latter being easier to visualize). In practice,

1456-457: A least element is called a well-order . The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to an initial segment of the other. So ordinal numbers exist and are essentially unique. Ordinal numbers are distinct from cardinal numbers , which measure the size of sets. Although the distinction between ordinals and cardinals is not always apparent on finite sets (one can go from one to

1560-577: A limit ordinal α {\displaystyle \alpha } is said to be unbounded (or cofinal) under α {\displaystyle \alpha } provided any ordinal less than α {\displaystyle \alpha } is less than some ordinal in the set. More generally, one can call a subset of any ordinal α {\displaystyle \alpha } cofinal in α {\displaystyle \alpha } provided every ordinal less than α {\displaystyle \alpha }

1664-399: A list of numbers, is the sum of all of the numbers divided by their count. Similarly, the mean of a sample x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} , usually denoted by x ¯ {\displaystyle {\bar {x}}} , is the sum of the sampled values divided by

1768-421: A measure of central tendency ). The mean of a set of observations is the arithmetic average of the values; however, for skewed distributions , the mean is not necessarily the same as the middle value (median), or the most likely value (mode). For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast,

1872-425: A natural number) there is another ordinal (natural number) larger than it, but still less than ω. Thus, every ordinal is either zero, or a successor (of a well-defined predecessor), or a limit. This distinction is important, because many definitions by transfinite recursion rely upon it. Very often, when defining a function F by transfinite recursion on all ordinals, one defines F (0), and F (α+1) assuming F (α)

1976-438: A rather surprising alternative solution to the Burali-Forti paradox of the largest ordinal). Rather than defining an ordinal as an equivalence class of well-ordered sets, it will be defined as a particular well-ordered set that (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number. For each well-ordered set T ,

2080-412: A slight modification, for classes of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any class of ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it in class-bijection with the class of all ordinals). So the γ {\displaystyle \gamma } -th element in the class (with

2184-485: A wide range of other notions of mean are often used in geometry and mathematical analysis ; examples are given below. In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music. The arithmetic mean (or simply mean or average ) of

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2288-428: Is ω ⋅ γ {\displaystyle \omega \cdot \gamma } (see ordinal arithmetic for the definition of multiplication of ordinals). Similarly, one can consider additively indecomposable ordinals (meaning a nonzero ordinal that is not the sum of two strictly smaller ordinals): the γ {\displaystyle \gamma } -th additively indecomposable ordinal

2392-400: Is ⁠ ω {\displaystyle \omega } ⁠ , which can be identified with the set of natural numbers (so that the ordinal associated with every natural number precedes ω {\displaystyle \omega } ). Indeed, the set of natural numbers is well-ordered—as is any set of ordinals—and since it is downward closed, it can be identified with

2496-437: Is a bijection f that preserves the ordering. That is, f ( a ) ≤' f ( b ) if and only if a ≤ b . Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: this makes it quite justifiable to consider the two sets as essentially identical, and to seek a "canonical" representative of the isomorphism type (class). This is exactly what the ordinals provide, and it also provides

2600-433: Is a proper subset of T . Moreover, either S is an element of T , or T is an element of S , or they are equal. So every set of ordinals is totally ordered . Further, every set of ordinals is well-ordered. This generalizes the fact that every set of natural numbers is well-ordered. Consequently, every ordinal S is a set having as elements precisely the ordinals smaller than S . For example, every set of ordinals has

2704-424: Is also referred to as "row numbering". This method corresponds to the "first", "last", and "random" methods in the R programming language to handle ties. Items that compare equal receive the same ranking number, which is the mean of what they would have under ordinal rankings; equivalently, the ranking number of 1 plus the number of items ranked above it plus half the number of items equal to it. This strategy has

2808-757: Is also the cardinality of ω or ε 0 (all are countable ordinals). So ω can be identified with ⁠ ℵ 0 {\displaystyle \aleph _{0}} ⁠ , except that the notation ℵ 0 {\displaystyle \aleph _{0}} is used when writing cardinals, and ω when writing ordinals (this is important since, for example, ℵ 0 2 {\displaystyle \aleph _{0}^{2}} = ℵ 0 {\displaystyle \aleph _{0}} whereas ω 2 > ω {\displaystyle \omega ^{2}>\omega } ). Also, ω 1 {\displaystyle \omega _{1}}

