standard deviation
. For example, a factory may
produce light bulbs packaged in large shipping cartons,
100 per carton. Even though the lifespan of individual
light bulbs may vary wildly following no recognizable
distribution of values, the central limit theorem asserts
that the average lifespan of the bulbs per carton is given
by a normal distribution. Since the height of an individual is the aggregate effect of the growth rate of a large
number of individual cells, the distribution of heights of
men and woman is essentially normal, as is the distribution of heights of most anything that grows—cats, maple
trees, or carrots, for example.
Another version of the central-limit theorem is useful when trying to ascertain what proportion p percent
of the entire population possesses a certain property
(such as “Has blood type AB” or “Will vote Republican next November”). As it is impossible to examine
every individual on the globe, or poll every individual
in the nation, one can examine a sample of individuals
and compute the percentage in this sample with the
desired property. The central-limit theorem also asserts:
If many different samples of N individuals are
examined, then the distribution of the percentages of those samples possessing a particular
property very closely follows a normal distribution (and the larger the value of N, the better is the approximation to a normal curve).
This distribution has mean p, the true percentage of the population with this property, and
standard deviation
.
Both versions of the central-limit theorem allow statisticians to make inferences and predictions based on statistical data.
See also GEORGE PÓLYA; STATISTICS: INFERENTIAL.
Ceva’s theorem Let P, Q, and R be, respectively,
points on sides BC, CA, and AB of a triangle ABC.
(One is permitted to extend one or more sides of the
triangle.) Then the lines connecting P to A, Q to B, and
R to C are CONCURRENT if, and only if:
Here BP, for instance, represents the distance between
points B and P, and the ratio
is considered positive
if the direction from B to P is the same as the direction from P to C, and negative if they are in opposite
directions.
This result is due to Italian mathematician Giovanni
Ceva (1647–1734). It is equivalent to the statement that
the operation of finding the CENTER OF GRAVITY of point
masses is ASSOCIATIVE. For instance, given three masses
at locations A, B, and C in a plane, one could locate the
center of mass of the entire system by first computing
the center of mass of just two points and then compute
the center of mass of that result with the third point.
Ceva proved that the same final result is produced no
matter which two points are chosen initially.
Ceva’s theorem can be proved mathematically by
making repeated use of MENELAUS’S THEOREM.
chain rule (function of a function rule) If y = f(u) is
a function of a quantity u, which in turn is a function
of another quantity x, u = g(x) say, then y itself can be
thought of as a function of x as a COMPOSITION of
functions: y = f(g(x)). The chain rule states that the rate
of change of y with respect to the quantity x is given by
the formula:
This can also be written: (f(g(x)))′ = f′(g(x))·g′(x). For
example, to differentiate y = (x
2 + 2)
100 , we can write
y = u
100
, where u = x
2 + 2 and so:
The chain rule can be proved by making use of the formal definition of a derivative as a LIMIT:
d
dx
f g x
f g x h
f g x
h
f g x h
f g x
g x h g x
g x h g x
h
f g x g x
h
h
( ( )) lim
( (
)) ( ( ))
lim
( (
)) ( ( ))
(
) ( )
(
) ( )
( ( ) ( )
=
+
−
=
+
−
+ −
⋅
+ −
= ′
⋅ ′
→
→
0
0
x x
=
+
(
)
2
9 9
200
1
dy
dx
dy
du
du
dx
d
du
u
d
dx
x
u
x
=
⋅
=
⋅
+ =
⋅
(
)
(
)
100
2
99
2 100
2
dy
dx
dy
du
du
dx
=
⋅
BP
PC
BP
PC
CQ
QA
AR
RB
⋅
⋅
=1
σ =
−
p
p
N
(
)
100
σ
N
70 Ceva’s theorem
. For example, a factory may
produce light bulbs packaged in large shipping cartons,
100 per carton. Even though the lifespan of individual
light bulbs may vary wildly following no recognizable
distribution of values, the central limit theorem asserts
that the average lifespan of the bulbs per carton is given
by a normal distribution. Since the height of an individual is the aggregate effect of the growth rate of a large
number of individual cells, the distribution of heights of
men and woman is essentially normal, as is the distribution of heights of most anything that grows—cats, maple
trees, or carrots, for example.
Another version of the central-limit theorem is useful when trying to ascertain what proportion p percent
of the entire population possesses a certain property
(such as “Has blood type AB” or “Will vote Republican next November”). As it is impossible to examine
every individual on the globe, or poll every individual
in the nation, one can examine a sample of individuals
and compute the percentage in this sample with the
desired property. The central-limit theorem also asserts:
If many different samples of N individuals are
examined, then the distribution of the percentages of those samples possessing a particular
property very closely follows a normal distribution (and the larger the value of N, the better is the approximation to a normal curve).
This distribution has mean p, the true percentage of the population with this property, and
standard deviation
.
Both versions of the central-limit theorem allow statisticians to make inferences and predictions based on statistical data.
See also GEORGE PÓLYA; STATISTICS: INFERENTIAL.
Ceva’s theorem Let P, Q, and R be, respectively,
points on sides BC, CA, and AB of a triangle ABC.
(One is permitted to extend one or more sides of the
triangle.) Then the lines connecting P to A, Q to B, and
R to C are CONCURRENT if, and only if:
Here BP, for instance, represents the distance between
points B and P, and the ratio
is considered positive
if the direction from B to P is the same as the direction from P to C, and negative if they are in opposite
directions.
This result is due to Italian mathematician Giovanni
Ceva (1647–1734). It is equivalent to the statement that
the operation of finding the CENTER OF GRAVITY of point
masses is ASSOCIATIVE. For instance, given three masses
at locations A, B, and C in a plane, one could locate the
center of mass of the entire system by first computing
the center of mass of just two points and then compute
the center of mass of that result with the third point.
Ceva proved that the same final result is produced no
matter which two points are chosen initially.
Ceva’s theorem can be proved mathematically by
making repeated use of MENELAUS’S THEOREM.
chain rule (function of a function rule) If y = f(u) is
a function of a quantity u, which in turn is a function
of another quantity x, u = g(x) say, then y itself can be
thought of as a function of x as a COMPOSITION of
functions: y = f(g(x)). The chain rule states that the rate
of change of y with respect to the quantity x is given by
the formula:
This can also be written: (f(g(x)))′ = f′(g(x))·g′(x). For
example, to differentiate y = (x
2 + 2)
100 , we can write
y = u
100
, where u = x
2 + 2 and so:
The chain rule can be proved by making use of the formal definition of a derivative as a LIMIT:
d
dx
f g x
f g x h
f g x
h
f g x h
f g x
g x h g x
g x h g x
h
f g x g x
h
h
( ( )) lim
( (
)) ( ( ))
lim
( (
)) ( ( ))
(
) ( )
(
) ( )
( ( ) ( )
=
+
−
=
+
−
+ −
⋅
+ −
= ′
⋅ ′
→
→
0
0
x x
=
+
(
)
2
9 9
200
1
dy
dx
dy
du
du
dx
d
du
u
d
dx
x
u
x
=
⋅
=
⋅
+ =
⋅
(
)
(
)
100
2
99
2 100
2
dy
dx
dy
du
du
dx
=
⋅
BP
PC
BP
PC
CQ
QA
AR
RB
⋅
⋅
=1
σ =
−
p
p
N
(
)
100
σ
N
70 Ceva’s theorem