106 coplanar

coplanar See

PLANE

.

coprime Another name for

RELATIVELY PRIME

.

correlation See

CORRELATION COEFFICIENT

;

SCATTER

DIAGRAM

.

correlation coefficient Any numerical value used to

indicate the extent to which two variables in a study are

associated is called a correlation coefficient. For exam-

ple, a medical study might record the height and shoe

size of adult participants suspecting that there might be

a relationship between these two features. If two vari-

ables are such that when one changes, then the other

does so in a related manner (generally the taller an indi-

vidual, the greater the shoe size on average, say) then

the two variables are said to be correlated. A

SCATTER

DIAGRAM

is used to detect possible correlations. If the

points of the scatter diagram tend to follow a straight

line, then the two variables are linearly correlated.

K

ARL

P

EARSON

(1857–1936) developed a measure

to specifically detect linear relationships. If the

DATA

values in a study are represented as pairs of values,

(x1,y1),…,(xN, yN), first define:

and

Here –

xis the

MEAN

of the x-values and –

ythe mean of

the y-values. The quantities Sxx and Syy are called the

VARIANCE

s and Sxy the

COVARIANCE

of the two vari-

ables. Then Pearson’s correlation coefficient, denoted

R2, is given by:

This quantity only adopts values between 0 and 1. A

value of R2= 1 indicates a perfect linear relationship

between the two variables, with the points in the asso-

ciated scatter diagram lying precisely on a straight line.

A value R2= 0 indicates that there is no relationship

between the two variables. (In particular, the covari-

ance of the two variables is zero.) All these claims can

be proved through a study of the

LEAST SQUARES

METHOD

. An R2value close to 1, say 0.9 or higher,

indicates that a linear correlation is very likely.

See also

RANK CORRELATION

;

REGRESSION

;

STATIS

-

TICS

:

DESCRIPTIVE

.

countable Any set, finite or infinite, whose elements

can be placed in a list is said to be countable. More

precisely, a set Sis countable if there is a one-to-one

correspondence between the elements of Sand a subset

of the

NATURAL NUMBERS

(that is, it is possible to

match each element of Swith a unique natural num-

ber). For example, the set {knife, fork, spoon} is count-

able because its elements can be matched with the

elements of the subset {1, 2, 3} of natural numbers: list

knife as first, fork as second, and spoon as third, for

instance. The set of all integers is countable, for its ele-

ments can be placed in the list:

0,1, – 1,2, – 2,3, – 3,…

The set of all English words that exist today and might

be of use in the future is countable: list all letters of the

alphabet (possible one-lettered words), then, in alpha-

betical order, all combinations of a pair of letters (the

two-lettered words), followed by all possible combina-

tions of three letters, and so forth, to produce a well-

defined list of all possible strings of letters.

The

DIAGONAL ARGUMENT

of the first kind shows

that the set of all

RATIONAL NUMBERS

is countable. The

diagonal argument of the second kind, however, estab-

lishes that the set of

REAL NUMBERS

is not. In a definite

sense then, the set of reals is an infinite set “larger”

than the set of rationals.

A set is called

DENUMERABLE

if it is infinite and

countable. Matters are a little confusing, however, for

some authors will interchangeably use the terms count-

able and denumerable for both finite and infinite sets.

The

CARDINALITY

of an infinite countable set is

denoted ℵ0. A countable set that is not infinite is said

to be

FINITE

.