2912-486: Is an average which is useful for sets of numbers which are defined in relation to some unit , as in the case of speed (i.e., distance per unit of time): For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is If we have five pumps that can empty a tank of a certain size in respectively 4, 36, 45, 50, and 75 minutes, then the harmonic mean of 15 {\displaystyle 15} tells us that these five different pumps working together will pump at

3016-403: Is an example: Suppose you have the data set 1.0, 1.0, 2.0, 3.0, 3.0, 4.0, 5.0, 5.0, 5.0. The ordinal ranks are 1, 2, 3, 4, 5, 6, 7, 8, 9. For v = 1.0, the fractional rank is the average of the ordinal ranks: (1 + 2) / 2 = 1.5. In a similar manner, for v = 5.0, the fractional rank is (7 + 8 + 9) / 3 = 8.0. Thus the fractional ranks are: 1.5, 1.5, 3.0, 4.5, 4.5, 6.0, 8.0, 8.0, 8.0 This method

3120-512: Is because the most involved parents will then avoid such schools, leaving only the children of non-ambitious parents to attend. In business, league tables list the leaders in the business activity within a specific industry, ranking companies based on different criteria including revenue, earnings, and other relevant key performance indicators (such as market share and meeting customer expectations) enabling people to quickly analyze significant data. The rank methodology based on some specific indices

3224-411: Is by the ranking numbers that would be produced for four items, with the first item ranked ahead of the second and third (which compare equal) which are both ranked ahead of the fourth. These names are also shown below. In competition ranking, items that compare equal receive the same ranking number, and then a gap is left in the ranking numbers. The number of ranking numbers that are left out in this gap

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3328-454: Is called "Low" by IBM SPSS and "min" by the R programming language in their methods to handle ties. Sometimes, competition ranking is done by leaving the gaps in the ranking numbers before the sets of equal-ranking items (rather than after them as in standard competition ranking). The number of ranking numbers that are left out in this gap remains one less than the number of items that compared equal. Equivalently, each item's ranking number

3432-417: Is called "Mean" by IBM SPSS and "average" by the R programming language in their methods to handle ties. In statistics , ranking is the data transformation in which numerical or ordinal values are replaced by their rank when the data are sorted. For example, if the numerical data 3.4, 5.1, 2.6, 7.3 are observed, the ranks of these data items would be 2, 3, 1 and 4 respectively. As another example,

3536-434: Is called "Sequential" by IBM SPSS and "dense" by the R programming language in their methods to handle ties. In ordinal ranking, all items receive distinct ordinal numbers, including items that compare equal. The assignment of distinct ordinal numbers to items that compare equal can be done at random, or arbitrarily, but it is generally preferable to use a system that is arbitrary but consistent, as this gives stable results if

3640-488: Is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality. Rather than formulating these definitions for (proper) classes of ordinals, one can formulate them for sets of ordinals below a given ordinal α {\displaystyle \alpha } : A subset of

3744-430: Is closed for the order topology on that ordinal, this is again equivalent). Of particular importance are those classes of ordinals that are closed and unbounded , sometimes called clubs . For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if

3848-600: Is defined, and then, for limit ordinals δ one defines F (δ) as the limit of the F (β) for all β<δ (either in the sense of ordinal limits, as previously explained, or for some other notion of limit if F does not take ordinal values). Thus, the interesting step in the definition is the successor step, not the limit ordinals. Such functions (especially for F nondecreasing and taking ordinal values) are called continuous. Ordinal addition, multiplication and exponentiation are continuous as functions of their second argument (but can be defined non-recursively). Any well-ordered set

3952-406: Is equal to {0, 1} and so it is a subset of {0, 1, 2, 3} . It can be shown by transfinite induction that every well-ordered set is order-isomorphic to exactly one of these ordinals, that is, there is an order preserving bijective function between them. Furthermore, the elements of every ordinal are ordinals themselves. Given two ordinals S and T , S is an element of T if and only if S

4056-589: Is equal to the number of items ranked equal to it or above it. This ranking ensures that a competitor only comes second if they score higher than all but one of their opponents, third if they score higher than all but two of their opponents, etc. Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B gets ranking number 3 ("joint third"), C also gets ranking number 3 ("joint third") and D gets ranking number 4 ("fourth"). In this case, nobody would get ranking number 2 ("second") and that would be left as

