civil

1. See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/284239308
Grain-Size Statistics I: Evaluation of the Folk and Ward Graphic Measures
Article in Journal of Sedimentary Research · January 1978
DOI: 10.1306/212F7595-2B24-11D7-8648000102C1865D
CITATIONS
37
READS
3,934
3 authors, including:
Some of the authors of this publication are also working on these related projects:
Climate change and glaciers View project
Cheekye Fan - Squamish Floodplain Interactions View project
John J. Clague
Simon Fraser University
536 PUBLICATIONS 19,248 CITATIONS
SEE PROFILE
All content following this page was uploaded by John J. Clague on 03 April 2018.
The user has requested enhancement of the downloaded file.

2. JOURNAL OF SEDIMENTARYPETROLOGY, VOL. 48, NO. 3, P, 863-878
FtGs. 1-8, SEPTEMaER, 1978
Copyright © [978, The Society of Economic Paleontologistsand Mineralogists
GRAIN-SIZE STATISTICS I:
EVALUATION OF THE FOLK AND WARD GRAPHIC MEASURES I
DAVIS SWAN, JOHN J. CLAGUE AND JOHN L. LUTERNAUER
Geological Survey of Canada
100 West Pender St., Vancouver
British Columbia V6B 1R8
AaSTR^CT: This study investigates the effectiveness of graphic statistical parameters as descriptors
of grain-size distributions. Grain-size distributions which cannot be described adequately using the
graphic technique are isolated by comparing the graphic parameters to moment measures calculated
for the ungrouped weight frequency data from hypothetical samples consisting of randomly generated
"grains" of known size, shape, and density.
Differences between graphic and ungrouped mean are insignificant, except for highly skewed
distributions. Ungrouped standard deviations generally are larger than their graphic counterparts: the
disparity is greatest for medium sorted samples which have long "tails" in the finest or coarsest
5% of the distribution. The respective skewness and kurtosis values are only weakly related, indicating
that the graphic measures respond erratically to significant deviations from normality in grain-size
distributions. Transformed graphic and ungrouped kurtosis values [kurtosis/(kurtosis + 1)] are more
strongly related than the corresponding nontransformed parameters. From these relationships it is
concluded that classification schemes of sediment types should make use of graphic parameters only
if the range in values of statistical parameters is large enough so that the limitations of the graphic
technique do not significantly affect the classification units.
It is also established that (a) obtaining data at intervals timer than whole phi is not justified if
graphic statistical parameters are to be used, (b) gaussian (probability) interpolations between known
points on a cumulative curve and extrapolations beyond the ends of the distribution are required
in the calculation of graphic parameters by computers (published computer programs employ linear
interpolations and extrapolations), and (c) ungrouped parameters calculated using weight and number
frequency are unrelated.
INTRODUCTION
Fundamental components of many classi-
fication schemes of sediment grain-size dis-
tributions are the statistical parameters mean,
standard deviation, skewness, and kurtosis.
Although different methods have been de-
vised to calculate these parameters, the
graphic technique of Folk and Ward (1957)
and the moment technique based upon
grouped data (for example, Friedman, 1967)
are the most widely used. Although statistical
parameters derived by the two methods are
not always comparable, the proponents of
both the graphic (e.g., Mason and Folk, 1958;
Folk and Robles, 1964) and moment (e.g.,
ChappeU, 1967) techniques claim to be able
~Manuscript received January 27, 1977; revised De-
cember 2, 1977.
to adequately differentiate sedimentary envi-
ronments. Kolclijk (1968), Davis and Ehrhch
(1970), and lsphording (1972) compared the
values derived by the two techniques and
attributed differences to the insensitivity of
the graphic measures (see also the regression
analyses of Sevon, 1968). However, Folk
(1966), Jones (1970), and Jaquet and Vernet
(1976) found that the moment measures can
themselves be significantly in error, even for
normally distributed samples.
Two possible sources of error in grain-size
statistical parameters are (a) grouping of data
into size classes (Kendall and Stuart, 1969,
p. 75) and truncating distributions (these
apply to both the grouped moment and
graphic techniques), and (b) application of
graphic statistical measures, which are de-
fined in terms of a normal distribution (Folk,
1974), to distributions which are non-normal.

3. 864 DA V1S SWAN, JOHN J. CLAGUE AND JOHN L. LUTERNA UER
These errors can be avoided if moment
calculations are made on a sample for which
the physical characteristics of each individual
grain are known. Ungrouped moment and
graphic parameters can then be compared
for both normal and non-normal samples to
isolate those types of distributions which
cannot be adequately described by the
graphic measures.
The present study develops a procedure
for digitally generating distributions of dis-
crete grains in order to identify the limitations
of the formulae defining graphic grain-size
statistical parameters. In the first part of this
report the relationship between conventional
and gram-size statistics is discussed and the
procedure for generating theoretical distribu-
tions of grain size summarized. The effects
on graphic measures of various possible
methods of interpolation between known
data points on a cumulative curve and ex-
trapolation at the ends of a distribution then
are assessed. Finally, Folk and Ward graphic
statistical parameters are evaluated m rela-
tion to moment measures calculated from
ungrouped frequency data.
CALCULATING UNGROUPED MOMENT
STATISTICAL PARAMETERS
Conventional vs. Grain-size Statistics
There are two important differences be-
tween conventional statistics and statistics
applied to grain-size distributions. The first
concerns the difference between weight fre-
quency and number frequency. Normally the
frequency distribution of a discrete random
variable x is described in terms of the mean
(Ix) of the distribution and the moments (tXr)
about that mean, where
1 N (1)
~, = S', (x, - Ix)'
N being the total number of objects for which
a parameter x is defined. If some of the
x values are not unique, eq. (1) can be
modified to produce moments for populations
in which only N' values are unique. That
is, ifx~ = x2 = ..- =x.,then (x~-it)r+
(x2 - Ix)' + ... + (x n - Ix)' = n(x, - ~)r.
By introducing the relative number frequency
n
f = --, eq: (1) becomes
N
N"
IX = E Xifi
1
N"
O~r "~ E (Xi -- I'L)r fi
I
,2>
Eq. (l) and (2) both produce parameters
which are easily interpreted in terms of the
variable x (i.e., ~ is the "central" value of
x values, a2 is a measure of the "dispersion"
of x around ~, etc.).
However, grain-size distributions are con-
ventionally plotted in terms of weight fre-
quency rather than number frequency, and
consequently the parameters p~and a r in eq.
(2) are calculated using fractional weight
frequencies (unit grain weight/total sample
weight), rather than number frequencies. It
is important to note that when ~ and a r are
determined from weight frequencies they can
no longer be interpreted in terms of individual
grains. For example, the mean diameter
calculated using weight frequency is not the
actual mean size of all the particles in a
sample. Rather it is the size around which
the weight of material in a sample is distrib-
uted. To illustrate the difference, the mean
size of four spherical particles can be calcu-
lated using weight and number frequency
(Table 1). It is apparent that the size calculat-
ed using weight frequency is neither the true
mean size of the particles, nor the size Of
the particle having the mean weight of grains
in the sample.
The geological significance of the discrep-
ancy between weight and number frequency
has long been recognized (Krumbein and
Pettijohn, 1938, p. 225-227). Physical inter-
pretation of weight frequency parameters is
difficult, if not impossible, because they do
not characterize any clear defined geological
population. The need to clearly identify such
populations has been stressed by Krumbein
and Graybill (1965, Chapter 4) who called
for "a sharper definition of what the concep-
tual populations being described actually
are" (p. 127). Clearly the population of

