Page 164 - Vogel's TEXTBOOK OF QUANTITATIVE CHEMICAL ANALYSIS

4 ERRORS AND STATlSTlCS

= k0.045 per cent

0.045 x 100
C. V. = = 0.63 per cent
7.13
The mean of several readings (X) will make a more reliable estimate of the
true mean (p) than is given by one observation. The greater the number of
measurements (n), the closer will the sample average approach the true mean.
The standard error of the mean s, is given by:

In the above example,

and if 100 measurements were made,

Hence the precision of a measurement may be improved by increasing the
number of measurements.

4.9 DISTRIBUTION OF RANDOM ERRORS
In the previous section (4.8) it has been shown that the spread of a series of
results obtained from a given set of measurements can be ascertained from the
value of the standard deviation. However, this term gives no indication as to
the manner in which the results are distributed.
If a large number of replicate readings, at least 50, are taken of a continuous
variable, e.g. a titrimetric end-point, the results attained will usually be
distributed about the mean in a roughly symmetrical manner. The mathematical
mode1 that best satisfies such a distribution of random errors is called the
Normal (or Gaussian) distribution. This is a bell-shaped curve that is symmetrical
about the mean as shown in Fig. 4.1.
The curve satisfies the equation:
1 -(x-,,)~

It is important to know that the Greek letters o and p refer to the standard
deviation and mean respectively of a total population, whilst the Roman letters
s and X are used for samples of populations, irrespective of the values of the
population mean and the population standard deviation.
With this type of distribution about 68 per cent of al1 values will fa11 within