4160-461: Is exactly what definition by transfinite recursion permits. In fact, F (0) makes sense since there is no ordinal β < 0 , and the set { F (β) | β < 0} is empty. So F (0) is equal to 0 (the smallest ordinal of all). Now that F (0) is known, the definition applied to F (1) makes sense (it is the smallest ordinal not in the singleton set { F (0)} = {0} ), and so on (the and so on is exactly transfinite induction). It turns out that this example

4264-399: Is greater than ⁠ ℵ 1 {\displaystyle \aleph _{1}} ⁠ , and so on, and ω ω {\displaystyle \omega _{\omega }} is the limit of the ω n {\displaystyle \omega _{n}} for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed

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4368-483: Is indexed as ⁠ ω γ {\displaystyle \omega ^{\gamma }} ⁠ . The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the γ {\displaystyle \gamma } -th ordinal α {\displaystyle \alpha } such that ω α = α {\displaystyle \omega ^{\alpha }=\alpha }

4472-631: Is less than or equal to some ordinal in the set. The subset is said to be closed under α {\displaystyle \alpha } provided it is closed for the order topology in ⁠ α {\displaystyle \alpha } ⁠ , i.e. a limit of ordinals in the set is either in the set or equal to α {\displaystyle \alpha } itself. There are three usual operations on ordinals: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents

4576-409: Is not always possible to assign rankings uniquely. For example, in a race or competition two (or more) entrants might tie for a place in the ranking. When computing an ordinal measurement , two (or more) of the quantities being ranked might measure equal. In these cases, one of the strategies below for assigning the rankings may be adopted. A common shorthand way to distinguish these ranking strategies

4680-403: Is not very exciting, since provably F (α) = α for all ordinals α, which can be shown, precisely, by transfinite induction. Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42 has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is no largest natural number. If an ordinal has a maximum α, then it

4784-453: Is one and only one function satisfying the recursion formula up to and including α. Here is an example of definition by transfinite recursion on the ordinals (more will be given later): define function F by letting F (α) be the smallest ordinal not in the set { F (β) | β < α} , that is, the set consisting of all F (β) for β < α . This definition assumes the F (β) known in the very process of defining F ; this apparent vicious circle

4888-818: Is one less than the number of items that compared equal. Equivalently, each item's ranking number is 1 plus the number of items ranked above it. This ranking strategy is frequently adopted for competitions, as it means that if two (or more) competitors tie for a position in the ranking, the position of all those ranked below them is unaffected (i.e., a competitor only comes second if exactly one person scores better than them, third if exactly two people score better than them, fourth if exactly three people score better than them, etc.). Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B gets ranking number 2 ("joint second"), C also gets ranking number 2 ("joint second") and D gets ranking number 4 ("fourth"). This method

4992-622: Is one of the most common systems used by policy makers and international organizations in order to assess the socio-economic context of the countries. Some notable examples include the Human Development Index (United Nations), Doing Business Index ( World Bank ), Corruption Perceptions Index (Transparency International), and Index of Economic Freedom (the Heritage Foundation). For instance, the Doing Business Indicator of

5096-511: Is similar (order-isomorphic) to a unique ordinal number α {\displaystyle \alpha } ; in other words, its elements can be indexed in increasing fashion by the ordinals less than ⁠ α {\displaystyle \alpha } ⁠ . This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some ⁠ α {\displaystyle \alpha } ⁠ . The same holds, with

5200-414: Is so important in relation to ordinals that it is worth restating here. That is, if P (α) is true whenever P (β) is true for all β < α , then P (α) is true for all α. Or, more practically: in order to prove a property P for all ordinals α, one can assume that it is already known for all smaller β < α . Transfinite induction can be used not only to prove things, but also to define them. Such

5304-422: Is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are not closed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality). Since the intersection of two closed unbounded classes

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5408-430: Is that a limit ordinal is the limit in a topological sense of all smaller ordinals (under the order topology ). When ⟨ α ι | ι < γ ⟩ {\displaystyle \langle \alpha _{\iota }|\iota <\gamma \rangle } is an ordinal-indexed sequence, indexed by a limit γ {\displaystyle \gamma } and

