MTH 108

left-skewed distribution

a distribution in which the longer tail of the histogram is on the left side. (Because negative numbers lie on the left side of a number line, such a distribution is also called "negatively skewed.")

right-skewed distribution

a distribution in which the longer tail of the histogram is on the right side. (Because positive numbers lie on the right side of a number line, such a distribution is also called "positively skewed.")

histogram

a graph of the distribution of outcomes (divided into consecutive nonoverlapping classes) for a single numerical variable. The height of each bar is the number of observations in the class of outcomes covered by the base of the bar. All classes should have the same width so that bars can be compared to each other accurately and readily.

range

a measure of variability of a set of observations. It is obtained by sub- tracting the smallest observation from the largest observation. Equivalently: range = maximum -minimum.

variable

a particular characteristic or trait that can take on different values for different individuals. A particular variable may be either qualitative (e.g., gender) or quantitative (e.g., age), though our focus here is the latter.

outlier

an individual value that falls outside the overall pattern.

linear growth

arithmetic growth

mean

average value

grouped frequency distribution

gives how many times in each interval of values the variable takes on a particular value.

distribution

gives information (as a table, graph, or formula) about how often the variable takes certain values or intervals of values.

symmetric distribution

if a vertical line could be superimposed on the histogram and have the left and right sides be approximate mirror images of each other.

Condorcet winner criterion (CWC)

if for every election in which there is a Condorcet winner, that candidate wins the election when that voting system is used

independence of irrelevant alternatives (IIA)

if the only way a candidate (called A ) can go from losing one election to being among the winners of a new election (with the same set of candidates and voters) is for at least one voter to reverse his or her ranking of A and the previous winner.

manipulability

if there are elections in which it is to a voter's advantage to submit a ballot that misrepre- sents his or her true preferences.

principal

initial balance

median

middle value

monotonicity

satisfies (blank) provided that ballot changes favorable to one candidate (and not favorable to any other candidate) can never hurt that candidate.

frequency distribution

states all observed values of the variable and how many times the variable takes on each of these values.

relative frequency distribution

states all observed values of the variable and what fraction (or percentage) of the time each value occurs.

individuals

the entities described by a set of data. Individuals may be people, but they may also be groups, animals, or things

annuity

A specified number of equal periodic payments

constant dollars

Costs are expressed in constant dol- lars if inflation or deflation has been taken into account by converting the costs to their equivalent in dollars of a particular year.

weight

Weight A number assigned to an edge of a graph that can be thought of as a cost, distance, or time associated with that edge.

standard deviation

"standard" or average amount that observed data values deviate from their mean. More precisely, it is the square root of what is roughly the mean of the squared deviations.

minimum cost hamiltonian circuit

A Hamiltonian c ircuit in a graph with weights on the edges, for which the sum of the weights of the edges of the Hamiltonian circuit is as small as possible.

preference list ballot

A ballot that ranks the candidates from most preferred to least preferred, with no ties.

universal product code (UPC)

A bar code and identification number that is used on most retail items. The UPC code detects 100 percent of all single-digit errors and most other types of errors.

euler circuit

A circuit that traverses each edge of a graph exactly once.

Hamiltonian Circuit

A circuit using distinct edges of a graph that starts and ends at a particular vertex of the graph and visits each vertex once and only once. A Hamiltonian circuit can start at any one of its vertices.

error detecting code

A code in which certain types of errors can be detected.

path

A connected sequence of edges in a graph.

check digit

A digit included in an identification number for the purpose of error detection.

order requirement digraph

A directed graph that shows which tasks precede other tasks among the collection of tasks making up a job.

management science

A discipline in which mathematical methods are applied to management problems in pursuit of optimal solutions that cannot readily be obtained by common sense. (operations research)

derivative

A financial instrument whose value "derives" from the value of an underlying asset.

digraph

A graph in which each edge has an arrow indi- cating the direction of the edge. Such directed edges are appropriate when the relationship is "one-sided" rather than symmetric (for instance, one-way streets as opposed to regular streets).

complete graph

A graph in which every pair of vertices is joined by an edge.

connected graph

A graph is connected if it is possible to reach any vertex from any specified starting vertex by traversing edges.

code

A group of symbols that represent information, together with a set of rules for interpreting the symbols.

edge

A link joining two vertices in a graph.

graph

A mathematical structure in which points (called vertices) are used to represent things of interest and in which links (called edges) are used to connect vertices, denoting that the connected vertices have a certain relationship.

fundamental principle of counting

A method for count ing outcomes of multistage processes.

circuit

A path that starts and ends at the same vertex.

vertex

A point in a graph where one or more edges end.

inflation

A rise in prices from a set base year

sinking fund

A savings plan to accumulate a fixed sum by a particular date, usually through equal periodic deposits.

minimum cost spanning tree

A spanning tree of a weighted connected graph having minimum cost. The cost of a tree is the sum of the weights on the edges of the tree.

nominal rate

A stated rate of interest for a specified length of time; a nominal rate does not take into account any compounding.

stemplot

A stemplot is a display of the distribution of a variable that attaches the final digits of each observation as a leaf on a stem made up of all but the final digit.

algorithm

A step-by-step description of how to solve a problem.

spanning tree

A subgraph of a connected graph that is a tree and includes all the vertices of the original graph.

