Page 16 - Statistics for Environmental Engineers

L1592_Frame_C02 Page 7 Tuesday, December 18, 2001 1:40 PM

2

A Brief Review of Statistics

KEY WORDS accuracy, average, bias, central limit effect, confidence interval, degrees of freedom,
dot diagram, error, histogram, hypothesis test, independence, mean, noise, normal distribution, param-
eter, population, precision, probability density function, random variable, sample, significance, standard
deviation, statistic, t distribution, t statistic, variance.

It is assumed that the reader has some understanding of the basic statistical concepts and computations.
Even so, it may be helpful to briefly review some notations, definitions, and basic concepts.

Population and Sample
The person who collects a specimen of river water speaks of that specimen as a sample. The chemist,
when given this specimen, says that he has a sample to analyze. When people ask, “How many samples
shall I collect?” they usually mean, “On how many specimens collected from the population shall we
make measurements?” They correctly use “sample” in the context of their discipline. The statistician
uses it in another context with a different meaning. The sample is a group of n observations actually
available. A population is a very large set of N observations (or data values) from which the sample of
n observations can be imagined to have come.

Random Variable
The term random variable is widely used in statistics but, interestingly, many statistics books do not give
a formal definition for it. A practical definition by Watts (1991) is “the value of the next observation in an
experiment.” He also said, in a plea for terminology that is more descriptive and evocative, that “A random
variable is the soul of an observation” and the converse, “An observation is the birth of a random variable.”

Experimental Errors
A guiding principle of statistics is that any quantitative result should be reported with an accompanying
estimate of its error. Replicated observations of some physical, chemical, or biological characteristic
that has the true value η will not be identical although the analyst has tried to make the experimental
conditions as identical as possible. This relation between the value η and the observed (measured) value
y i is y i = η + e i , where e i is an error or disturbance.
Error, experimental error, and noise refer to the fluctuation or discrepancy in replicate observations
from one experiment to another. In the statistical context, error does not imply fault, mistake, or blunder.
It refers to variation that is often unavoidable resulting from such factors as measurement fluctuations
due to instrument condition, sampling imperfections, variations in ambient conditions, skill of personnel,
and many other factors. Such variation always exists and, although in certain cases it may have been
minimized, it should not be ignored entirely.

© 2002 By CRC Press LLC