Encyclopedia of Environmental Science and Engineering, Volume I and II

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STATISTICAL METHODS FOR ENVIRONMENTAL SCIENCE

All measurement involves error. Any field which uses empir-

ical methods must therefore be concerned about variability

in its data. Sometimes this concern may be limited to errors

of direct measurement. The physicist who wishes to deter-

mine the speed of light is looking for the best approximation

to a constant which is assumed to have a single, fixed true

value.

Far more often, however, the investigator views his data

as samples from a larger population, to which he wishes to

apply his results. The scientist who analyzes water samples

from a lake is concerned with more than the accuracy of

the tests he makes upon his samples. Equally crucial is the

extent to which these samples are representative of the lake

from which they were drawn. Problems of inference from

sampled data to some more general population are omni-

present in the environmental field.

A vast body of statistical theory and procedure has been

developed to deal with such problems. This paper will con-

centrate on the basic concepts which underlie the use of

these procedures.

DISTRIBUTIONS

Discrete Distributions

A fundamental concept in statistical analysis is the probabil-

ity of an event. For any actual observation situation (or exper-

iment) there are several possible observations or outcomes.

The set of all possible outcomes is the sample space. Some

outcomes may occur more often than others. The relative

frequency of a given outcome is its probability; a suitable set

of probabilities associated with the points in a sample space

yield a probability measure. A function x, defined over a

sample space with a probability measure, is called a random

variable, and its distribution will be described by the prob-

ability measure.

Many discrete probability distributions have been stud-

ied. Perhaps the more familiar of these is the binomial dis-

tribution. In this case there are only two possible events; for

example, heads and tails in coin flipping. The probability

of obtaining x of one of the events in a series of n trials is

described for the binomial distribution by where u is the

probability of obtaining the selected event on a given trial.

The binomial probability distribution is shown graphically

in Figure 1 for u = 0.5, n = 20.

fx n

n

x

(;, )uuu
⎛ xnx( ) ,

⎝⎜

⎞

⎠⎟

1

(1)

It often happens that we are less concerned with the prob-

ability of an event than in the probability of an event and

all less probable events. In this case, a useful function is the

cumulative distribution which, as its name implies gives for

any value of the random variable, the probability for that

and all lesser values of the random variable. The cumulative

distribution for the binomial distribution is

Fx n f x n

i

x

(;, )uu
 (;, ).


 0

∑

(2)

It is shown graphically in Figure 2 for u = 0.5, n = 20.

An important concept associated with the distribution

is that of the moment. The moments of a distribution are

defined as

mkik i

i

n


 xfx

()

1

∑

(3)

NUMBER OF X

5 10 15 20

0

.05

.10

.15

.20

f(X)

FIGURE 1

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