of each entrant and then comparing the result with the
maximum sum possible.
To illustrate: in the above example, apple received
ranks of 5 and 4. The rank difference squared is (5 – 4)
2
= 1
2 = 1. Similarly, the rank difference squared for blueberry is (2 – 1)
2 = 1, and for the remaining pies, in
alphabetical order, (3 – 2)
2 = 1, (1 – 3)
2 = 4, and (4 – 5)
2
= 1. Summing all five rank differences squared yields the
total score: D
2 = 1 + 1 + 1 + 4 + 1 = 8.
With n entrants, it can be shown that the maximal
sum possible is
. It occurs when the ranks
assigned by one judge are in reverse order to the ranks
assigned by the second. (To see this, observe how the
quantity D
2 changes when the ranks of two entrants
are switched. This will show that the quantity D
2 is
largest when, for any pair of entrants, their assigned
ranks are in reverse order.) For n = 5, the maximum
sum possible is 40.
Spearman’s rank correlation coefficient is given by:
Notice that a value ρ = 1 occurs when D
2 = 0, that is,
there is perfect agreement. A value ρ = –1 occurs when
D
2 has the maximum value, i.e., perfect disagreement.
In our example, we have ρ = 1 – 2 ·
= 0.6. This value
again indicates some disagreement and an inconsistency in the two ranking schemes.
rate of change See DIFFERENTIAL CALCULUS.
ratio The relationship of one numerical quantity to
another expressed as a QUOTIENT so as to indicate their
relative sizes is called a ratio. We write a:b for the ratio
of a quantity a to a quantity b, or, alternatively, express
the ratio as a FRACTION a/b. The first term of a ratio
a:b is called the antecedent and the second term the
consequent. For example, the ratio of 10 to 4 is written
10:4 or 10/4, which can be simplified to 5/2. (Thus the
number 10 is 2 1/2 times as large as 4.) Here, 10 is the
antecedent, and 4 is the consequent.
The value of a ratio is unchanged if both terms in
the ratio are multiplied or divided by the same quantity. For example, 10:4, 30:12, and 2.5:1, are equivalent ratios. A ratio for which one of its terms is equal
to 1 is called a unitary ratio. Thus, 2.5:1, for instance,
is a unitary ratio.
If the ratio of two variables y and x is constant,
then the two variables are said to be PROPORTIONAL.
For example, if y/x has constant value k, then it must
be the case that y = kx.
In PROBABILITY theory, the ODDS of a game are
expressed as a ratio. For example, if in a game of
chance a of the possible outcomes are deemed favorable and the remaining b possible outcomes unfavorable, then the odds in favor for the game is the ratio
a:b. Thus for a game with favorable odds of 3:2, for
instance, three of the five outcomes are deemed favorable, and there is 3/5, or 60 percent, chance of winning the game.
A number that can be expressed as a ratio of two
integers is called a RATIONAL NUMBER. Not all numbers
are rational. The FUNDAMENTAL THEOREM OF ARITHMETIC establishes, for example, that √
–
2 is not rational.
The RATIO TEST is often used to determine the convergence of an infinite SERIES.
In the theory of VECTORs, the ratio theorem (also
known as the section formula) states the following:
If a point P divides a line segment AB in the
ratio a:b, then the position vector p of P is
given in terms of the position vectors a and b
of A and B, respectively, by the formula:
The ratio notation for numbers can be extended to
indicate the relative sizes of more than two quantities.
For instance, the ratio a:b:c states that the ratio of the
first quantity to the second is a:b, that the ratio of the
second quantity to the third is b:c, and that the ratio of
the first quantity to the third is a:c. For example, the
numbers 40, 80, and 200 are in the ratio 1:2:5.
See also GOLDEN RATIO.
rational function (rational expression) Any function
f(x) =
given as a RATIO of two polynomials p(x) and
q(x) is called a rational function. Such a function is
defined at all values x for which q(x) ≠ 0. For example,
is a rational function with a domain
f x
x
x
x
( ) =
−
+
−
2
2
2
3
4
p(x)
––
q(x)
p
a
b
=
+
+
a b
a b
8
–
40
ρ = − ⋅
1 2
2
D
maximum score possible
n n
(
)
2 1
3
−
rational function 439
maximum sum possible.
