hcju/proofwikips huggingface dataset:
Definition:Pythagorean Quadrilateral
A '''Pythagorean quadrilateral''' is a convex quadrilateral whose diagonals intersect at right angles, and is formed by fitting four right triangles with integer-valued sides. The interest is finding solutions where no two triangles are similar. The smallest Pythagorean quadrilateral is formed by the triangles with side lengths: :$\tuple {25, 60, 65}, \tuple {91, 60, 109}, \tuple {91, 312, 325}, \tuple {25, 312, 313}$ The smallest primitive Pythagorean quadrilateral, where each Pythagorean triple is primitive is: :$\tuple {28435, 20292, 34933}, \tuple {284795, 20292, 285517}, \tuple {284795, 181908, 337933}, \tuple {28435, 181908, 184117}$ The smallest anti-primitive Pythagorean quadrilateral, where no Pythagorean triples are primitive is: :$\tuple {1209, 6188, 6305}, \tuple {10659, 6188, 12325}, \tuple {10659, 23560, 25859}, \tuple {1209, 23560, 23591}$ with common divisors: :$13, 17, 19, 31$
Definition:EPORN
An '''EPORN''' is a natural number which can be expressed as the product of a number and its reversal in two different ways.
Definition:Dudeney Number
A '''Dudeney number''' is a natural number which is equal to the sum of the digits of its cube when expressed in decimal notation.
Definition:Supremum Norm/Continuous on Closed Interval Real-Valued Function
Let $I = \closedint a b$ be a closed real interval. Let $\map \CC I$ be the space of real-valued functions, continuous on $I$. Let $f \in \map \CC I$. Let $\size {\, \cdot \,}$ denote the absolute value. Suppose $\sup$ denotes the supremum of real-valued functions. Then the '''supremum norm''' over $\map \CC I$ is defined as :$\displaystyle \norm {f}_\infty := \sup_{x \mathop \in I} \size {\map f x}$
Definition:Think of a Number
A '''think of a number''' puzzle is usually in the form of a game between two players. Player '''A''' asks player '''B''' to: :''Think of a number'' perhaps with constraints. Let this number be referred to as $n$. Player '''A''' asks player '''B''' to perform certain arithmetical manipulations on $n$. As a result, player '''B''' is left with another number, which we will refer to as $m$. The game now goes one of $2$ ways: :$(1): \quad$ Player '''A''' announces: ::::''The number you have been left with is $m$.'' :$(2): \quad$ Player '''A''' asks what $m$ is, and on learning what it is, instantaneously replies: ::::''The number you first thought of was $n$.''
Definition:Ancient Egyptian Papyrus
Most of our knowledge of the mathematics of {{WP|Ancient_Egypt|ancient Egypt}} has come to us through a number of documents written on {{WP|Papyrus|papyrus}}.
Definition:Babylonian Mathematics
'''Babylonian mathematics''' is the name given to the mathematics of the {{WP|First_Babylonian_Dynasty|ancient Babylon}}. Its content was arithmetical and algebraical, and considerably in advance of the mathematics of the approximately contemporaneous {{WP|Ancient_Egypt|ancient Egypt}}.
Definition:Loculus of Archimedes
The '''loculus of Archimedes''' is a dissection of the square into $14$ pieces: :500px
Definition:Rusty Compass
A '''rusty compass''' is (despite what its name may make it seem) an ideal tool for drawing circles. It differs from a normal compass in that, once it has been opened, it is too stiff and rusty to be changed. Hence it can draw circles with only one radius.
Definition:Historic Measures
Various historic systems of measurement that are no longer in common use are gathered here.
Definition:Historic Measures/Length
Various historic systems of measurement of length that are no longer in common use are gathered here.
Definition:Historic Measures/Length/Leuca
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''leuca''' }} {{eqn | r = 1500       | c = paces }} {{eqn | r = 7500       | c = feet }} {{eqn | r = 90 \, 000       | c = inches }} {{end-eqn}}
Definition:Heap Problem
A '''heap problem''' is one where: :an unknown $x$ is established :$x$ is expressed as the sum of one or more fractions of $x$ and one or more other whole number. The object is to find $x$.