4. EVALUATION OF GRAPHIC STATISTICAL MEASURES 865
TABLE l,--Calculation of raean size and weight for Jour
spherical particles
Generating Theoretical Distributions of
Grain Sizes
Size Diameter Masst
(~) (ram) (rn~) f w
0,5 0.7071 0.490568 0.25 0.8752
1.5 0.3536 0.061321 0.25 0.1094
2.5 0.1768 0.007665 0.25 0,0137
3.5 0.0884 0.000958 0.25 0.0017
Mean Unit Type of Mean
2.0000 6
0.3315 mm
0.6419 6
0.1650 mm
0.1401 nag
I. 1026
0.4657 mm
number frequency mean size
number frequency mean size
weight frequency mean size
weight frequency mean size
number frequency mean weight
size of particle having mean weight
size of particle having mean weight
tP = 2.65 gcm 3
objects described by weight frequency sta-
tistical parameters is not that of the individual
particles in a sample. No population of
numbers representing measures of some
attribute of discrete particles (such as grain
sizes or grain weights) can be associated with
weight frequency statistical parameters. Be-
cause sedimentation occurs particle by parti-
cle (and, in fact, the hydrometer and pipette
techniques of size analysis are based upon
settling velocities of discrete particles),
weight frequency statistics may not provide
the best parameters with which to charac-
terize a sediment. For example, Griffiths
(1967, p. 66-69; 1969, p. 95) suggested that
grain-size statistical calculations be based on
measurements of individual grams. An al-
ternative approach would require the cal-
culation of statistical parameters using an
estimated fractional number frequency
(Swan et al., 1976).
The second difference between conven-
tional statistics and statistics applied to
•grain-size distributions relates to the fact that
size is described by a transformed variable
dprather than diameter. Although parameters
calculated from transformed and diameter
data can be simply related for normally
distributed samples (Sahu, 1965a), such is
not the case for non-normal samples. Be-
cause the use of weight frequency makes
physical interpretation of derived statistical
parameters difficult anyway, the use of the
phi scale can perhaps be justified for the
sake of convenience, although not on strictly
mathematical grounds.
In order to eliminate inaccuracies intro-
duced by grouping data prior to calculation
of statistical parameters, it is necessary to
have measurements for each particle in a
sample. Consequently, the authors first con-
sidered thin section point counts as a possible
standard against which the graphic measures
could be evaluated. However, the many
problems associated with sampling tech-
niques and grain exposures (see review article
by KeUerhals et aL, 1975) indicated that thin
sections would not provide a rigorous stan-
dard. It was then recognized that numbers
(representing grain sizes in phi units) can
be generated digitally to produce specified
distributions. These numbers are the raw,
ungrouped data from which a reference set
of statistical measures can be calculated.
However, many reasonable number fre-
quency distributions produce gram-size dis-
tributions which are not representative of
natural sediments because of the non-linear
transformation from number to weight fre-
quency. Thus it is first necessary to investi-
gate the relationships between number and
weight frequency distributions.
Relationships between Number and Weight
Frequency Distributions.--In a series of ex-
cellent articles, Sahu (1964, 1965a, 1965b,
1968) discussed the relationships between
log-normal distributions based upon number
and weight frequency. He theoretically veri-
fied the empirical results of Hatch (1933)
who found that only the mean of a perfect
log-normal distribution is changed when
transforming from weight to number fre-
quency (Fig. 1). The standard deviation,
skewness, and kurtosis are not affected by
the transformation. However, this simple
relationship is contingent on the fact that
the distributions are defined throughout their
particle-size ranges, rather than being
"open-ended". Frequencies for grain sizes
at any distance from either mean (it, or ~x~)
can be calculated because the density func-
tions for the distributions are known. In
practice it is difficult to define the tails of
a distribution consisting of a finite number
of discrete grains accurately enough to pro-
duce a reasonable weight frequency distribu-
tion. However, if the same range of phi values