5512-487: Is the probability density function . In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the Lebesgue integral of the random variable with respect to its probability measure . The mean need not exist or be finite; for some probability distributions the mean is infinite ( +∞ or −∞ ), while for others the mean is undefined . The generalized mean , also known as

5616-446: Is the next ordinal after α, and it is called a successor ordinal , namely the successor of α, written α+1. In the von Neumann definition of ordinals, the successor of α is α ∪ { α } {\displaystyle \alpha \cup \{\alpha \}} since its elements are those of α and α itself. A nonzero ordinal that is not a successor is called a limit ordinal . One justification for this term

5720-426: Is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-ordering defines a countable ordinal, and ω 1 {\displaystyle \omega _{1}} is the order type of that set), ω 2 {\displaystyle \omega _{2}} is the smallest ordinal whose cardinality

5824-440: Is to inform potential applicants about British universities based on a range of criteria. Similarly, in countries like India, league tables are being developed and a popular magazine, Education World, published them based on data from TheLearningPoint.net . It is complained that the ranking of England's schools to rigid guidelines that fail to take into account wider social conditions actually makes failing schools even worse. This

5928-511: Is variously called "Ord", "ON", or "∞". An ordinal is finite if and only if the opposite order is also well-ordered, which is the case if and only if each of its non-empty subsets has a greatest element . There are other modern formulations of the definition of ordinal. For example, assuming the axiom of regularity , the following are equivalent for a set x : These definitions cannot be used in non-well-founded set theories . In set theories with urelements , one has to further make sure that

6032-600: Is written ⁠ ε γ {\displaystyle \varepsilon _{\gamma }} ⁠ . These are called the " epsilon numbers ". A class C {\displaystyle C} of ordinals is said to be unbounded , or cofinal , when given any ordinal ⁠ α {\displaystyle \alpha } ⁠ , there is a β {\displaystyle \beta } in C {\displaystyle C} such that α < β {\displaystyle \alpha <\beta } (then

6136-430: Is written ⁠ ω α {\displaystyle \omega _{\alpha }} ⁠ , it is always a limit ordinal. Its cardinality is written ⁠ ℵ α {\displaystyle \aleph _{\alpha }} ⁠ . For example, the cardinality of ω 0 = ω is ⁠ ℵ 0 {\displaystyle \aleph _{0}} ⁠ , which

6240-530: Is ω+ω), ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set of ordinals formed in this way (the ω· m + n , where m and n are natural numbers) must itself have an ordinal associated with it: and that is ω . Further on, there will be ω , then ω , and so on, and ω , then ω , then later ω , and even later ε 0 ( epsilon nought ) (to give a few examples of relatively small—countable—ordinals). This can be continued indefinitely (as every time one says "and so on" when enumerating ordinals, it defines

6344-704: The Principia Mathematica , defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in ZF and related systems of axiomatic set theory because these equivalence classes are too large to form a set. However, this definition still can be used in type theory and in Quine's axiomatic set theory New Foundations and related systems (where it affords

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6448-418: The equivalence relation of "being order-isomorphic". There is a technical difficulty involved, however, in the fact that the equivalence class is too large to be a set in the usual Zermelo–Fraenkel (ZF) formalization of set theory. But this is not a serious difficulty. The ordinal can be said to be the order type of any set in the class. The original definition of ordinal numbers, found for example in

6552-501: The Fréchet mean is defined on a space whose elements cannot necessarily be added together or multiplied by scalars. It is sometimes also known as the Karcher mean (named after Hermann Karcher). In geometry, there are thousands of different definitions for the center of a triangle that can all be interpreted as the mean of a triangular set of points in the plane. This is an approximation to

6656-585: The World Bank measures business regulations and their enforcement in 190 countries. Countries are ranked according to ten indicators that are synthesized to produce the final rank. Each indicator is composed of sub-indicators; for instance, the Registering Property Indicator is composed of four sub-indicators measuring time, procedures, costs, and quality of the land registration system. These kinds of ranks are based on subjective criteria for assigning

6760-399: The age of 19, now called definition of von Neumann ordinals : "each ordinal is the well-ordered set of all smaller ordinals". In symbols, ⁠ λ = [ 0 , λ ) {\displaystyle \lambda =[0,\lambda )} ⁠ . Formally: The natural numbers are thus ordinals by this definition. For instance, 2 is an element of 4 = {0, 1, 2, 3}, and 2

