2.3STATISTICS IN CHEMICAL ANALYSIS

2.118

SECTION 2

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 tem-

perature. 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 sys-

tematic 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.

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

is defined as:

andx

x 1 x2 x3 · · · x N

(2.4)

x

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 1 (x i

x )2

s 2

(2.5)

N 1

The values of

and

s 2 vary from sample set to sample set. However, as

N

increases, they may be

x

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)