method of trees

A visual method of carrying out the fundamental principle of counting.

condorcet method

A voting system for elections with several candidates in which a candidate is a winner precisely when he or she would, on the basis of the ballots cast, defeat every other candidate in a one-on-one contest.

hare system

A voting system for elections with several candidates in which candidates are successively eliminated in an order based on the number of first-place votes.

sequential pairwise voting

A voting system for elections with several candidates in which one starts with an agenda and pits the candidates against each other in one-on-one contests (based on preference list ballots), with losers being eliminated as one moves along the agenda.

borda count

A voting system for elections with several candidates in which points are assigned to voters' prefer- ences; these points are summed for each candidate to determine a winner. The actual point totals are referred to as a candidate's Borda score

plurality voting

A voting system for elections with several candidates in which the candidate with the most first-place votes wins.

plurality runoff method

A voting system for elections with several candidates in which, assuming there are no ties, there is a runoff between the two candidates receiving the most first-place votes.

majority rule

A voting system for elections with two candidates (and an odd number of voters) in which the candidate preferred by more than half the voters is the winner

pareto condition

A voting system satisfies the (blank) provided that every voter's ranking of one candidate higher than another precludes the possibility of this latter candidate winning.

eulerizing

Adding new edges which duplicate existing edges to a connected graph so as to make a graph that possesses an Euler circuit.

Kruskal's Algorithm

An algorithm developed by Joseph Kruskal that solves the minimum-cost spanning-tree problem by selecting edges in order of increasing cost, but in such a way that no edge forms a circuit with edges chosen earlier. It can be proved that this algorithm always produces an optimal solution.

nearest neighbor algorithm

An algorithm for attempting to solve the TSP that begins at a "home" vertex and visits next that vertex not already visited that can be reached most cheaply. When all other vertices have been visited, the tour returns to home. This method may not give an optimal answer.

sorted edges algorithm

An algorithm for attempting to solve the TSP where the edges added to the circuit being built up are selected in order of increasing cost, but no edge is chosen that would prevent a Hamiltonian circuit from forming. These edges must all be connected at the end, but not necessarily at earlier stages. The tour obtained may not have the lowest possible cost.

greedy algorithm

An approach for solving an optimization problem, where at each stage of the algorithm the best (or cheapest) action is taken. Unfortunately, greedy algorithms do not always lead to optimal solutions.

codabar scheme

An error-detecting method used by credit cards, libraries, blood banks, the Canadian Social Insurance program, and others.

international standard book number (ISBN)

An identification number used on books throughout the world that contains a check digit for error detection.

agenda

An ordering of the candidates to be considered, which is often used in sequential pairwise voting.

exponential decay

Geometric growth at a negative rate.

arithmetic growth

Growth by a constant amount in each time period.

geometric growth

Growth proportional to the amount present.

compound interest

Interest that is paid on both the original principal and the accumulated interest.

simple interest

Interest that is paid on the original principal only, not on any accumulated interest.

arrow's impossibility theorem

Kenneth J. Arrow's discovery that any voting system can give undesirable outcomes.

may's theorem

Kenneth May's discovery that, for two alternatives and an odd number of voters, majority rule is the only voting system satisfying three natural properties.

five number summary

Minimum Q1 M Q3 Maximum

interest

Money earned on a savings account or a loan.

weights

Numbers used in the calculation of check digits.

continuous compounding

Payment of interest in an amount toward which compound interest tends with more and more frequent compounding.

current dollars

The actual cost of an item at a point in time; inflation or deflation before or since then has not been taken into account.

annual rate of inflation

The additional proportionate cost of goods one year later.

e

The base for continuous compounding, geometric (expo- nential) growth, and natural logarithms; e=2.71828

annual percentage yield (APY)

The effective interest rate per year.

compound interest formula

The formula for the amount in an account that pays compound interest periodically. For an initial principal (P) and an effective rate (i) per compounding period, the amount after (n) compounding periods is A=P (1+i ) ^n

compounding period

The fundamental interval on which compounding is based, within which no compounding is done. Also called simply period .

critical path

The longest path in an order-requirement digraph. The length of this path gives the earliest comple- tion time for all the tasks making up the job consisting of the tasks in the digraph

brute force method

The method that solves the traveling salesman problem (TSP) by enumerating all the Hamiltonian circuits and then selecting the one with minimum cost.

mode

The mode is the most frequently occurring value in a set of numerical observations. It is possible for a dataset to have no mode, one mode, or more than one mode.

valence (of a vertex)

The number of edges touching that vertex.

condorcet voting paradox

The observation that there are elections in which Condorcet's method yields no winner.

Chinese postman problem

The problem of finding a circuit on a graph that covers every edge of the graph at least once and that has the shortest possible length.

Traveling Salesman Problem

The problem of finding a minimum-cost Hamiltonian circuit in a complete graph where each edge has been assigned a cost (or weight).

effective rate

The rate of simple interest that would realize exactly as much interest over the same period of time.

present value

The value today of an amount to be paid or received at a specific time in the future, as determined from a given interest rate and compounding period.

optimal solution

When a problem has various solutions that can be ranked in preference order (perhaps according to some numerical measure of "goodness"), the optimal solution is the best-ranking solution.

tree

a connected graph with no circuits