# Plate Theory and Extensions - The Summation of Variances > Page 67

# The Summation of Variances

Peak
dispersion can occur in any part of the chromatographic system, from the point
of injection to the detector sensing cell. The width of the eluted peak will be
the net effect of all the dispersion processes, not merely those that take place
in the column, but also those that occur in all the other parts of the mobile
phase conduit system. Thus, to determine the ultimate peak width, the
contribution from all the extra-column dispersion processes must be identified
and then added to the dispersion that occurs in the column itself. It is not
possible to sum the standard deviations of a series of random processes, but it
is possible to sum their variances. However, all the various processes must be *random*
and *non-interactive*; that is, the extent to which one dispersion process
proceeds does not affect the progress of another dispersion process.
Experimental evidence indicates that all dispersion processes that occur in a
chromatographic system are, indeed, random in nature. Assuming there are (N)
non-interacting, random dispersive processes occurring in the chromatographic
system, then any process (p) acting alone will produce a Gaussian curve having
a variance .

Hence,

where, () is the variance of the peak as sensed by the

detector.

The equation expresses the concept of the summation of variances in an algebraic form. It follows, that if the individual dispersion processes that occur in a column are identified and an expression for the magnitude of their resulting variances derived, then they can be summed to provide an expression for the total column variance. The following determination of the maximum sample volume illustrates the use of this concept.