Instant Notes: Analytical Chemistry

confidence interval, CI, and the upper and lower limits of the range as confi-

dence limits, CL. Confidence limits can be calculated using the standard devia-

tion, s, if it is known, or the estimated standard deviation, s, for the data. In

either case, a probability levelmust be defined, otherwise the test is of no

value.

When the standard deviation is already known from past history, the confi-

dence limits are given by the equation

CL(m) = x

_

± (5)

where z is a statistical factorrelated to the probability level required, usually

90%, 95% or 99%. The values of zfor these levels are 1.64, 1.96 and 2.58, respec-

tively, and correspond to the multiples of the standard deviation shown in

Figure 3.

Where an estimated standard deviation is to be used, sis replaced by s,

which must first be calculated from the current data. The confidence limits are

then given by the equation

CL(m) = x

_

± (6)

where zis replaced by an alternative statistical factor, t, also related to the prob-

ability level but in addition determined by the number of degrees of freedom

for the set of data, i.e. one lessthan the number of results. It should be noted

that (i) the confidence interval is inversely proportional to N, and (ii) the

higher the selected probability level, the greater the confidence interval becomes

as both zand tincrease. A probability level of 100 percent is meaningless, as the

confidence limits would then have to be ±•.

The following examples demonstrate the calculation of confidence limits

using each of the two formulae.

Example 3

The chloride content of water samples has been determined a very large number

of times using a particular method, and the standard deviation found to be

7 ppm. Further analysis of a particular sample gave experimental values of

350 ppm for a single determination, for the mean of two replicates and for the

mean of four replicates. Using equation (5), and at the 95% probability level,

z =1.96 and the confidence limits are:

1 determinations CL(m)= 350 ±= 350 ±14 ppm

2 determinations CL(m)= 350 ±= 350 ±10 ppm

4 determinations CL(m)= 350 ±= 350 ±7 ppm 7

Example 4

The same chloride analysis as in Example 3, but using a new method for which

the standard deviation was not known, gave the following replicate results,

mean and estimated standard deviation:

1.96 ×^7

 4 

1.96 ×^7

 2 

1.96 × 7

 1 

ts

N

zs

N

32 Section B – Assessment of data