Definition:Sterling
'''Sterling''' is the system of monetary units that England (later Britain and the United Kingdom) inherited from the Roman system. Until $1971$ its main units were pounds, shillings and pence (that is, "pennies"). From $15$ February $1971$ its main units have been pounds and (new) pence.
Definition:Money
'''Money''' is a measure of purchasing power.
Definition:Penny
A '''penny''' is one of two coins of the Sterling monetary system: === New Penny === {{:Definition:Sterling/Penny/New}} === Old Penny === {{:Definition:Sterling/Penny/Old}}
Definition:Donkey and Mule Problem
A '''Donkey and Mule problem''' is of the form: :$2$ participants are comparing their quantity of goods they own relative to the other. :Participant $1$ suggests transferring some of the goods of one of them to the other, and remarks on their relative quantities after that exchange. :Participant $2$ then suggests the same (or a similar) thing, but transferring the goods in the other direction, and again remarks on their relative quantities after that exchange. From these remarks, $2$ simultaneous linear equations can be set up and (assuming the question is well-crafted) solved. The name comes from the original setting of this problem concerning a '''donkey''' and a '''mule''' each carrying a number of sacks.
Definition:Troy/Talent
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''talent''' }} {{eqn | r = 75       | c = troy pounds }} {{eqn | r = 900       | c = troy ounces }} {{end-eqn}}
Definition:Box-Jenkins Model
The '''Box-Jenkins model''' is a very general mathematical model for time series analysis in forecasting and prediction. === ARMA (Autoregressive Moving Average) === {{:Definition:Box-Jenkins Model/ARMA}} === ARIMA (Autoregressive Integrated Moving Average) === {{:Definition:Box-Jenkins Model/ARIMA}}
Definition:Recreational Chess
'''Recreational chess''' is a subgenre of recreational mathematics which explores the geometry of the chessboard and the moves of its pieces outside of the context of an actual game of chess itself. The name amusingly belies the fact chess is itself a recreation.
Definition:Heronian Triangle/Definition 2
A '''Heronian triangle''' is a triangle whose side lengths and altitudes are all rational numbers.
Definition:Proper Heronian Triangle
A '''proper Heronian triangle''' is a Heronian triangle which is specifically not right-angled.
Definition:Integer Heronian Triangle
An '''integer Heronian triangle''' is a Heronian triangle whose sides are integers.
Definition:Independence System
Let $S$ be a finite set. Let $\mathscr F$ be a set of subsets of $S$ satisfying the independence system axioms: {{:Definition:Independence System Axioms}} The ordered pair $I = \struct{S, \mathscr F}$ is called an '''independence system on $S$''', or simply an '''independence system''' when the context is obvious.
Definition:Independence System Axioms
{{begin-axiom}} {{axiom | n = \text I 1         | mr= \O \in \mathscr F  }} {{axiom | n = \text I 2         | q = \forall X \in \mathscr F: \forall Y \subseteq S         | mr= Y \subseteq X \implies Y \in \mathscr F }} {{end-axiom}}
Definition:Matroid Axioms/Axioms 1
{{begin-axiom}} {{axiom | n = \text I 1         | mr= \O \in \mathscr I  }} {{axiom | n = \text I 2         | q = \forall X \in \mathscr I: \forall Y \subseteq S         | mr= Y \subseteq X \implies Y \in \mathscr I }} {{axiom | n = \text I 3         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}}
Definition:Matroid Axioms/Axioms 2
{{begin-axiom}} {{axiom | n = \text I 1         | mr= \O \in \mathscr I  }} {{axiom | n = \text I 2         | q = \forall X \in \mathscr I: \forall Y \subseteq S         | mr= Y \subseteq X \implies Y \in \mathscr I }} {{axiom | n = \text I 3'         | q = \forall U, V \in \mathscr I         | mr= \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}}
Definition:Matroid Axioms/Axioms 3
