Instant Notes: Analytical Chemistry

Note that Nin the denominator is replaced by N-1, which is known as the

number of degrees of freedomand is defined as the number of independent

deviations(xi−x

_

) used to calculate s. For single sets of data, this is always one

less than the number in the set because when N- 1 deviations are known the last

one can be deduced as, taking sign into account, 

i=N

i= 1

(xi−x

_

) must be zero (see

Example 1below).

In summary, the calculation of an estimated standard deviation, s, for a small

number of values involves the following steps:

● calculation of an experimental mean;

● calculation of the deviations of individual xivalues from the mean;

● squaring the deviations and summing them;

● dividing by the number of degrees of freedom, N-1,and

● taking the square root of the result.

Note that if Nwere used in the denominator, the calculated value of swould be

an underestimate of s.

Estimated standard deviations are easily obtained using a calculator that

incorporates statistical function keys or with one of the many computer soft-

ware packages. It is, however, useful to be able to perform a stepwise arithmetic

calculation, and an example using the set of five replicate titers by analyst A

(Fig. 1) is shown below.

Example 1

xi/cm^3 (xi−x)(xi−x)^2

20.16 -0.04 1.6 × 10 -^3

20.22 +0.02 4 × 10 -^4

20.18 -0.02 4 × 10 -^4

20.20 0.00 0

20.24 +0.04 1.6 × 10 -^3

 101.00 4 ¥ 10 -^3

x

_

20.20

s
^4 ×

4

10



− 3


=0.032 cm^3

The relative standard deviation,RSDor sr,is also known as the coefficient of

variation, CV. It is a measure of relative precisionand is normally expressed as

a percentage of the mean value or result

sr(s/x

_

) × (^100) (3)
It is an example of arelative error(Topic B1) and is particularly useful for
comparisons between sets of data of differing magnitude or units, and in calcu-
lating accumulated (propagated) errors. The RSDfor the data in Example 1is
given below.
Example 2
sr
^0
2

0

0

.

3

2

2

0

× 100 =0.16%

Relative
standard
deviation

30 Section B – Assessment of data