5. 866 DA VIS SWAN, JOHN J. CLA GUE AND JOHN L. LUTERNA UER
0.5"
Z 0.3-
tu
0
tu 0.2-
A /~
//
ol- / //
o.o 1/ /
/ ,
. ~
4:o 5~.o
~ ~o i
GRAIN SIZE {PHI UNITS)
Flo. l.--Relationshipbetween weight(A) and number
(B) frequency for a normal sample (1~ = 3.0, ~w =
1.0) using the density functions of Sahu (1965b).
is considered for corresponding weight and
number frequency distributions, one finds
that the significant portion of the former is
defined by a portion of the coarse "tail"
of the latter (Fig. l). This tail has an approxi-
mately exponential form with negative
skewness. Consequently, a reasonable
weight frequency distribution can be pro-
duced by generating phi sizes and frequencies
corresponding to the coarse tail of a number
frequency distribution. The range of data will
then be the same for both.
Producing Specified Distributions.--The
unit weight of spherical grains decreases
exponentially on the phi scale (Table 1) so
that, in order to produce even a uniform
weight frequency, the number of grains must
increase by an order of magnitude in each
consecutive whole phi class. For example,
a phi-normal sample having measurable
weight percentages in each of the phi classes
- 1to 6 would require the generation of about
106 grains. This problem was resolved by
applying a weighting function [n (qb)] to each
grain, as described below.
Sahu (1965b) has shown that the density
function for a phi-normal number frequency
distribution is
1
f = -- ln(qb) g (6)1 (3)
N
6WlO 3
where n(~) - - - exp [3 (:n2)6]
p'rr
and g (~) =
1 [,
crwX/2~r exp 2
and N and W are the total number and weight
of particles in the sample, respectively. The
function n(6) compensates for the exponen-
tial decrease in the unit weight of spherical
particles on the phi scale, and hence produces
a uniform distribution of weight frequency.
The normal probability function g('b) trans-
forms the uniform distribution to a normal
one.
In the present study g(qb) was approxi-
mated by generating standardized normal
distributions (mean = 0.0, standard deviation
= 1.0) on an IBM 370/168 computer using
Marsaglia's Rectangle-Wedge-Tail method
(Marsaglia and Bray, 1968; Knuth, 1969).
Each distribution, representing a sample of
1000random numbers (z, or normal deviates),
then was transformed into non-normal form
using a technique outlined by Kendall and
Stuart (1969) by specifying skewness and
kurtosis. A random number generator was
used to produce values within the limits
-3 < Skw < 3 and0 < Kw < 20. These limits
were chosen on the basis of values found
for approximately 500 sediment samples from
a variety of marine environments off the west
coast of Canada (and are similar to the
published values of Jaquet and Vernet, 1976,
for fine sediments). Setting ./3 = SK~,/4 =
Kw-3, and all :s having subscripts greater
than four in Kendall and Stuart's eq. 6.56
to zero, the non-normal deviates z' corre-
sponding to each of the normal deviates were
calculated from the equation
z'=z+:3( z2-1)+ ~
6 7;(z 3 - 3z)
:~ (2z 3 - 5z) - 6:,
- 3--6- ~ (z' - 5z2 + 2)
:33
+ -- (12Z4 -- 53Z2 + 17)
324
:4
384
---(3z 5 - 24z3 + 29z)

6. EVA L UA TION OF GRA PHIC STA TISTICA L MEA S URES 867
+ - (14ze - 103z3 + 107z)
288
- -- (252z s - 1688z3 + 1511z) (4)
7776
The deviates calculated in eq. (4) then were
transformed into non-standard form (phi
sizes) using eq. (5).
= CrwZ' + ~w (5)
The means and standard deviations for the
desired distributions were randomly generat-
ed using the limits -3 < ~w < 8 and 0 < cr
< 5. Grain sizes outside the limits -6 < ~b <
14 were discarded as not being representa-
tive of real systems.
The number of grains of each phi size was
then calculated using the weighting function
n(~b), and the fractional number frequency
(f) of eq. (3) was obtained by dividing by
the total number of particles in the sample.
Finally, ungrouped statistics were calcu-
lated for each of the generated distributions
using the equations
IOOO
~n = E fi~bi
I
IOOO
E -
Crn = fi(qbi ~)2
I
10OO
Sk. = E fi(~i- ~°?/~'~n
(6)
I
IO00
K, = '~",f,(,b,- " "
I
Distributions having unreasonable values for
these statistical parameters were discarded.
Despite certain theoretical limitations on
the above procedure (Kendall and Stuart,
1969, p. 162), the authors were able to gener-
ate distributions having specified charac-
teristics. As all the parameters for the dis-
tributions were generated randomly, there
is no operator bias in the distributions pro-
duced.
Because the graphic parameters of Folk
and Ward are determined from cumulative
weight frequencies, a weight was calculated
for each grain size assuming spherical parti-
cles with density p (see Hunter, 1967).
p'rr 4 3
m = -- exp [-3(fn2)+] = -- 'rrr p
6 3
The fractional weight frequency (w) of parti-
cles having size 6 is the product of the
number of particles and the unit weight,
divided by the total sample weight. 2
n (~) m
w -- - - (7)
W
The particles in each hypothetical sample
were digitally "sieved" on the basis of size
at 0.25-phi intervals, and the number and
weight of particles in each size class were
accumulated.
CALCULATING FOLK AND WARD STATISTICAL
PARAMETERS
Several factors affect the values derived
from graphic statistical measures. Of these,
the methods of interpolation between known
points and extrapolation at the ends of the
distribution are most important.
Interpolations
Graphic estimates of grain-size parameters
are based upon certain percentiles which are
interpolated from cumulative curves plotted
on probability paper by j oining known points
with straight lines. As the divisions on proba-
bility paper are based upon the integral of
the normal curve (gaussian function), a nu-
merical computation of graphic parameters
must incorporate a function
l ¢_1/2zz
gauss(z) = -~- -~ dz (8)
where z is the normal deviate (z score) of
the distribution. In plotting known points of
a cumulative curve on probability paper, it
is assumed that these points lie on a gaussian
curve such as that defined by eq. (8); that
is, the cumulative weight frequency percent
Zln fact, w is identical for every 6-size in a given
sample, because n(~) represents a uniform weight fre-
quency distribution.

