This key result, needed for computing the derivative of
the cosine function, can also be established geometri-
cally by making use of the squeeze rule and the dia-
gram above. (For xmeasured in degrees we also have
limx→0= 0.)
Finally, we note that limx→0= limx→0
· = 1 · = 1 (For xmeasured in degrees we have
limx→0= .)
See also
APOTHEM
;
E
;
LIMIT
.
standard deviation See
STATISTICS
:
DESCRIPTIVE
.
statistics Statistics is the branch of mathematics con-
cerned with the methods of collecting, tabulating, and
summarizing numerical facts (this is called descriptive
statistics), and for making inferences and predictions
based on these facts (inferential statistics). The numeri-
cal information gathered is called
DATA
, and an individ-
ual numerical fact about the data is called a statistic.
For example, a medical study might record the
heights of 100 children, all age 8. The average height of
the children would be an example of a statistic.
Another statistic would be the tallest height recorded
or the range of heights observed. Making a judgment
based on the data that another child outside of the
study is of abnormal height would be an example of
using data for inferential purposes.
The word statistik was coined by the German
political scientist Gottfried Achenwall (1719–72) to
mean “a summary of how things stand.” It is based on
the Latin verb stare meaning “to stand.”
Statistics is an indispensable tool used in practically
every aspect of life today. Weather predictions are based
on methods of statistical inference, for example, as are
the assessed effectiveness of new drugs, new medical
procedures, and other health practices. Statistics is used
extensively in government, business, and commerce to
analyze opinion polls, campaign and advertising strate-
gies, business operations, pollution control, and other
environmental concerns, for example, and as well as in
scientific research and economic, political, and sociolog-
ical studies. Insurance companies analyze
LIFE TABLES
to
make inferences and to set insurance rates. At the turn
of each decade, every household in the United States is
required to complete a short census questionnaire. Gov-
ernment decisions on the apportionment of representa-
tion and of funds are based on the census results. In
addition, a small percentage of households must com-
plete a longer questionnaire, from which further statisti-
cal inferences about the entire population are made. In
leisure, many sports fans follow statistical analyses to
assess team and player performance.
Because statistics pervades so many areas of life,
study of the subject is now a standard part of many
high-school curricula.
See also
HISTORY OF PROBABILITY AND STATISTICS
(essay);
POPULATION AND SAMPLE
;
STATISTICS
:
DESCRIP
-
TIVE
;
STATISTICS
:
INFERENTIAL
.
statistics: descriptive The science of collecting, tabu-
lating, and summarizing numerical information
obtained from observational or experimental studies is
called descriptive statistics. For example, a medical
study might record the blood types of 100 army
inductees and present the information obtained as lists
or tables, or perhaps visually via charts, graphs, or fre-
quency diagrams as described below. General features
of the
DATA
, such as the most common blood group
observed, or the shape of the frequency distribution
observed, can be used to describe and summarize the
information. Providing general descriptions of data
allows one to draw conclusions about a particular pop-
ulation as a whole. As a simple example, one might
extrapolate and deduce that a certain percentage of the
entire world’s population has a particular blood type.
In the example above, the measurements taken are
descriptive and fall into precise categories: type A, type
B, type AB, and type O. Such a study is said to be cate-
gorical. A numerical study, however, collects numerical
information about participants (such as height, age, or
weight), and placing the data into categories is a matter
of choice. For example, one might wish to arrange the
ages of in-patients at a busy metropolitan hospital into
categories of decade (ages 0–10 years, 11–20 years, and
so on) or some other convenient division. Measure-
ments of height, for example, can adopt a continuous
array of values, including fractional values. A study on
human growth rates might have its data organized into
categories of height ranges, for example, 60.1–65.0 in.,
65.1–70.0 in., and so on.
π
–––
180
tan x
–––––
x
1
–
1
1
–––
cos x
sin x
–––
x
tan x
–––––
x
cos x– 1
––––––
x
478 standard deviation