Statistical analysis - Concrete Mix Design, Quality Control and Specification - page 218

x m = Mean of all results
n = Number of results
Another method of determining the standard deviation is from the differ-
ence between successive results: Average difference/1.13. This method gives
the same answer as the earlier “standard method” if the data analysed is a
true normal distribution. However, there is a very useful significant differ-
ence if the data analysed is a time sequence of results having a change of
mean somewhere in the sequence. In such a situation the standard method
gives an inflated value for the standard deviation because it effectively
involves a change of the true mean of the results both before and after the
change to a new intermediate mean. We do not need to go into the math-
ematics of this (although they are quite simple); it is sufficient to realise
that it occurs and to take it into account. The difference method is almost
totally unaffected by such a change. It is particularly useful in assessing the
variability of multigrade results since it is quite easy for the computer to be
programmed to average differences from the last result in the same grade.
In this way a much more meaningful SD can be obtained from a relatively
small number of results scattered over a large number of grades.
The UK QSRMC quality control system uses the difference method since
it assumes that change points will be relatively rare and effectively restarts
the analysis after one has been experienced. Ken Day's QC system prints
out the SD from the difference method at the top of its result table display,
but then gives the SD by the standard method for each separate grade of
concrete in the table itself. Of course grades with few results are likely to
show large fluctuations in SD, but looking at grades with say 20 or more
results, a standard method SD much in excess of the difference method SD
at the top of the table usually indicates that there has been a change point
in that grade, which should be investigated. However, it could also indicate
that there are particular problems causing high variability in that grade
(also requiring investigation).
The difference method SD can also be applied to the within-sample (or
testing error) SD. Where pairs of specimens are tested, the within-sample
SD is given by Average pair difference/1.13. Where three specimens are
tested at the same age, the SD is given by Average range (difference between
highest and lowest)/1.69.
Returning now to illustrating the principles of statistics, Figure 9.2
shows three distributions with the same standard deviation but different
mean values. Figure 9.3 shows three distributions with the same mean but
different values of standard deviation.
We are interested in the percentage of results less than a certain strength
(i.e., the percentage defective). Looking again at Figure 9.1, the distance
below the mean (or above, the curve is symmetrical) can be expressed as a
parameter k (i.e., a variable number) times σ and the area as a percentage