7. 868 DA VIS SWAN, JOHN J. CLA GUE AND JOHN L. LUTERNA UER
(c) is related to the normal deviate by the
equation
c = 100gauss(z) (9)
It follows from eq. (9) that the z score
corresponding to a given cumulative fre-
quency is
Consider a size class interval having lower
and upper bounds 6~ and '52, respectively,
with corresponding cumulative weight fre-
quency percents c~ and c2 (Fig. 2). Assume
that * is the value corresponding to some
percentile c required for the calculation of
a graphic statistical parameter. By defimtion
Z -- - -
O"
Assuming a phi-normal distribution
Z 2 -- Z t Z -- Z t
Therefore
1~ "~ '1 + (*2 -- *1) -- -- (11)
--Z 1
Substituting from eq. (10) into eq. (11) and
generalizing to the ith size fraction, where
Ci_l (C(C i
, = ~_. + (,~ - ,,_,).
i c (c,
L gauss 1 (l~O) i (Ci 1~/ (12)
gauss 100 ] _J
From eq. (12) percentile phi values, and
hence graphic statistical parameters compa-
rable to those found using probability graph
paper, can be calculated. Computation of the
inverse gauss function (gauss t) has been
discussed at length by Strecok (1968). A
FORTRAN function subroutine based upon
a University of Chicago routine (Kuki, 1966)
was used to compute the inverse gauss func-
tion and is available from the authors upon
request.
'°'i A
z !
/
o0 ,~ ,e ~a ,o ,o o~ ,e
GRAIN SIZE [PHIUNITS)
~°i // ,
j
lo ,'o ~,
Fro. 2.--Cumulative curves for two typical normal
distributions demonstratingthe differencebetweenlin-
ear interpolations(dottedlines)and gaussianinterpola-
tions (solidlines).
Extrapolations
If neither the coarsest nor finest size frac-
tion contains more than 5% by weight of
a sample, then the graphic parameters can
be readily calculated without the necessity
of extrapolating beyond known data points.
However, in a gravelly sand a few very large
grains in the coarsest fraction may constitute
more than 5% of the sample weight (this
problem sometimes can be avoided by sam-
piing a large volume of material), whereas
in a mud sample less than 95 weight percent
may be analysed because of the length of
time required for fine particles to settle out
of suspension. In any event, if either the
coarsest or finest fraction contains more than
5 weight percent of the sample, some form
of extrapolation must be made in order to
calculate graphic grain-size statistical param-
eters.
Folk (1966) has suggested that a linear
extrapolation be made from the last known
percentile at 10, to 100% at 14,, allotting
equal weight percentages to each of the four
classes between 10~ and 14d~. Jaquet and
Vernet (1976) distributed unanalysed material
logarithmically into size fractions beyond the
finest analysed fraction until the remainder
was less than 5%. After examining the cu-
mulative curves for a large number of marine
sediment samples, the present authors decid-
ed to use a gaussian function to extrapolate
to 0.01% at one size fraction coarser than
the coarsest analysed, and to 99.99% at 14*.
For most samples this procedure produces
reasonable results. However, it must be
noted that this, like all other extrapolation
techniques, is totally arbitrary and does not
take into account the character of individual

8. EVA L UA TION OF GRA PHIC STA T1STICA L MEA SURES 869
samples. No extrapolation technique can be
considered a substitute for an adequate size
analysis or proper sampling procedures.
Because there are no practical limitations
on the digital sieving procedure used in this
study, few samples contain more than 5%
by weight in the first or last size fractions.
Comparison of Graphic Parameters
Calculated Using Linear and Gaussian
Interpolations
The graphic parameters of Folk and Ward
(1957) were calculated from percentiles
computed using gaussian interpolations, as
described above. Similar parameters were
also calculated using percentiles interpolated
linearly between known data points. Al-
though this procedure is not that advocated
by Folk and Ward, it is commonly used in
computer programs to calculate graphic mea-
sures (for example, see Koldijk, 1968; Is-
phording, 1970).
If the percentiles calculated using linear
interpolations between known points are
denoted by primed values, and those calcu-
lated using probability interpolations are
denoted by non-primed values, some general
comments on the differences resulting from
A95 ~ A 5 > A84 ~- AI6 > A75 ~ A25
A95 > A 5, A84 > Ai6, and Av5 > A25
A75 = A25 ) A95 = A 5
the two techniques can be made (Fig. 2).
Note that on a probability scale a linear
interpolation represents a curved line of equal
weight percent per phi unit.
~; = +, - A,
+',6 = +,, - a,6
+':5 = +25 - a2~
+'50 = .5o
~'s4 = d~84+ A84
(13)
75
FIG. 3.--Cumulative curves (phi scale) after Folk
(1974, p. 52), having dominant characteristics A) Skw
< 0.0, B) Sk~ = 0.0, C) Skw > 0.0, D) K~ > 3.0
(leptokurtic), E) Kw < 3.0 (platykurtic), and F) K W
> 3.0 and Skw > 0.0. Dotted lines represent frequency
curves; solid lines are cumulative curves.
where all the A's are positive.
By comparing A and B in Figure 2, it can
be seen that the errors increase with
steepness of the slope of the cumulative
curve. In other words, for phi-normal curves
the errors in individual percentiles increase
with decreasing standard deviation. This, in
turn, implies some general relationships be-
tween the errors for each interpolated per-
centile and the various types of grain-size
distributions.
I for phi-normal or symmetrical
distributions (Figs. 2 and 3B)
for negatively skewed distribu-
tions (Fig. 3A); the reverse for
positively skewed distributions
(Fig. 3C, F)
for symmetrical leptokurtic dis-
tributions (Fig. 3D)
(14)
The error for a given percentile also
depends on where the interpolated percentile
falls in the phi interval, as errors increase
towards the center of the interval. The theo-
retical maximum possible error for a given
percentile is equal to the sieving interval.
By applying eq. (13) to the formulae for
graphic measures, values calculated using
linear and probability interpolations can be
compared.
Mean. --
1
_- (+ ,~ + *,o ÷ *'s,)
3