6864-728: The axiom of choice, a cardinal may be represented by the set of sets with that cardinality having minimal rank (see Scott's trick ). One issue with Scott's trick is that it identifies the cardinal number 0 {\displaystyle 0} with ⁠ { ∅ } {\displaystyle \{\emptyset \}} ⁠ , which in some formulations is the ordinal number ⁠ 1 {\displaystyle 1} ⁠ . It may be clearer to apply Von Neumann cardinal assignment to finite cases and to use Scott's trick for sets which are infinite or do not admit well orderings. Note that cardinal and ordinal arithmetic agree for finite numbers. The α-th infinite initial ordinal

6968-534: The class must be a proper class, i.e., it cannot be a set). It is said to be closed when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function F {\displaystyle F} is continuous in the sense that, for δ {\displaystyle \delta } a limit ordinal, F ( δ ) {\displaystyle F(\delta )} (the δ {\displaystyle \delta } -th ordinal in

7072-419: The class) is the limit of all F ( γ ) {\displaystyle F(\gamma )} for γ < δ {\displaystyle \gamma <\delta } ; this is also the same as being closed, in the topological sense, for the order topology (to avoid talking of topology on proper classes, one can demand that the intersection of the class with any given ordinal

7176-412: The convention that the "0-th" is the smallest, the "1-st" is the next smallest, and so on) can be freely spoken of. Formally, the definition is by transfinite induction: the γ {\displaystyle \gamma } -th element of the class is defined (provided it has already been defined for all β < γ {\displaystyle \beta <\gamma } ), as

7280-490: The definition excludes urelements from appearing in ordinals. If α is any ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X . This concept, a transfinite sequence (if α is infinite) or ordinal-indexed sequence , is a generalization of the concept of a sequence . An ordinary sequence corresponds to the case α = ω, while a finite α corresponds to a tuple , a.k.a. string . Transfinite induction holds in any well-ordered set, but it

7384-444: The elements of any given well-ordered set (the smallest element being labelled 0, the one after that 1, the next one 2, "and so on"), and to measure the "length" of the whole set by the least ordinal that is not a label for an element of the set. This "length" is called the order type of the set. Any ordinal is defined by the set of ordinals that precede it. In fact, the most common definition of ordinals identifies each ordinal as

7488-470: The expense of continuity. Interpreted as nimbers , a game-theoretic variant of numbers, ordinals can also be combined via nimber arithmetic operations. These operations are commutative but the restriction to natural numbers is generally not the same as ordinary addition of natural numbers. Each ordinal associates with one cardinal , its cardinality. If there is a bijection between two ordinals (e.g. ω = 1 + ω and ω + 1 > ω ), then they associate with

7592-425: The first cardinal after all the ω n {\displaystyle \omega _{n}} ). Mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency ") in mathematics , especially in statistics . Each attempts to summarize or typify

7696-401: The importance of well-ordering is justified by the possibility of applying transfinite induction , which says, essentially, that any property that passes on from the predecessors of an element to that element itself must be true of all elements (of the given well-ordered set). If the states of a computation (computer program or game) can be well-ordered—in such a way that each step is followed by

7800-443: The integral converges. But the mean may be finite even if the function itself tends to infinity at some points. Angles , times of day, and other cyclical quantities require modular arithmetic to add and otherwise combine numbers. In all these situations, there will not be a unique mean. For example, the times an hour before and after midnight are equidistant to both midnight and noon. It is also possible that no mean exists. Consider

7904-536: The least natural number that has not been previously used. To extend this process to various infinite sets , ordinal numbers are defined more generally using linearly ordered greek letter variables that include the natural numbers and have the property that every set of ordinals has a least or "smallest" element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number ω {\displaystyle \omega } (omega) to be

8008-430: The least element that is greater than every natural number, along with ordinal numbers ⁠ ω + 1 {\displaystyle \omega +1} ⁠ , ⁠ ω + 2 {\displaystyle \omega +2} ⁠ , etc., which are even greater than ⁠ ω {\displaystyle \omega } ⁠ . A linear order such that every non-empty subset has

8112-411: The lowest and the highest quarter of values. assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights. In some circumstances, mathematicians may calculate a mean of an infinite (or even an uncountable ) set of values. This can happen when calculating the mean value y avg {\displaystyle y_{\text{avg}}} of

