preface to fifteenth edition

2.118 SECTION 2
2.3.2 Errors in Quantitative Analysis
Two broad classes of errors may be recognized. The first class, determinate or systematic errors, is
composed of errors that can be assigned to definite causes, even though the cause may not have been
located. Such errors are characterized by being unidirectional. The magnitude may be constant from
sample to sample, proportional to sample size, or variable in a more complex way. An example is
the error caused by weighing a hygroscopic sample. This error is always positive in sign; it increases
with sample size but varies depending on the time required for weighing, with humidity and temperature.
An example of a negative systematic error is that caused by solubility losses of a precipitate.
The second class, indeterminate or random errors, is brought about by the effects of uncontrolled
variables. Truly random errors are as likely to cause high as low results, and a small random error
is much more probable than a large one. By making the observation coarse enough, random errors
would cease to exist. Every observation would give the same result, but the result would be less
precise than the average of a number of finer observations with random scatter.
The precision of a result is its reproducibility; the accuracy is its nearness to the truth. A systematic
error causes a loss of accuracy, and it may or may not impair the precision depending upon
whether the error is constant or variable. Random errors cause a lowering of reproducibility, but
by making sufficient observations it is possible to overcome the scatter within limits so that the
accuracy may not necessarily be affected. Statistical treatment can properly be applied only to
random errors.
2.3.3 Representation of Sets of Data
Raw data are collected observations that have not been organized numerically. An average is a value
that is typical or representative of a set of data. Several averages can be defined, the most common
being the arithmetic mean (or briefly, the mean), the median, the mode, and the geometric mean.
The mean of a set of N numbers, x 1 , x 2 , x 3 ,...,x N , is denoted by x and is defined as:
x x x ··· x
1 2 3 N
x (2.4)
N
It is an estimation of the unknown true value of an infinite population. We can also define the
sample variance s 2 as follows:
N
i
i1
(x x) 2
2
s
N 1
(2.5)
The values of x and s 2 vary from sample set to sample set. However, as N increases, they may be
expected to become more and more stable. Their limiting values, for very large N, are numbers
characteristic of the frequency distribution, and are referred to as the population mean and the
population variance, respectively.
The median of a set of numbers arranged in order of magnitude is the middle value or the
arithmetic mean of the two middle values. The median allows inclusion of all data in a set without
undue influence from outlying values; it is preferable to the mean for small sets of data.
The mode of a set of numbers is that value which occurs with the greatest frequency (the most
common value). The mode may not exist, and even if it does exist it may not be unique. The empirical
relation that exists between the mean, the mode, and the median for unimodal frequency curves
which are moderately asymmetrical is:
Mean mode 3(mean median) (2.6)