9. 870 DA VIS SWAN, JOHN J. CLA GUE AND JOHN L. LUTERNA UER
1 1
(4,~ + ,1,5o+ +~,) + _- (a~, - a,~)
3 3
1
= ~ + 7- (a.,, -a,~) = Ix + ~,
3
From the relationships (14) it follows that
e, = 0 for symmetrical distributions. In such
cases the two interpolation techniques yield
the same values for graphic mean. For nega-
tively skewed distributions, e~ > 0 (that is,
probability mean < linear mean). For posi-
tively skewed distributions, e ~< 0 (probabil-
ity mean > linear mean).
Standard Deviation.-
'584- ~;6 '5;5- '5~ '584- '516
~' _ +
4 6.6 4
'595 -- qb5 A~,1 "t- ~16 A95 "Jr A5
+ - - + + - -
6.6 4 6.6
=(l+e 2
The sign of e2is always positive, thus the
standard deviation calculated using a linear
interpolation technique is always greater than
that using a gaussian interpolation. The dif-
ference increases as the standard deviation
decreases so that the relative error can be
significant in well sorted sediments.
Kurtosis. --
K, =
,5;, - 4;
2.44(4;, - 4;,)
49s - (55 + A~5 + A5
2.44[('5, - G,) + (a,, + a:,)l
4~, - ,55
2.44[(*75 -- '525) + (A75 + a25)1
A95 + A5
+
2"44[(475 -- '525) 4" (a75 @ a25)]
495 - G
2.44[('5,~ - +~,) + (a,5 + a~,)l
+ e3
Under most circumstances A,s and A2s are
small so that the term g3tends to make values
of linear graphic kurtosis slightly larger than
those of probability graphic kurtosis.
However, in highly leptokurtic distributions
A75 and A25 are large whereas Ags and A5
are small. Thus e3 tends to approach zero,
and the denominator of the first term in-
creases. The result is a smaller value for
K' than K for this type of distribution.
Skewness. --
*',~ + G4 - 2,L
Sk' =
2('5'a,- qbi16)
4~ + *;, - 2qb;o
+
2(+;s - 4;)
*,6 + *84 - 2450 + (A84- a,6)
=
2I(+~4 - +,6) + (a84 + a,6)]
45 + 695 - 2'550 + (a95 - As)
+
2[(+95 - +5) + (a95 + a,)l
The complexity of the expression relating
skewness calculated using the two techniques
does not lend itself to simple interpretations.
The difference between probability and linear
graphic skewness is extremely sensitive to
the characteristics of a given distribution,
and is strongly influenced by the chance
positions of the required percentiles within
their respective phi intervals.
In distributions which are poorly sorted
the error terms in the denominator become
insignificant relative to the terms '595-'55 and
'584-'5~6. This leaves error terms similar to
e i which produce differences between Sk and
Sk' of the same sign as those between Ix
and ~'. In samples which are well sorted
the terms in the denominator dominate so
that Sk' < Sk, provided that neither A~ or
&5 vanish due to proximity to a known point
on the cumulative curve. Finally, the two
terms in the expression for linear skewness
usually operate in sympathy to make Sk'
either larger or smaller than Sk, as both are
influenced by the same characteristics of a
given distribution. Because the skewness
values themselves are generally low, dif-
ferences resulting from the two techniques
can be highly significant.
Numerical deferences between graphic
statistical parameters calculated using linear
and gaussian interpolations have been deter-
mined for 100 computer generated samples

10. EVALUATION OF GRAPHIC STATISTICAL MEASURES 871
sieved at whole-phi intervals (Fig. 4). The
standard deviations of the distributions of
these differences decrease in proportion to
the sieving interval (Fig. 5), indicating that
the error in a given percentile decreases as
a smaller sieving interval is used.
EVALUATION OF FOLK AND WARD STATISTICAL
MEASURES
Relationships between Folk and Ward
Measures and Ungrouped Number
Frequency Measures
As discussed above, the ungrouped
number frequency statistical parameters pro-
duced by eq. (6) describe the actual distribu-
tion of particle sizes within a sample. Krum-
bein and Pettijohn (1938, p. 226) concluded
that "It will not be true in general, however,
that there is any necessary simple relation
between the measures defined on a weight
basis and the measures defined by number."
This statement is supported by the data from
100 hypothetical samples presented in Figure
6, showing the relationships between Folk
and Ward graphic measures and the un-
grouped size parameters. Clearly the two sets
of parameters describe different charac-
teristics of a sample.
3O
24
>. 21t
151
12}
u- 1
9~
6i
I
,1
>-
u
Z
o
kkl
t~
t.t,.
-10 -.08 -.06 -04 -.02 0.00 0.02 00a 0.06 008 0.10
PHI UNITS
-.i8 -.16 -.14 -.12 -.10 -.08 -.06 -.04 -.02 .00
PHi UNITS
3°t
C
-20 -16 -.12 -08 -.04 0.00 0.04 0.08 0.12 016 0.20
5o1 O II
'°i II
I
2°1 II
-01 0.0 0.1 0.2 0.3 0.4
Ft6.4.--Distributions of differences between graphic measures using linear and gaussian interpolations for
100 hypothetical samples. Differences were determined by subtracting linear values from gaussian values. A)
mean, B) standard deviation, C) skewness, D) kurtosis.

