Beckman Coulter LS 13 320, User Manual

Page: 233 / 254

PN B05577AC C-5
Optical Models
Statistics
C
Statistics
The LS13 320 program calculates a number of statistical quantities to characterize the particle size
distribution. The statistics are calculated as they would be for a frequency distribution, with the
volume percent (or surface area percent or number percent) in a certain size channel being
analogous to the frequency of occurrence of a certain value.
Size channels in the LS 13 320 are spaced logarithmically, and are therefore progressively wider in
span toward larger sizes. Statistical calculations are made based on the logarithmic center of each
channel.
Statistics can be calculated either arithmetically or geometrically (logarithmically); that is, the
statistics are based on either the value of a channel center or the logarithm of that value. Arithmetic
statistics are more appropriate for particle size distributions that are similar to a Gaussian, or
normal distribution. Geometric statistics are more appropriate for particle size distributions that
are closer to log-normal. A log-normal distribution is symmetrical when plotted on a logarithmic
size scale.
The statistical formulas used by the LS 13 320 software, and a brief description of each, are
presented in the next sections.
User-Defined Mean
There are two general formulas for the user-defined mean diameter D(p,q), one for arithmetic
means and one for geometric means. These functions allow you to view one type of weighted
distribution while obtaining a mean normally displayed only with another weighting. For example,
the surface-area weighted arithmetic mean diameter can be available when you are viewing a
volume weighted distribution. You can also define weightings other than the three standard ones.
The definitions are:
for p = q (arithmetic mean diameter), and
for p = q (geometric mean diameter),
If D(p,q) is chosen in the Size Statistics Preferences dialog box, and integer values for p and q are
entered in the provided fields, D(p,q) will appear with other statistics when
Analyze
,
Statistics
is
chosen.
Where:
xi = the channel center
ni = the percentage of particles in the i'th channel
D p q ( , )
n
i
x
i
p
∑
n
i
x
i
q
∑
--------------------
1
p q – ( )
-----------------
=