To illustrate: in the above example, apple received
ranks of 5 and 4. The rank difference squared is (5 – 4)
2
= 1
2 = 1. Similarly, the rank difference squared for blueberry is (2 – 1)
2 = 1, and for the remaining pies, in
alphabetical order, (3 – 2)
2 = 1, (1 – 3)
2 = 4, and (4 – 5)
2
= 1. Summing all five rank differences squared yields the
total score: D
2 = 1 + 1 + 1 + 4 + 1 = 8.
With n entrants, it can be shown that the maximal
sum possible is
. It occurs when the ranks
assigned by one judge are in reverse order to the ranks
assigned by the second. (To see this, observe how the
quantity D
2 changes when the ranks of two entrants
are switched. This will show that the quantity D
2 is
largest when, for any pair of entrants, their assigned
ranks are in reverse order.) For n = 5, the maximum
sum possible is 40.
Spearman’s rank correlation coefficient is given by:
Notice that a value ρ = 1 occurs when D
2 = 0, that is,
there is perfect agreement. A value ρ = –1 occurs when
D
2 has the maximum value, i.e., perfect disagreement.
In our example, we have ρ = 1 – 2 ·
= 0.6. This value
again indicates some disagreement and an inconsistency in the two ranking schemes.
rate of change See DIFFERENTIAL CALCULUS.
ratio The relationship of one numerical quantity to
another expressed as a QUOTIENT so as to indicate their
relative sizes is called a ratio. We write a:b for the ratio
of a quantity a to a quantity b, or, alternatively, express
the ratio as a FRACTION a/b. The first term of a ratio
a:b is called the antecedent and the second term the
consequent. For example, the ratio of 10 to 4 is written
10:4 or 10/4, which can be simplified to 5/2. (Thus the
number 10 is 2 1/2 times as large as 4.) Here, 10 is the
antecedent, and 4 is the consequent.
The value of a ratio is unchanged if both terms in
the ratio are multiplied or divided by the same quantity. For example, 10:4, 30:12, and 2.5:1, are equivalent ratios. A ratio for which one of its terms is equal
to 1 is called a unitary ratio. Thus, 2.5:1, for instance,
is a unitary ratio.
If the ratio of two variables y and x is constant,
then the two variables are said to be PROPORTIONAL.
For example, if y/x has constant value k, then it must
be the case that y = kx.
In PROBABILITY theory, the ODDS of a game are
expressed as a ratio. For example, if in a game of
chance a of the possible outcomes are deemed favorable and the remaining b possible outcomes unfavorable, then the odds in favor for the game is the ratio
a:b. Thus for a game with favorable odds of 3:2, for
instance, three of the five outcomes are deemed favorable, and there is 3/5, or 60 percent, chance of winning the game.
A number that can be expressed as a ratio of two
integers is called a RATIONAL NUMBER. Not all numbers
are rational. The FUNDAMENTAL THEOREM OF ARITHMETIC establishes, for example, that √
–
2 is not rational.
The RATIO TEST is often used to determine the convergence of an infinite SERIES.
In the theory of VECTORs, the ratio theorem (also
known as the section formula) states the following:
If a point P divides a line segment AB in the
ratio a:b, then the position vector p of P is
given in terms of the position vectors a and b
of A and B, respectively, by the formula:
The ratio notation for numbers can be extended to
indicate the relative sizes of more than two quantities.
For instance, the ratio a:b:c states that the ratio of the
first quantity to the second is a:b, that the ratio of the
second quantity to the third is b:c, and that the ratio of
the first quantity to the third is a:c. For example, the
numbers 40, 80, and 200 are in the ratio 1:2:5.
See also GOLDEN RATIO.
rational function (rational expression) Any function
f(x) =
given as a RATIO of two polynomials p(x) and
q(x) is called a rational function. Such a function is
defined at all values x for which q(x) ≠ 0. For example,
is a rational function with a domain
f x
x
x
x
( ) =
−
+
−
2
2
2
3
4
p(x)
––
q(x)
p
a
b
=
+
+
a b
a b
8
–
40
ρ = − ⋅
1 2
2
D
maximum score possible
n n
(
)
2 1
3
−
rational function 439