11. 872 DA VIS SWAN, JOHN £ CLA GUE AND JOHN L. LUTERNA UER
.08
Z .06-
o
m
1--
.04-
<
z
<
.02-
.0(3
standard deviat ion
• skewness
kurtosis
0.25 0'.5 1.0
SIEVING INTERVAL (PHI UNITS)
FIo. 5.--Relationship between sieving interval and
standard deviations of differences between linear and
gaussian graphic statistical measures for 100 hypothet-
ical samples.
Relationships between Folk and Ward
Measures and Ungrouped Weight
Frequency Measures
The graphic statistical parameters must be
compared to standard parameters based upon
the weight distribution of individual particles.
The standard ungrouped weight frequency
moment parameters are those calculated
using eq. (6), using fractional weight fre-
quency [as calculated from eq. (7)], rather
than number frequency.
The relationships between Folk and Ward
measures and the ungrouped measures, illus-
trated in Figure 7, are approximately the same
as those found by Davis and Ehrfich (1970),
Jones (1970), Isphording (1972), and Jaquet
and Vernet (1976) for grouped moment and
graphic measures. This suggests that the
grouped weight frequency moment measures
are good estimates of the corresponding
ungrouped parameters.
The distribution of differences between
graphic and ungrouped values for each of
the statistical parameters is shown in Figure
8. It is interesting to note that the standard
deviations of these distributions increase
sfightly as the sieving interval is decreased,
indicating that graphic measures for quarter-
phi data are no more accurate than those
for whole-phi data. This is because the per-
centiles used to calculate graphic measures
are sensitive to minor inflections in cumula-
tive curves, which are more pronounced m
data obtained at quarter-phi intervals.
Mean.--For most samples the graphic
mean closely approximates the ungrouped
measure (Fig. 7A, Table 2). As the mean
is a first-order measure, the central portion
of the grain-size distribution is the dominant
determinant of its value. The graphic measure
"samples" this portion of the distribution
at three points (616, ~b~o, and ~bs4) which
proves to be adequate for most distributions.
Exceptions include samples which are highly
skewed with significant tails occurring above
~s4 or below qbl6.
Standard Deviation.--The differences be-
tween graphic and ungrouped standard de-
viation (Fig. 7B) are perhaps best visualized
by recognizing that the contribution of each
grain size to the ungrouped value of crw is
the product of weight frequency and the
square of the difference between the phi size
and the mean size [eq. (6)]. Although fre-
quencies, in general, decrease away from the
mean, this tendency is counterbalanced by
increases in differences between the mean
and the individual particle sizes. In effect,
there is a delicate balance between these two
factors, and consequently the ungrouped
standard deviation is sensitive to the exis-
tence of extreme grain sizes, even though
TABLE2.--Linear regression resultsof graphic statisticalparameters(dependent variable) against corresponding ungrouped
parameters (independent variable)for a suite of 100 hypothetical samples
VariableCorrelated
Regression
Parameter P-w ~w Skw Kw K~'~
Slope 1.050 I. 114 O.177 0.061 0.880
Intercept -0.239 -0.,567 -0.018 0.966 -0.168
Correlation Coefficient 0.998 0.990 0.670 0.298 0.715
K.
tTransformed kurtosis, K~* -
K~+I

12. EVALUATION OF GRAPHIC STA TIST1CAL MEASURES 873
9.0 ,
A
7.0-
Z
<
~ 5.0-
U lO-
ft.
< 1.0-
ix
O
-1.0-
-30
0.0
i I I I
• • e•
• "~
)e •
• ••edl ~
:1
• |
' ' 8'.0 ' ' '
' 4.0 !2.0
UNGROUPED MEAN
16.0
Z
o
I--
<
LU
8.0 I
B
Z0-
6.0-
5.0-
4.0-
Z 3.0:
~') 2.0=
U
I 1.o-
Q.
< 0.0 "
O o.o
ffl
• e •
e• o •
• •j,qr •
• •• #e
d4 ' o'8 ' 1'.2 ' 2.0
| i
1.6
UNGROUPED STANDARD DEVIATION
1.0
IAJ
Z 0.5-
~ 0.0-
U
• -o.5-
o
- 1.0
-4.0
C
• e i ~ )
• . ~11 "~
• eo mb • •
to • ~ •oe
eo
-3:o -2'.o -1% o'.o 1.o
UNGROUPED SKEWNESS
i
o
U
<
o
2.0
D
1.5-
e
1.0- •
0.5-
0.0
0.0
I ' I
#
" ......
• | °)• I,~ ,
Q
g
go 16o 1~o 2oo
UNGROUPED KURTOSIS
FtG.&--Relationships between Folk and Ward graphic measures and ungrouped number frequencysize
parameters for 100hypotheticalsamples;A)mean, B)standarddeviation,C) skewness, D)kurtosis.
they may occur with very low frequencies.
In contrast, the graphic standard deviation
ignores sediment coarser than 45 and finer
than 695.
Samples with low ungrouped standard de-
viations (< 1.56) have normal or near normal
particle-size distributions and no "tails," and
consequently the graphic and ungrouped pa-
rameters are approximately equal. However,
for ungrouped standard deviations above
about 1.54 there is a large discrepancy (up
to 0.54) between graphic and ungrouped
parameters. The difference is greatest for
samples which are well sorted except for
a free or coarse tail representing less than
5% of the sample weight. This material is
ignored by the graphic calculation so that,
in general, the ungrouped standard deviation
is larger than its graphic counterpart. As the
material in the tails of the distribution begins
to exceed 5%, the graphic measure responds,
so that the deviations between graphic and
ungrouped standard deviation tend to be less.
This accounts for the tendency of points lying
below a one-to-one line in Figure 7B to
converge on this line between standard de-
viations of 1 and 4.
Points located above a one-to-one line in
Figure 7B represent poorly sorted bimodal
or multimodal distributions. Because the
central portion of a multimodal sample is
very poorly sorted (for example, see Fig.
3E), and because this central portion is
emphasized in the graphic calculation, the
graphic standard deviation exceeds the un-
grouped value.

