(c) The normal distribution is described by its mean and standard 

 deviation. Since the distribution of sand size is approximately lognormal, 

 then individual sand size distributions can be more easily described by 

 units based on the logarithm of the diameter rather than the absolute diam- 

 eter. Comparison with the theoretical lognormal distribution is also a 

 convenient way of characterizing and comparing the size distribution of 

 different samples. 



Of these three advantages, only (a) is unique to the phi units. The 

 other two, (b) and (c) , would be valid for any unit based on the logarithm 

 of size. 



Disadvantages of phi units are: 



(a) Phi units increase as absolute size in millimeters decreases. 



(b) Physical appreciation of the size involved is easier when the 

 units are millimeters rather than phi units. 



(c) The median diameter can be easily obtained without phi units. 



(d) Phi units are dimensionless, and are not usable in physically 

 related quantities where grain size must have units of length such as 

 grain-size, Reynolds number, or relative roughness. 



Size distributions of samples of littoral materials vary widely. 

 Qualitatively, the size distribution of a sample may be characterized by 

 a diameter that is in some way typical of the sample, and by the way that 

 the sizes coarser and finer than the typical size are distributed. (Note 

 that size distributions are generally based on weight, rather than number 

 of particles.) 



A size distribution is described qualitatively as well-sorted if all 

 particles have sizes that are close to the typical size. If all the par- 

 ticles have exactly the same size, then the sample is perfeatly sorted. 

 If the particle sizes are distributed evenly over a wide range of sizes, 

 then the sample is said to be well-graded. A well-graded sample is poorly 

 sorted; a well-sorted sample is poorly graded. 



The median (Mj) and the mean (M) define typical sizes of a sample of 

 littoral materials. The median size, N^ in millimeters, is the most com- 

 mon measure of sand size in engineering reports. It may be defined as 



M^ = d,„ (4-2) 



where dso is the size in millimeters that divides the sample so that half 

 the sample, by weight, has particles coarser than the dso size. An equiv- 

 alent definition holds for the median of the phi size distribution, using 

 the symbol M^,), instead of M^f. 



Several formulas have been proposed to compute an approximate mean 

 (M) from the cumulative size distribution of the sample. (Otto, 1939; 



4-15 