{{begin-axiom}} {{axiom | n = \text I 1         | mr= \O \in \mathscr I  }} {{axiom | n = \text I 2         | q = \forall X \in \mathscr I: \forall Y \subseteq S         | mr= Y \subseteq X \implies Y \in \mathscr I }} {{axiom | n = \text I 3''         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren{V \cup Z \in \mathscr I} \land \paren{\size{V \cup Z} = \size U} }} {{end-axiom}}
Definition:Matroid Axioms/Axioms 4
{{begin-axiom}} {{axiom | n = \text I 1         | mr= \O \in \mathscr I  }} {{axiom | n = \text I 2         | q = \forall X \in \mathscr I: \forall Y \subseteq S         | mr= Y \subseteq X \implies Y \in \mathscr I }} {{axiom | n = \text I 3'''         | q = \forall A \subseteq S         | mr= \text{ all maximal subsets } Y \subseteq A \text{ with } Y \in \mathscr I \text{ have the same cardinality} }} {{end-axiom}}
Definition:P-adic Norm/Definition 1
Let $p \in \N$ be a prime. Let $\nu_p: \Q \to \Z \cup \set {+\infty}$ be the $p$-adic valuation on $\Q$. The '''$p$-adic norm''' on $\Q$ is the mapping $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ defined as: :$\forall q \in \Q: \norm q_p := \begin{cases} 0 & : q = 0 \\ p^{-\map {\nu_p} q} & : q \ne 0 \end{cases}$
Definition:P-adic Norm/Definition 2
Let $p \in \N$ be a prime. Let $k, m, n \in \Z : p \nmid m, n$. Let $\displaystyle r := p^k \frac m n$. The '''$p$-adic norm''' on $\Q$ is the mapping $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ defined as: :$\forall r \in \Q: \norm r_p := \begin{cases} 0 & : r = 0 \\ p^{-k} & : r \ne 0 \end{cases}$
Definition:Meridian
=== Terrestrial Meridian === {{:Definition:Meridian (Terrestrial)}} === Celestial Meridian === {{:Definition:Celestial Meridian}}
Definition:Geography Puzzle
A '''geography puzzle''' is a puzzle whose solution is dependent upon some particular aspect of geography or geodesy.
Definition:Chess/Chessboard/Square/White
A '''white square''' of a chessboard is one of the squares which is of the lighter colour.
Definition:Chess/Chessboard/Square/Black
A '''black square''' of a chessboard is one of the squares which is of the darker colour.
Definition:Formation of Ordinary Differential Equation by Elimination
Let $\map f {x, y, C_1, C_2, \ldots, C_n} = 0$ be an equation: :whose dependent variable is $y$ :whose independent variable is $x$ :$C_1, C_2, \ldots, C_n$ are constants which are deemed to be arbitrary. A '''differential equation''' may be '''formed''' from $f$ by: :differentiating $n$ times {{WRT|Differentiation}} $x$ to obtain $n$ equations in $x$ and $\dfrac {\d^k y} {\d x^k}$, for $k \in \set {1, 2, \ldots, n}$ :'''eliminating''' $C_k$ from these $n$ equations, for $k \in \set {1, 2, \ldots, n}$.
Definition:Real Number/Equality
Two real numbers are defined as being '''equal'''  they correspond to the same point on the real number line.
Definition:Nonzero Integer
A '''nonzero integer''' is an element of the set: :$\Z \setminus \set 0$ where: :$\Z$ denotes the set of integers :$\setminus$ denotes set difference. Thus: :$\Z \setminus \set 0 = \set {\ldots, -3, -2, -1, 1, 2, 3, \ldots}$ and can be denoted $\Z_{\ne 0}$.
Definition:Basis Expansion/Termination
Let the basis expansion of $x$ in base $b$ be: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$ Let it be the case that: :$\exists m \in \N: \forall k \ge m: d_k = 0$ That is, every digit of $x$ in base $b$ after a certain point is zero. Then $x$ is said to '''terminate'''.
Definition:Basis Expansion/Recurrence
Let the basis expansion of $x$ in base $b$ be: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$ Let there be a finite sequence of $p$ digits of $x$: :$\tuple {d_{r + 1} d_{r + 1} \ldots d_{r + p} }$ such that for all $k \in \Z_{\ge 0}$ and for all $j \in \set {1, 2, \ldots, p}$: :$d_{r + j + k p} = d_{r + j}$ where $p$ is the smallest $p$ to have this property. That is, let $x$ be of the form: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} \ldots}_b$ That is, $\tuple {d_{r + 1} d_{r + 2} \ldots d_{r + p} }$ repeats from then on, or '''recurs'''. Then $x$ is said to '''recur'''.