13. 874 DA VIS S WA N, JOHN J. CLA G UE A ND JOHN L. L UTERNA UER
8.0
6.0-
z
ell 4.0-
U 2.0-
o.o-
0 - 2.0-
!
A
I I I I i. e
e
#"
11,
- 4.0
-4.o-2'.o 0'.0 2b 4'.0 6'.0
UNGROUPED MEAN
8.0
Z
O
im
>
I.LI
t~
t~
Z
u
1
i
<
o
7.O
6.0"
5.0"
4.0-,,
3.0-
2.0-
1.0"
0.0
0.0
!
B
I ! I t
°
e.
4'
e•
.4
iu
i
Q,t
11o 2b 3'.o 4'.o 5'.o 6'.o 7o
UNGROUPED STANDARD DEVIATION
0.8-
u~ O.6-
Z o.4-
~ 0.2-
0.0-
u_
I
o. -0.2-
~ -0.4-
0
I I I I I I
C
0
°
• . .~'t: "."
• •
O •
-0.6
' i ' ' ' .'o
- 2.0 - 1.5 -I 0 -0.5 0.0 0.5 I 1.5
UNGROUPED SKEWNESS
5.5-
5.0-
¢¢J
m 4.5-
O 4.0-
I--
3.5-
3.0-
U 2.5-
2.0-
1.5-
1.0-
0.5
0.0
|
D
i
.!.:. •
dllt,,l, g°'• "• • .•
2'.0 4'.0 6'.0 8'.0 io'.o 12.0
UNGROUPED KURTOSIS
Ftc;. 7.--Relationships between Folk and Ward graphic measures and ungrouped weight frequency size parameters
for 100 hypothetical samples; A) mean, B) standard deviation, C) skewness, D) kurtosis.
Skewness and Kurtosis. --Graphic and un-
grouped measures of skewness and kurtosis
are not comparable for many samples (Fig.
7C, D, Table 2). An inherent difference
between graphic and ungrouped kurtosis
arises because the former is defined such
that a normal curve has a value of 1, whereas
the corresponding ungrouped value is 3.
Alternatively, the ungrouped kurtosis can be
defined such that a normal curve has a value
of 0 (Kendall and Stuart, 1969, p. 85). Such
a definition can result in negative kurtosis
values (for example, see Thomas et aL, 1972).
Low ungrouped values of either skewness
or kurtosis indicate high standard deviation
(poor sorting) and / or a lack of tails extending
over a large particle-size range [eq. (6)]. For
samples of this type graphic measures yield
approximately the same values as the un-
grouped measures, but only because the tails,
to which skewness and kurtosis are especially
sensitive, are absent. In other words, the
graphic and ungrouped measures are only
comparable for samples which approach
normality. However, skewness and kurtosis
determine the degree of non-normality of a
sample (Agterberg, 1974, p. 168), and hence
the graphic measures would seem to be rather
inefficient descriptors of the weight fre-
quency distributions of individual samples.
High values of skewness and kurtosis re-
quire both a low standard deviation and a
large particle-size range in the tails of the
distribution. This second condition, requiring
large deviations from the mean in at least
a few sizes, tends to increase the standard

14. EVALUATION OF GRAPHIC STATISTICAL MEASURES 875
20, A
16.
Z 127
o
tl.
45 -035"025-015-005 005 0.13 025 0.35 0.45 055
PHi UNITS PHI UNITS
25~ C
2o!
3°1 O
r
8 -I 4 -10 -06 -0 2 02 06 10 14 1.8 2 2 0.0 10 2.0 30 40 50 60 70 80 90 100 I10
FIG. 8.--Distributions of differences between Folk and Ward graphic measures and ungrouped weight frequency
size parameters for 100 hypothetical samples. Differences were determined by subtracting graphic values from
ungrouped values. A) mean, B) standard deviation, C) skewness, D) kurtosis.
deviation, so that only samples that have
well sorted central portions and long tails
with low frequencies produce high values
of skewness and kurtosis. In highly leptokur-
tic samples the sorting of the central portion
must be extremely good to compensate for
the very long tails that must be present. The
most extreme values input to the graphic
calculations are ~5 and Cbgs,and consequently
large size ranges outside these limits are
ignored. The result is that as samples become
more skewed and/or leptokurtic the rela-
tionships between the graphic and ungrouped
parameters break down (Fig. 7C, D).
From Figure 7 and the above discussion,
it is obvious that graphic skewness and
kurtosis respond only erratically to signifi-
cant deviations from normality in grain-size
distributions. This affects the usefulness of
these measures in two important ways:
(1) It may be impossible to differentiate
two samples which have significantly dif-
ferent grain-size distributions using graphic
statistical parameters.
(2) The distributions of skewness and
kurtosis for the samples in a suite may be
altered significantly if graphic values are
used. This is shown in Table 3 which summa-
rizes the distributions of the various graphic
and ungrouped statistical parameters ob-
tained from a suite of 100 digitally generated
samples. The statistical parameters describ-
ing the distributions of graphic skewness and
kurtosis differ radically from those of un-
grouped skewness and kurtosis. In contrast,

15. 876 DAVIS SWAN, JOHN £ CLA GUE AND JOHN L. LUTERNA UER
TASTE 3.--Statistical parameters derived from distributions of mean, standard deviation, skewness, and knrtosis calculated from
a suite of 100 hypothetical samples
Ungrouped Graphic
Sample Suite
Parameter tt~ ~r~ Sk~ Kw K *~'~ tt~ aw Skw Kw K ~,t
Mean 3.309 3.527 0.187 4.736 0.769 3.236 3.413 0.015 1.254 0.509
Standard Deviation 2.547 1.506 0.897 4.494 0.102 2.680 1.696 0.237 0.916 0.126
Skewness -0.423 0.127 0.295 4.271 -0.150 -0.394 0.381 0.318 2.273 0.902
Kurtosis 2.297 2.331 2.580 25.934 1.659 2.215 2.140 3.559 8.449 2.820
K~
tTransformed kurtosis, K ~ = - -
K=+I
the distributions of graphic mean and stan-
dard deviation approximate those of the
corresponding ungrouped measures. Note
that by transforming both graphic and un-
grouped kurtosis by the equation
K
K* _ m
K+I
the relationship between the two parameters
is significantly improved, although it is still
much weaker than that between graphic and
ungrouped mean and between graphic and
ungrouped standard deviation (Table 2).
CONCLUSIONS AND RECOMMENDATIONS
Because ungrouped number frequency
statistical parameters are unrelated to corre-
sponding weight frequency parameters, the
following conclusions are based upon the
relationships between graphic and ungrouped
weight frequency measures.
(1) Differences between graphic and un-
grouped means are small and can be ignored
for most sample types. However, the graphic
measure should be used with discretion for
samples which are highly skewed with signif-
icant tails occurring above des4or below ~b,6.
(2) The relationship between graphic and
ungrouped standard deviation is not as strong
as that between the means. Deviations are
largest for medium sorted samples where
there are significant tails in the finest or
coarsest 5% of the sample. The ungrouped
standard deviation generally is larger than
the graphic measure, although the reverse
commonly is true for multimodal samples.
(3) Relationships between the two
skewness and two kurtosis measures are
poorly defined, in large part because the
ungrouped measures are sensitive to long tails
containing low weight percents, whereas the
corresponding graphic measures ignore such
tails. Transformed graphic and ungrouped
kurtosis are more strongly related than the
corresponding nontransformed parameters.
(4) In calculating graphic statistical pa-
rameters, linear interpolations between
known points and linear extrapolations at the
ends of particle-size distributions are unwar-
ranted. Gaussian interpolations and extrapo-
lations should be used to maintain consis-
tency with techniques involving the use of
probability graph paper.
(5) Graphic measures are not very sensi-
tive to significant deviations from normality
in grain-size distributions. Classification
schemes of sediment types should make use
of graphic parameters only if the range in
values of statistical parameters is sufficiently
large such that the limitations of the graphic
technique do not significantly affect the
classification units.
(6) If graphic measures are to be used,
samples should be sieved at whole-phi inter-
vals. Obtaining data at finer intervals does
not improve the accuracy of graphic statisti-
cal measures and, in fact, tends to make
them slightly less accurate.
ACKNOWLEDGMENTS
The authors gratefully acknowledge J. Sy-
vitski, a graduate student in the Department
of Geological Sciences at the University of
British Columbia, who originally suggested
generating individual grains to produce fre-
quency distributions, and M. Greig of the
University of British Columbia Computing
Center who was instrumental in working
through the details of the technique. We also
thank the following individuals for comments
on early versions of the manuscript: F. P.

