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14:7:40 Page 125
measured variable tends to follow. In turn, the standard distribution is used to interpret the data. Of
course, the list in Table 4.2 is not inclusive, and the reader is referred to more complete treatments of this
subject (3–5).
Regardless of its probability density form, a variable that shows a central tendency can be
described and quantified through its mean value and variance. In the absence of systematic errors,
the true mean value or central tendency of a random variable, x(t), which is continuous in either
time or space, and having a probability density function p(x), is given by
x
0
¼ lim
T!1
1
T
Z T
0
xðtÞdt
ð4:4aÞ
which for any continuous random variable x is equivalent to
x
0
¼
Z 1
À1
xpðxÞdx
ð4:4bÞ
If the measured variable is described by discrete data, the mean value of measured variable, x i ,
where i ¼ 1, 2, . . . , N, is given by
x
0
¼ lim
N!1
1
N
X N
i¼1
x i
ð4:5Þ
Physically, the width of the density function reflects the data variation. For a continuous random
variable, the true variance is given by
s
2
¼ lim
T!1
1
T
Z T
0
xðtÞ À x
0
½
2 dt
ð4:6aÞ
which is equivalent to
s
2
¼
Z 1
À1
x À x
0
ð
Þ
2 pðxÞdx
ð4:6bÞ
or for discrete data, the variance is given by
s
2
¼ lim
N!1
1
N
X N
i¼1
x i À x
0
ð
Þ
2
ð4:7Þ
The standard deviation, s, a commonly used statistical parameter, is defined as the square root of
the variance, that is s ¼
ffiffiffiffiffi
s 2
p
.
The fundamental difficulty in using Equations 4.3 to 4.7 is that we assume a knowledge of the
values of the entire population of the variable. But what if the data-set represents only a finite portion
of that population? Do these relations change? Real data-set sizes may range from a single value to a
large finite number. The next section discusses the behavior of infinite data sets (entire populations)
and introduces the connection between probability and statistics. After that, we will turn our
attention to the practical statistical treatment of finite data sets.
4.3 DESCRIBING THE BEHAVIOR OF A POPULATION
This section discusses the relation between probability and statistics. To do this, we assume a
particular distribution for p(x) to characterize the behavior of the population of x. Table 4.2 lists a
few of the many distributions we could use, but to develop our relations we will use the normal
4.3 Describing the Behavior of a Population 125
14:7:40 Page 125
measured variable tends to follow. In turn, the standard distribution is used to interpret the data. Of
course, the list in Table 4.2 is not inclusive, and the reader is referred to more complete treatments of this
subject (3–5).
Regardless of its probability density form, a variable that shows a central tendency can be
described and quantified through its mean value and variance. In the absence of systematic errors,
the true mean value or central tendency of a random variable, x(t), which is continuous in either
time or space, and having a probability density function p(x), is given by
x
0
¼ lim
T!1
1
T
Z T
0
xðtÞdt
ð4:4aÞ
which for any continuous random variable x is equivalent to
x
0
¼
Z 1
À1
xpðxÞdx
ð4:4bÞ
If the measured variable is described by discrete data, the mean value of measured variable, x i ,
where i ¼ 1, 2, . . . , N, is given by
x
0
¼ lim
N!1
1
N
X N
i¼1
x i
ð4:5Þ
Physically, the width of the density function reflects the data variation. For a continuous random
variable, the true variance is given by
s
2
¼ lim
T!1
1
T
Z T
0
xðtÞ À x
0
½
2 dt
ð4:6aÞ
which is equivalent to
s
2
¼
Z 1
À1
x À x
0
ð
Þ
2 pðxÞdx
ð4:6bÞ
or for discrete data, the variance is given by
s
2
¼ lim
N!1
1
N
X N
i¼1
x i À x
0
ð
Þ
2
ð4:7Þ
The standard deviation, s, a commonly used statistical parameter, is defined as the square root of
the variance, that is s ¼
ffiffiffiffiffi
s 2
p
.
The fundamental difficulty in using Equations 4.3 to 4.7 is that we assume a knowledge of the
values of the entire population of the variable. But what if the data-set represents only a finite portion
of that population? Do these relations change? Real data-set sizes may range from a single value to a
large finite number. The next section discusses the behavior of infinite data sets (entire populations)
and introduces the connection between probability and statistics. After that, we will turn our
attention to the practical statistical treatment of finite data sets.
4.3 DESCRIBING THE BEHAVIOR OF A POPULATION
This section discusses the relation between probability and statistics. To do this, we assume a
particular distribution for p(x) to characterize the behavior of the population of x. Table 4.2 lists a
few of the many distributions we could use, but to develop our relations we will use the normal
4.3 Describing the Behavior of a Population 125