Malvern Mastersizer 2000, User Manual

Page: 85 / 154

Understanding the views Chapter 6
Mastersizer 2000 Page 6-13
Derived distribution parameters
The third point is that the analysed distribution is expressed in a set of size classes
which are optimised to match the detector geometry and optical configuration giv-
ing the best resolution. All parameters are derived from this fundamental distribu-
tion.
Distribution parameters and derived diameters are calculated from the fundamen-
tal distribution using the summation of the contributions from each size band. In
performing this calculation the representative diameter for each band is taken to be
the geometric mean of the size band limits:
This number will be slightly different to the arithmetic mean:
For example, the size band 404.21 - 492.47 microns has a geometric mean of 446.16
microns and an arithmetic mean of 448.34 microns. In most cases the difference is
small but the geometric mean is chosen in these calculations as more appropriate to
the logarithmic spacing of the fundamental size classes.
The same principle of calculation applies to the distribution statistics standard devi-
ation, skewness and kurtosis, as shown below:
standard deviation
skewness
kurtosis
d
i l –
di
d
i l –
d
i
+
2
---------------------
σ
X
i
d
i
d – ( )
2
∑
X
i
∑
---------------------------------- D K 2 + K [ , ]
2
D K 1 + K [ , ]
2
+ = =
Skew
X
i
d
i
d –
( )
3 ∑
σ
3
X
i
∑
----------------------------------
D K 3 K
,
+
[ ]
3
d d
2
3
σ
2
+
( )
–
( )
σ
3
------------------------------------------------------------------------- = =
Kurt
X
i
d
i
d –
( )
4
∑
σ
4
X
i
∑
---------------------------------- 3 –
D K 4 K , + [ ]
4
d 4 D K 3 K , + [ ]
3
3 d
3
6 d σ
2
+ + ( ) – ( )
σ
4
-------------------------------------------------------------------------------------------------------------------------- 3 – = =
MAN0384-1.0 Mastersizer 2000.book Page 13 Tuesday, March 20, 2007 8:56 AM