8216-422: The mean and size of sample i {\displaystyle i} respectively. In other applications, they represent a measure for the reliability of the influence upon the mean by the respective values. Sometimes, a set of numbers might contain outliers (i.e., data values which are much lower or much higher than the others). Often, outliers are erroneous data caused by artifacts . In this case, one can use

8320-598: The mean is given by ∑ x P ( x ) {\displaystyle \textstyle \sum xP(x)} , where the sum is taken over all possible values of the random variable and P ( x ) {\displaystyle P(x)} is the probability mass function . For a continuous distribution , the mean is ∫ − ∞ ∞ x f ( x ) d x {\displaystyle \textstyle \int _{-\infty }^{\infty }xf(x)\,dx} , where f ( x ) {\displaystyle f(x)}

8424-409: The median income is the level at which half the population is below and half is above. The mode income is the most likely income and favors the larger number of people with lower incomes. While the median and mode are often more intuitive measures for such skewed data, many skewed distributions are in fact best described by their mean, including the exponential and Poisson distributions. The mean of

8528-430: The notion of size, which leads to cardinal numbers , and the notion of position, which leads to the ordinal numbers described here. This is because while any set has only one size (its cardinality ), there are many nonisomorphic well-orderings of any infinite set, as explained below. Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with

8632-429: The number of items in the sample. For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is: The geometric mean is an average that is useful for sets of positive numbers, that are interpreted according to their product (as is the case with rates of growth) and not their sum (as is the case with the arithmetic mean): For example, the geometric mean of five values: 4, 36, 45, 50, 75 is: The harmonic mean

8736-475: The numbers are from observing a sample of a larger group , the arithmetic mean is termed the sample mean ( x ¯ {\displaystyle {\bar {x}}} ) to distinguish it from the group mean (or expected value ) of the underlying distribution, denoted μ {\displaystyle \mu } or μ x {\displaystyle \mu _{x}} . Outside probability and statistics,

8840-550: The operation or by using transfinite recursion. The Cantor normal form provides a standardized way of writing ordinals. It uniquely represents each ordinal as a finite sum of ordinal powers of ω. However, this cannot form the basis of a universal ordinal notation due to such self-referential representations as ε 0 = ω . Ordinals are a subclass of the class of surreal numbers , and the so-called "natural" arithmetical operations for surreal numbers are an alternative way to combine ordinals arithmetically. They retain commutativity at

8944-434: The ordinal associated with it. Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, ... After all natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them as names.) After all of these come ω·2 (which

9048-725: The ordinal data hot, cold, warm would be replaced by 3, 1, 2. In these examples, the ranks are assigned to values in ascending order, although descending ranks can also be used. League tables are used to compare the academic achievements of different institutions. College and university rankings order institutions in higher education by combinations of factors. In addition to entire institutions, specific programs, departments, and schools are ranked. These rankings usually are conducted by magazines, newspapers, governments and academics. For example, league tables of British universities are published annually by The Independent , The Sunday Times , and The Times . The primary aim of these rankings

9152-486: The other just by counting labels), they are very different in the infinite case, where different infinite ordinals can correspond to sets having the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated , although none of these operations are commutative . Ordinals were introduced by Georg Cantor in 1883 in order to accommodate infinite sequences and classify derived sets , which he had previously introduced in 1872 while studying

9256-517: The parameter m , the following types of means are obtained: This can be generalized further as the generalized f -mean and again a suitable choice of an invertible f will give The weighted arithmetic mean (or weighted average) is used if one wants to combine average values from different sized samples of the same population: Where x i ¯ {\displaystyle {\bar {x_{i}}}} and w i {\displaystyle w_{i}} are

9360-462: The partner of the second element in the second set, and vice versa. Such a one-to-one correspondence is called an order isomorphism , and the two well-ordered sets are said to be order-isomorphic or similar (with the understanding that this is an equivalence relation ). Formally, if a partial order ≤ is defined on the set S , and a partial order ≤' is defined on the set S' , then the posets ( S ,≤) and ( S' ,≤') are order isomorphic if there

9464-511: The power mean or Hölder mean, is an abstraction of the quadratic , arithmetic, geometric, and harmonic means. It is defined for a set of n positive numbers x i by x ¯ ( m ) = ( 1 n ∑ i = 1 n x i m ) 1 m {\displaystyle {\bar {x}}(m)=\left({\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{m}\right)^{\frac {1}{m}}} By choosing different values for

