Page 162 - Vogel's TEXTBOOK OF QUANTITATIVE CHEMICAL ANALYSIS

4 ERRORS AND STATlSTlCS

normal arithmetic functions, a suitable calculator for statistical work should
enable the user to evaluate the mean and standard deviation (Section 4.8), linear
regression and correlation coefficient (Section 4.16). The results obtained by
the use of the calculator must be carefully scrutinised to ascertain the
number of significant figures to be retained, and should always be checked
against a 'rough' arithmetical calculation to ensure there are no gross
computational errors. Microcomputers are used for processing large amounts
of data. Although computer programming is outside the scope of this book it
should be pointed out that standard programs now exist in BASIC, and other
high-level computer languages (see Bibliography, Section 5.7).
The microcomputer may also be interfaced with most types of electronic
equipment used in the laboratory. This facilitates the collection and processing
of the data, which may be stored on floppy or hard discs for later use.
There is a large amount of commercial software available for performing the
statistical calculations described later in this chapter, and for more advanced
statistical tests beyond the scope of this text.

4.8 MEAN AND STANDARD DEVlATlON
When a quantity is measured with the greatest exactness of which the instrument,
method, and observer are capable, it is found that the results of successive
determinations differ among themselves to a greater or lesser extent. The average
value is accepted as the most probable. This may not always be the true value.
In some cases the difference may be small, in others it may be large; the reliability
of the result depends upon the magnitude of this difference. It is therefore of
interest to enquire briefly into the factors which affect and control the
trustworthiness of chemical analysis.
The absolute error of a determination is the difference between the observed
or measured value and the true value of the quantity measured. It is a measure
of the accuracy of the measurement.
The relative error is the absolute error divided by the true value; it is usually
expressed in terms of percentage or in parts per thousand. The true or absolute
value of a quantity cannot be established experimentally, so that the observed
result must be compared with the most probable value. With pure substances
the quantity will ultimately depend upon the relative atomic mass of the
constituent elements. Determinations of the relative atomic mass have been
made with the utmost care, and the accuracy obtained usually far exceeds that
attained in ordinary quantitative analysis; the analyst must accordingly accept
their reliability. With natural or industrial products, we must accept provisionally
the results obtained by analysts of repute using carefully tested methods. If
several analysts determine the same constituent in the same sample by different
methods, the most probable value, which is usually the average, can be deduced
from their results. In both cases, the establishment of the most probable value
involves the application of statistical methods and the concept of precision.
In analytical chemistry one of the most common statistical terms employed
is the standard deviation of a population of observations. This is also called
the root mean square deviation as it is the square root of the mean of the sum
of the squares of the differences between the values and the mean of those values
(this is expressed mathematically below) and is of particular value in connection
with the normal distribution.