recorded via a histogram. The categories considered are
conveniently chosen intervals of height ranges, such as
36.1–40.0 in., 40.1–44.0 in., and so on, for example.
If the
DATA
gathered is numerical and can adopt a
continuous array of values, including fractional values
(for example, height can adopt fractional values—48
3/4 in., or 52.837 in. are possible measurements), then
one can choose narrower and narrower interval ranges
for categories. In the
LIMIT
, the histogram becomes
then the graph of a smooth curve representing the dis-
tribution of measurements over a continuous spectrum
of values. In some sense, the total area under the curve
represents the total number of measurements observed,
and the area above an interval [a,b] represents the
number of measurements that have value greater than a
and less than b. To make this more precise, it is appro-
priate to scale the distribution so that the total area
under the curve is one (that is, one draws histograms
with vertical bars of heights representing the percentage
of measurements recorded within that category—”rela-
tive frequencies”—with the total area under such a his-
togram representing 100 percent). The ideal curve
obtained in the limit is called a “probability density
function.” The area under the curve above an interval
[a,b] here represents the
PROBABILITY
that a measure-
ment taken at random falls within the range [a,b].
A numerical quantity that can adopt a continuous
array of values (such as height, weight, or temperature)
is usually called a continuous random variable. One
ascertains the distribution (probability density func-
tion) of a random variable by conducting experiments
or studies—for example by recording the heights of
1,000 8-year-old children—or from mathematical rea-
soning, making use of the
CENTRAL
-
LIMIT THEOREM
,
the
NORMAL DISTRIBUTION
, or perhaps the
BINOMIAL
DISTRIBUTION
, for example. Often the distribution of a
random, variable is unknown, and “hypothesis testing”
is used to check the validity of an assumption that a
SAMPLE
follows a particular distribution. This is part of
inferential statistics.
If a random variable has probability density func-
tion given by a formula f(x),then the area under the
curve to the left of a value xgives a new function F(x)
called the cumulative distribution function. The quan-
tity F(x) represents the probability that a measurement
taken at random has value less than or equal to x.
See also
STATISTICS
:
INFERENTIAL
.
distributive property Given a mathematical system
with two operations, such as addition and multiplica-
tion, or union and intersection, we say that one opera-
tion distributes over the second if applying the first
operation to a set of elements combined via the second
produces the same result as applying the first operation
to the individual members of the combination, and
then combining them via the second. For example, in
ordinary arithmetic, multiplication distributes over addi-
tion. We have, for instance:
3 ×(2 + 5 + 4) = 3 ×2 + 3 ×5 + 3 ×4
that is, tripling a sum of numbers produces the same
result as tripling each individual number and then sum-
ming. (In arithmetic, the distributive property corre-
sponds to the operation of
EXPANDING BRACKETS
.)
Notice that addition is not distributive over multiplica-
tion, however. For instance: 4 + (6 ×7)≠(4 + 6) ×(4 + 7).
(The first quantity equals 4 + 42 = 46, whereas the sec-
ond is 10 ×11 = 110.)
In arithmetic, the distributive property is usually
expressed as a multiplication applied to the sum of just
two terms:
a×(b+ c) = a×b+ a×c
That it applies to a sum of three or more terms follows
from applying this basic law more than once. For
instance:
a×(b+ c+ d) = a×((b+ c) + d)
= a×(b+ c) + a×d
= a×b+ a×c+ a×d
Multiplication also distributes over addition “from the
right.” We have:
distributive property 143
A distribution as a limit