classes produces a systematic and logical division of particle sizes, 

 this same property can also create some unique problems for the statis- 

 tical analysis and graphing of size data. In statistics, sample size 

 often affects analysis results; therefore, it is desirable to have a size 

 scale with class limits that can be easily halved or quartered in order 

 to provide an adequate number of experimentally determined points for 

 analytic purposes. Geometric scales can be subdivided into smaller 

 equal-sized classes but the class limits produced are often irrational 

 rather than of integer value and more difficult to handle quantita- 

 tively. An arithmetic-size scale would be easy to subdivide and could 

 be derived from an existing geometric scale through the use of an 

 appropriate logarithmic transformation. 



Graphing techniques are commonly used for comparing the grain-size 

 distributions Cgsd) of different sediment samples. Plots of cumulative 

 proportion (usually weight percent) of sediment coarser than a series of 

 size classes tend to be fairly straight and steep in the less than 

 1-millimeter class size, and then to "tail out" toward the coarser sizes. 

 The shapes of plots for different sample gsds might appear similar even 

 though there are important textural differences. If the differences 

 occur in the finer sizes, this kind of diagram tends to push these sizes 

 together rather than to accentuate them (Fig. l,a). This graphing prob- 

 lem, like the statistical problem above, could also be solved by using 

 logarithms to transform the geometric-size scale into an arithmetic 

 scale. 



2 I 0.5 



Groin Size (mm) 



-2 



12 3 

 Gram Size((/)) 



Figure 1. Cumulative size-frequency plots comparing 

 (a) millimeter and (b) phi-size scales. 



2. Phi Grade Scale . 



The phi notation, introduced by Krumbein (1934, 1958), is used to 

 transform the geometric VVentworth scale into an arithmetric scale where 



= -log„ (d(mm)/lmm) , 



(1) 



10 



