Where replicate samples are analyzed on a number of occasions under the same

prescribed conditions, an improved estimate of the standard deviation can be

obtained by pooling the data from the individual sets. A general formula for the

pooled standard deviation, spooled, is given by the expression

spooled=





(4)

where N 1 , N 2 , N 3 ...Nkare the numbers of results in each of the ksets, and x

_

1 , x

_

2 ,

x

_

3 ,... x

_

k, are the means for each of the k sets.

The squareof the standard deviation, s^2 , or estimated standard deviation, s^2 , is

used in a number of statistical computations and tests, such as for calculating

accumulated (propagated) errors (Topic B1 and below) or when comparing the

precisions of two sets of data (Topic B3).

Overall precision Random errors accumulated within an analytical procedure contribute to the
overall precision. Where the calculated result is derived by the addition or
subtraction of the individual values, the overall precision can be found by
summing the variancesof all the measurements so as to provide an estimate of
the overall standard deviation, i.e.

soverall=



s 12 +s 2  (^2) +s 32 +...
Example
In a titrimetric procedure, the buret must be read twice, and the error associated
with each reading must be taken into account in estimating the overall preci-
sion. If the reading error has an estimated standard deviation of 0.02 cm^3 , then
the overall estimated standard deviation of the titration is given by
soverall=

0.02 (^2) +0.02 2 0.028 cm 3
Note that this is less than twice the estimated standard deviation of a single
reading. The overall standard deviation of weighing by difference is estimated
in the same way.
If the calculated result is derived from a multiplicative expression, the overall
relative precision is found by summing the squares of the relativestandard devia-
tions of all the measurements, i.e.

sr(overall)=s^2 r 1 +s^2 r 2 +s^2 r 3 +...

The true or accepted mean of a set of experimental results is generally

unknown except where a certified reference material is being checked or

analyzed for calibration purposes. In all other cases, an estimate of the accu-

racyof the experimental mean, x

_

, must be made. This can be done by defining a

range of values on either side of x

_

within which the true mean, m, is expected to

lie with a defined level of probability. This range, which ideally should be as

narrow as possible, is based on the standard deviation and is known as the

Confidence
interval

Variance



i=N 1

i= 1

(xi−x

_

1 )^2 +

i=N 2

i= 1

(xi−x

_

2 )^2 +

i=N 3

i= 1

(xi−x

_

3 )^2 +... +

i=Nk

i= 1

(xi−x

_

k)^2



i=k

i= 1

Ni=k

Pooled standard
deviation

B2 – Assessment of accuracy and precision 31