Page 163 - Vogel's TEXTBOOK OF QUANTITATIVE CHEMICAL ANALYSIS

MEAN AND STANDARD DEVlATlON 4.8

If we consider a series of n observations arranged in ascending order of
magnitude:

the arithmetic mean (often called simply the mean) is given by:

The spread of the values is measured most efficiently by the standard deviations
defined by:
/(XI-%) . - , (x,, - x)~
S =

In this equation the denominator is (n - 1) rather than n when the number
of values is small.
The equation may also be written as:

The square of the standard deviation is called the variance. A further measure
of precision, known as the Relative Standard Deviation (R.S.D.), is given by:

S
R.S.D. = y
X
This measure is often expressed as a percentage, known as the coefficient of
variation (C.V.):
s x 100
C.V. =
X
Example 1. Analyses of a sample of iron ore gave the following percentage
values for the iron content: 7.08, 7.21, 7.12, 7.09, 7.16, 7.14, 7.07, 7.14, 7.18, 7.11.
Calculate the mean, standard deviation and coefficient of variation for the values.

Results (x) x - X (x -

Xx = 71.30 X(x - 2)' = 0.0182
Mean X 7.13 per cent