Definition:Basis Expansion/Recurrence/Notation
Let the basis expansion of $x$ in base $b$ be: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$ such that $x$ is recurring. Let the non-recurring part of $x$ be: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r}_b$ Let the recurring part of $x$ be: :$\sqbrk {\ldots d_{r + 1} d_{r + 2} \ldots d_{r + p} \ldots}_b$ Then $x$ is denoted: :$x = s.d_1 d_2 d_3 \ldots d_r \dot d_{r + 1} d_{r + 2} \ldots \dot d_{r + p}$ That is, a dot is placed over the first and last digit of the first instance of the recurring part.
Definition:Basis Expansion/Recurrence/Non-Recurring Part
Let the basis expansion of $x$ in base $b$ be recurring: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} d_{r + 2} \ldots d_{r + p } d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} \ldots}_b$ The '''non-recurring part''' of $x$  is: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r}$
Definition:Closed Ball/Metric Space
Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\epsilon \in \R_{>0}$ be a positive real number. The '''closed $\epsilon$-ball of $a$ in $M$''' is defined as: :$\map { {B_\epsilon}^-} a := \set {x \in A: \map d {x, a} \le \epsilon}$ where $B^-$ recalls the notation of topological closure. If it is necessary to show the metric itself, then the notation $\map { {B_\epsilon}^-} {a; d}$ can be used.
Definition:Multiplicative Inverse/Number
Let $\Bbb F$ be one of the standard number fields: $\Q$, $\R$, $\C$. Let $a \in \Bbb F$ be any arbitrary number. The '''multiplicative inverse''' of $a$ is its inverse under addition and can be denoted: $a^{-1}$, $\dfrac 1 a$, $1 / a$, and so on. :$a \times a^{-1} = 1$
Definition:Instant of Time
An '''instant''' is a specific point in time.
Definition:Time Series
A '''time series''' is a sequence of observations taken at a sequence of instants of time.
Definition:Time Series Analysis
'''Time series analysis''' is the discipline of analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Its main activity is concerned with techniques for determining the nature of the dependence of adjacent observations of such time series.
Definition:Observation
An '''observation''' is an item of raw data that has been obtained as the result of measurement or counting. Hence it is a particular value taken by a variable in statistics.
Definition:Torsion Element
=== Torsion Element of Group === Let $G$ be a group. {{:Definition:Torsion Element of Group}} === Torsion Element of Module === {{:Definition:Torsion Element of Module}}
Definition:Time Series/Adjacent Observations
Two observations of a '''time series''' are '''adjacent'''  the index of one of them is the immediate predecessor of the other (and the other is the immediate successor of the one).
Definition:Time Series/Dependence of Adjacent Observations
A defining characteristic of a '''time series''' is that every pair of adjacent observations is dependent.
Definition:Time Series/Timestamp
A '''timestamp''' of an '''observation''' $x$ of a '''time series''' is the instant with which $x$ is associated.
Definition:Forecasting
In the context of the discipline of '''time series analysis''', '''forecasting''' is the estimation of future observations of a given time series based on knowledge of current and past such observations.
Definition:Time Series/Current Value
The '''current value''' of a '''time series''' $T$ is an '''observation''' $x$ of $T$ whose timestamp is the '''current time'''. That is, it is the most recent observation.
Definition:Past
The '''past''' is the portion of time previous to the present. That is, the '''past''' is what has already happened.
Definition:Present
The '''present''' is the time at this point '''now'''.
Definition:Future
The '''future''' is the portion of time subsequent to the present. That is, the '''future''' is what has not happened yet.
Definition:Transfer Function (Time Series Analysis)
A '''transfer function''', in the context of time series analysis, is a function of time which theoretically models the future output for each possible input.
Definition:Intervention Event
An '''intervention event''' is an event whose occurrence is not characteristic of a particular time series such that it affects the behaviour of that time series. As a result, forecasting future values based upon the past values is likely to be compromised.
