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

 a sample of littoral materials. The median size Mj , in millimeters, is the 

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



\ = So (^-2) 



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

 the sample, by weight, has particles coarser than the dcQ size. An equiva- 

 lent definition holds for the median of the phi-size distribution, using the 

 symbol M,<t) instead of Mj . 



Several formulas have been proposed to compute an approximate mean (M) 

 from the cumulative size distribution of the sample (Otto, 1939; Inman, 1952; 

 Folk and Ward, 1957; McCammon, 1962). These formulas are averages of 2, 3, 5, 

 or more symmetrically selected percentiles of the phi frequency distribution, 

 such as the formula of Folk and Ward. 



^ ' ^16 ^ ^50 + '"84 ,, ,, 



M = (4-3) 



9 J 



where (j) is the particle size in phi units from the distribution curve at the 

 percentiles equivalent to the subscripts 16, 50, and 84 (Fig. 4-8); ^ is the 

 size in phi units that is exceeded by x percent (by dry veight) of the total 

 sample. These definitions of percentile (after Griffiths, 1967, p. 105) are 

 known as graphic measures. A more complex method — the method of moments — can 

 yield more precise results when properly used. 



To a good approximation, the median V\^ is interchangeable with the 

 mean (M) for most beach sediment. Since the median is easier to determine, 

 it is widely used in engineering studies. For example, in one CERC study of 

 465 sand samples from three New Jersey beaches, the mean computed by the 

 method of moments averaged only 0.01 millimeter smaller than the median for 

 sands whose average median was 0.30 millimeter (1.74 phi) (Ramsey and Galvin, 

 1971). 



Since the actual size distributions are such that the log of the size is 

 approximately normally distributed, the approximate distribution can be 

 described (in phi units) by the two parameters that describe a normal dis- 

 tribution — the mean and the standard deviation. In addition to these two 

 parameters, skewness and kurtosis describe how far the actual size distri- 

 bution of the sample departs from this theoretical lognormal distribution. 



Standard deviation is a measure of the degree to which the sample spreads 

 out around the mean (i.e., its sorting) and can be approximated using Inman' s 

 (1952) definition by 



'"84 ~ '"16 ,, ,. 



a, = X (4-4) 



4-15 



