Global Positioning System Reference
In-Depth Information
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measurements have been made and that they have been grouped into bins of length
∆
x
, with bin
i
containing
n
i
observations. Graphing the bins in a coordinate system of
relative frequency
n
i
/n
versus
x
i
gives the histogram. For surveying measurements,
the smoothed step function of the rectangular bins typically has a bell-like shape. The
maximum occurs around the sample mean. The larger the deviation from the mean,
the smaller the relative frequency, i.e., the probability that such a measurement will
actually be obtained. A goodness-of-fit test would normally confirm the hypothe-
sis that the observations have a normal distribution. Thus, the typical measurement
process in surveying follows the statistical law of normal distribution.
4.
1.2 Elementary Statistical Concepts
Several concepts from statistics are required in least-squares adjustment. The follow-
ing is a partial listing of frequently used concepts and terminology:
[94
Observation: An observation, or a statistical event, is the outcome of a statistical
experiment, e.g., throwing a dice or measuring an angle or a distance.
•
Lin
—
4.1
——
Cus
PgE
Random Variable: A random variable is the outcome of an event. The random
variable is denoted by a tilde. Thus,
•
x
is a random variable and
x
is a vector of
random variables. However, we will frequently not use the tilde to simplify the
notation when it is unambiguous which symbol represents the random variable.
˜
Population: The population is the totality of all events. It includes all possible
values that the random variable can have. The population is described by a finite
set of parameters, called the population parameters. The normal distribution,
e.g., describes such a population and is completely specified by the mean and
the variance.
•
[94
Sample: A sample is a subset of the population. For example, if the same distance
is measured ten times, then these ten measurements are a sample of all the
possible measurements.
•
•
A statistic represents an estimate of the population parameters or functions of
these parameters. It is computed from a sample. For example, the ten measure-
ments of the same distance can be used to estimate the mean and the variance of
the normal distribution.
•
Probability: Probability is related to the frequency of occurrence of a specific
event. Each value of the random variable has an associated probability.
•
Probability Density Function: The probability density function relates the prob-
ability to the possible values of the random variable. If
f(x)
denotes the proba-
bility density function, then
b
P(a
≤˜
x
≤
b)
=
f(x)dx
(4.1)
a
is the probability that the random variable
x
assumes a value in the interval [
a, b
].
˜