Definition:Control Scheme
A '''control scheme''' is a system whereby potential deviations of future values from a desired target may be compensated for, by adjustment of input values.
Definition:Lead Time
When forecasting on a time series, the '''lead time''' is the future time over which forecasts are needed. The '''lead time''' depends upon the problem for which the forecasting is being performed.
Definition:Time Series/Discrete
A '''discrete time series''' is such that the timestamps of the observations occur at well-defined instants, separated one from another by a time interval.
Definition:Time Series/Equispaced
A '''time series''' is '''equispaced'''  the time intervals between the timestamps is equal for all pairs of adjacent observations.
Definition:Time Series/Origin
The '''origin''' of a '''time series''' is an arbitrary timestamp of an observation which is chosen in order that all other timestamps can be measured from that '''origin'''.
Definition:Forecast Function
Let $T$ be a time series. Let $S$ denote the range of $T$. Let $L$ denote the set of lead times of $T$. Let $t$ denote the origin of $T$. A '''forecast function''' $\hat z_t : L \to S$ is a function which provides forecasts of the future values of $T$ at the lead times $L$. The objective is to make the mean squares of the deviations $z_{t + l} - \map {\hat z_t} l$ as small as possible for each lead time $l$. In the above: :$\map {\hat z_t} l$ denotes the forecast value of the observation at the timestamp of lead time $l$ :$z_{t + l}$ denotes the actual value of the observation at the timestamp of $l$.
Definition:Time Series/Forecast Value
A '''forecast value''' of a time series $T$ is an estimate of a future value at some lead time $t + l$.
Definition:Time Series/Actual Value
An '''actual value''' of a time series $T$ is the result of a measurement of an observation at some time $t$. The term is usually made in reference to a forecast value made after its lead time has elapsed, and its timestamp is now the current time.
Definition:Hyperbolic Sine/Real
The '''real hyperbolic sine''' function is defined on the real numbers as: :$\sinh: \R \to \R$: :$\forall x \in \R: \sinh x := \dfrac {e^x - e^{-x} } 2$
Definition:Hyperbolic Cosine/Real
The '''real hyperbolic cosine''' function is defined on the real numbers as: :$\cosh: \R \to \R$: :$\forall x \in \R: \cosh x := \dfrac {e^x + e^{-x} } 2$
Definition:Real Hyperbolic Tangent/Definition 2
The '''real hyperbolic tangent''' function is defined on the real numbers as: :$\tanh: \R \to \R$: :$\forall x \in \R: \tanh x := \dfrac {\sinh x} {\cosh x}$ where: :$\sinh$ is the real hyperbolic sine :$\cosh$ is the real hyperbolic cosine
Definition:Real Hyperbolic Tangent/Definition 1
The '''real hyperbolic tangent''' function is defined on the real numbers as: :$\tanh: \R \to \R$: :$\forall x \in \R: \tanh x := \dfrac {e^z - e^{-x} } {e^z + e^{-x} }$
Definition:Real Hyperbolic Cotangent/Definition 1
The '''real hyperbolic cotangent''' function is defined on the real numbers as: :$\coth: \R_{\ne 0} \to \R$: :$\forall x \in \R_{\ne 0}: \coth x := \dfrac {e^x + e^{-x} } {e^x - e^{-x} }$ where it is noted that at $x = 0$: :$e^x - e^{-x} = 0$ and so $\coth x$ is not defined at that point.
Definition:Real Hyperbolic Cotangent/Definition 2
The '''real hyperbolic cotangent''' function is defined on the real numbers as: :$\coth: \R_{\ne 0} \to \R$: :$\forall x \in \R_{\ne 0}: \coth x := \dfrac {\cosh x} {\sinh x}$ where: :$\sinh$ is the real hyperbolic sine :$\cosh$ is the real hyperbolic cosine It is noted that at $x = 0$ we have that $\sinh x = 0$, and so $\coth x$ is not defined at that point.