9568-428: The property that the sum of the ranking numbers is the same as under ordinal ranking. For this reason, it is used in computing Borda counts and in statistical tests (see below). Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B and C each get ranking number 2.5 (average of "joint second/third") and D gets ranking number 4 ("fourth"). Here

9672-645: The ranking is done multiple times. An example of an arbitrary but consistent system would be to incorporate other attributes into the ranking order (such as alphabetical ordering of the competitor's name) to ensure that no two items exactly match. With this strategy, if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first") and D gets ranking number 4 ("fourth"), and either B gets ranking number 2 ("second") and C gets ranking number 3 ("third") or C gets ranking number 2 ("second") and B gets ranking number 3 ("third"). In computer data processing, ordinal ranking

9776-404: The same cardinal. Any well-ordered set having an ordinal as its order-type has the same cardinality as that ordinal. The least ordinal associated with a given cardinal is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, and no other ordinal associates with its cardinal. But most infinite ordinals are not initial, as many infinite ordinals associate with

9880-519: The same cardinal. The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In theories with the axiom of choice, the cardinal number of any set has an initial ordinal, and one may employ the Von Neumann cardinal assignment as the cardinal's representation. (However, we must then be careful to distinguish between cardinal arithmetic and ordinal arithmetic.) In set theories without

9984-407: The same rate as much as five pumps that can each empty the tank in 15 {\displaystyle 15} minutes. AM, GM, and HM satisfy these inequalities: Equality holds if all the elements of the given sample are equal. In descriptive statistics , the mean may be confused with the median , mode or mid-range , as any of these may incorrectly be called an "average" (more formally,

10088-471: The score. Sometimes, the adopted parameters may produce discrepancies with the empirical observations, therefore potential biases and paradox may emerge from the application of these criteria. Ordinal numbers In set theory , an ordinal number , or ordinal , is a generalization of ordinal numerals (first, second, n th, etc.) aimed to extend enumeration to infinite sets . A finite set can be enumerated by successively labeling each element with

10192-528: The sequence is increasing , i.e. α ι < α ρ {\displaystyle \alpha _{\iota }<\alpha _{\rho }} whenever ι < ρ , {\displaystyle \iota <\rho ,} its limit is defined as the least upper bound of the set { α ι | ι < γ } , {\displaystyle \{\alpha _{\iota }|\iota <\gamma \},} that is,

10296-428: The set of ordinals that precede it. For example, the ordinal 42 is generally identified as the set {0, 1, 2, ..., 41}. Conversely, any set S of ordinals that is downward closed — meaning that for any ordinal α in S and any ordinal β < α, β is also in S — is (or can be identified with) an ordinal. This definition of ordinals in terms of sets allows for infinite ordinals. The smallest infinite ordinal

10400-401: The smallest element greater than the β {\displaystyle \beta } -th element for all ⁠ β < γ {\displaystyle \beta <\gamma } ⁠ . This could be applied, for example, to the class of limit ordinals: the γ {\displaystyle \gamma } -th ordinal, which is either a limit or zero

10504-437: The smallest ordinal (it always exists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself). Put more directly, it is the supremum of the set of smaller ordinals. Another way of defining a limit ordinal is to say that α is a limit ordinal if and only if: So in the following sequence: ω is a limit ordinal because for any smaller ordinal (in this example,

10608-449: The special kind of sets that are called well-ordered . A well-ordered set is a totally ordered set (an ordered set such that, given two distinct elements, one is less than the other) in which every non-empty subset has a least element. Equivalently, assuming the axiom of dependent choice , it is a totally ordered set without any infinite decreasing sequence — though there may be infinite increasing sequences. Ordinals may be used to label

10712-412: The terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of ε ⋅ {\displaystyle \varepsilon _{\cdot }} ordinals, or the class of cardinals , are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded. A class

10816-434: The uniqueness of trigonometric series . A natural number (which, in this context, includes the number 0 ) can be used for two purposes: to describe the size of a set , or to describe the position of an element in a sequence. When restricted to finite sets, these two concepts coincide, since all linear orders of a finite set are isomorphic . When dealing with infinite sets, however, one has to distinguish between

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