the mean (y) ■ Thus, a can be used to measure both the spread of phi 

 sizes under the distribution curve and the areas under the curve; e.g., 

 68 percent of the area under a normal curve lies between ±1 a from the 

 mean, or between the 16th and 84th percentiles of the cumulative plot. 

 These relationships can be adapted to describe sediment and one estimate 

 of phi sorting (a) commonly encountered is: 



s = $Zk ~ <t>l6 . (3) 



If an actual distribution were completely symmetrical, the mean (y) 

 would be located at the 50th percentile phi size (<f>5o) or be equal to 

 the median size (Md) . However, it is common practice to select an esti- 

 mate of the mean that is statistically more efficient than the median as 

 well as being less biased than the median for cases where the actual gsd is 

 not completely symmetrical. 



M = * 8 » I *™ . (4) 



For a symmetrical distribution, equation (4) will produce the same value 

 as the median but for an asymmetrical distribution the mean estimate, M, 

 is more reliable. Thus, S and M are probably the best estimates of a 

 and |i for describing unimodal sedimentary grain-size distributions (eqs. 

 3 and 4 ) . 



A common way to calculate these parameters, using a graphical tech- 

 nique, is shown in Figure 3. Here, the sample size data are plotted as a 

 cumulative distribution on log (phi) probability paper in such a way that 

 the phi and percent coordinates of a point on the curve indicate the per- 

 cent of the sample coarser than that particular phi size. The sizes 

 associated with the 84th and 16th percentiles may be interpreted directly 

 from the plot and used to calculate M and S. 



A sample that is lognormally distributed will appear as a straight 

 line on phi probability paper. However, most sample distributions are 

 somewhat asymmetric and their plots are not straight (Fig. 3). The degree 

 of asymmetry, or nonnormality of the observed sample distribution, can 

 then be determined by comparing this curve with a straight "approximation" 

 curve which is constructed by drawing a straight line through the 84th and 

 16th percentile intercepts of the observed curve. The comparison can 

 either be made qualitatively by noting the size of the "gap" between the 

 curves along the phi size equal to the mean, or quantitatively by comput- 

 ing an estimate of the skewness parameter. 



Sk = & ~ Md ) . (5) 



o 



In both cases, the difference between the mean and median sizes is reflected 

 by the observed asymmetry. For example, a negative skewness exists when 

 the observed distribution lengthens or tails out toward the coarser, 



13 