Definition:Real Hyperbolic Cosecant/Definition 2
The '''real hyperbolic cosecant''' function is defined on the real numbers as: :$\csch: \R_{\ne 0} \to \C$: :$\forall x \in \R_{\ne 0}: \csch x := \dfrac 1 {\sinh x}$ where $\sinh$ is the real hyperbolic sine. It is noted that at $x = 0$ we have that $\sinh x = 0$, and so $\csch x$ is not defined at that point.
Definition:Real Hyperbolic Cosecant/Definition 1
The '''real hyperbolic cosecant''' function is defined on the real numbers as: :$\csch: \R_{\ne 0} \to \R$: :$\forall x \in \R_{\ne 0}: \csch x := \dfrac 2 {e^x - e^{-x} }$ where it is noted that at $x = 0$: :$e^x - e^{-x} = 0$ and so $\csch x$ is not defined at that point.
Definition:Mean Square
Let $S$ be a set of numbers. The '''mean square''' of $S$ is the arithmetic mean of the squares of the elements of $S$: :$s^2 = \dfrac 1 n \displaystyle \sum_{i \mathop = 1}^n {x_i}^2$ where $S = \set {x_1, x_2, \ldots, x_n}$.
Definition:Deviation
Let $S$ be a set of observations. Let $x \in S$. The '''deviation''' of $x$ is the difference between $x$ and some other value, whose nature depends upon the context.
Definition:Deviation from Forecast
Let $T$ be a time series. Let $S$ denote the range of $T$. Let $L$ denote the set of lead times of $T$. Let $\hat z_t$ be a forecast function on $L$. Let $\map {\hat z_t} l$ denote the forecast value of the observation at the timestamp of lead time $l$. Let $z_{t + l}$ denote the actual value of the observation at the timestamp of $l$. The '''deviation (from forecast)''' is the difference between $\map {\hat z_t} l$ and $z_{t + l}$: :$\Delta_l := z_{t + l} - \map {\hat z_t} l$
Definition:Probability Limit/Upper
Let $\Delta_p$ be a probability limit for $\map {\hat z_t} l$. The '''upper probability limit''' of $\map {\hat z_t} l$ is the value $\map {\hat z_t} l + \Delta_p$.
Definition:Probability Limit
Let $T$ be a time series. Let $S$ denote the range of $T$. Let $L$ denote the set of lead times of $T$. Let $\hat z_t$ be a forecast function on $L$. Let $\map {\hat z_t} l$ denote the forecast value of the observation at the timestamp of lead time $l$. A '''probability limit''', for a given probability $p$, is the deviation $\Delta_p$, either positive or negative, from $\map {\hat z_t} l$ such that the probability that the actual value lies within $\Delta_p$ of $\map {\hat z_t} l$ is greater than $p$. That is, the actual value of the time series at $l$, when it occurs, will be within those '''probability limits''' within that stated probability $p$.
Definition:Probability Limit/Lower
Let $\Delta_p$ be a probability limit for $\map {\hat z_t} l$. The '''lower probability limit''' of $\map {\hat z_t} l$ is the value $\map {\hat z_t} l - \Delta_p$.
Definition:Stochastic Model
A '''stochastic model''' is a mathematical model which incorporates stochastic (random) processes in its definition.
Definition:Measure of Central Tendency
A '''measure of central tendency''' is a central or typical value for a probability distribution or set of sample data.
Definition:Median (Statistics)
Let $S$ be a set of quantitative data. Let $S$ be arranged in order of size. The '''median''' is the element of $S$ that is in the '''middle''' of that ordered set. Suppose there are an odd number of element of $S$ such that $S$ has cardinality $2 n - 1$. The '''median''' of $S$ in that case is the $n$th element of $S$. Suppose there are an even number of element of $S$ such that $S$ has cardinality $2 n$. Then the '''middle''' of $S$ is not well-defined, and so the '''median''' of $S$ in that case is the arithmetic mean of the $n$th and $n + 1$th elements of $S$.
Definition:Mode
Let $S$ be a set of quantitative data. The '''mode''' of $S$ is the element of $S$ which occurs most often in $S$. If there is more than one such element of $S$ which occurs equally most often, it is then understood that each of these is a '''mode'''. If there is no element of $S$ which occurs more often (in the extreme case, all elements are equal) then $S$ has '''no mode'''