16. EVALUATION OF GRAPHIC STATISTICAL MEASURES 877
Agterberg, M. Church, M, W. Davis, R. L.
Folk, M. Greig, J. C. Griffiths, J. M. Jaquet,
and T. A. Jones.
REFERENCES
AOTERBERG,F. P., 1974, Geomathematics: Elsevier Sci-
entific Publishing Co., Amsterdam, 596 p.
CnAPPELL,J., !967, Recognizing fossil strand lines from
grain-size analysis: Jour. Sed. Petrology, v. 37, p.
157-165.
D^vJs, M. W., AND R. EHRLJCH, 1970, Relationships
between measures of sediment-size-frequency dis-
tfibutions and the nature of sediments: Geol. Soc.
America Bull., v. 81, p. 3537-3548.
FOLK, R. L., 1966, A review of grain-size parameters:
Sedimentology, v. 6, p. 73-93.
, 1974, Petrology of sedimentary rocks: Hemphill
Publishing Co., Austin, Texas, 182 p.
, ANDR. ROeLES, 1964, Carbonate sands of Isla
Perez, Alacran reef complex, Yucat~in: Jour. Geol.,
v. 72, p. 255-292.
- - , ANDW. C. WARD, 1957, Brazos River Bar--a
study in the significance of grain size parameters:
Jour. Seal. Petrology, v. 27, p. 3-26.
FRmOMAN,G. M., 1967, Dynamic processes and statisti-
cal parameters compared for size frequency distribu-
tion of beach and river sands: Jour. Sed. Petrology,
v. 37, p. 327-354.
GRIrFIT8S, J. C., 1967, Scientific method in analysis
of sediments: McGraw-Hill Book Co., New York,
508 p.
, ANDC. W. ONDRtCK,1969, Modelling the petrol-
ogy of detrital sediments: In Merriam, D. F. (ed.),
Computer applications in the earth sciences, Plenum
Press, New York, p. 73-97.
HATCH, T., 1933, Determination of "average particle
size" from the screen-analysis of non-uniform par-
ticulate substances: Jour. Franklin Inst., v. 215, p.
27-37.
HUNTER, R. E., 1967, A rapid method for determining
weight percentages of unsieved heavy minerals: Jour.
Sed. Petrology, v. 37, p. 521-529.
ISPHOROtNG, W. C., 1970, FORTRAN IV program for
calculation of measures of central tendency and dis-
persion on IBM 360 computer: Jour. Geol., v. 78,
p. 626-628.
- - , 1972, Analysis of variance applied to measures
of central tendency and dispersion in sediments: Jour.
Sed. Petrology, v. 42, p. 107-121.
JAQUET,J. -M., ANDJ. -P. VERNET, 1976, Moment and
graphic size parameters in sediments of Lake Geneva
(Switzerland): Jour. Sed. Petrology, v. 46, p. 305-312.
.JONES, T. A., 1970, Comparison of the descriptors of
sediment grain-size distributions: Jour. Sed. Petrol-
ogy, v. 40, p. 1204-1215.
KELLERHALS,R., J. SHAW,AND V. K. ARORA,1975, On
grain size from thin sections: Jour. Geol., v. 83, p.
79-96.
KENDALL, M. G., ANDA. STUART,1969, The advanced
theory of statistics. Volume l--distribution theory
[3rd edition] : Hafner Publishing Co., New York, 439
p.
KNUTH,D. E., 1969, The art of computer programming.
Volume 2--seminumerical algorithms: Addison-
Wesley, Reading, Massachusetts, 624 p.
KOLDIJK,W. S., 1968, On environment-sensitive grain-
size parameters: Sedimentology, v. 10, p. 57--69.
KRUMBEJN,W. C., ANDF. A. GRAYSILL,1965, An intro-
duction to statistical models in geology: McGraw-Hill
Book Co., New York, 475 p.
- - , ANDF. J. PETrlJOHN, 1938, Manual of sedimen-
tary petrography: Appleton-Century-Crofts, New
York, 549 p.
Kozl, H., 1966, Mathematical functions--a description
of the Center's 7094 FORTRAN II mathematical
function library: Univ. Chicago, Computation Center
Report, 255 p.
MARSAGLIA,G., ANDT. A. BRAY,1968, One-line random
number generators and their use in combinations:
Communications of the Assoc. for Computing Ma-
chinery (ACM), v. 11, p. 757-759.
MASON, C. C., AND R. L. FOLK, 1958, Differentiation
of beach, dune, and aeolian fiat environments by size
analysis, Mustang Island, Texas: Jour. Sed. Petrology,
v. 28, p. 2l 1-226.
SAHU,B. K., 1964, Transformation of weight frequency
and number frequency data in size distribution studies
of elastic sediments: Jour. Sed. Petrology, v. 34, p.
768-773.
- - , 1965a, Transformation of arithmetic and phi
size-distribution moments: Jour. Sed. Petrology, v.
35, p. 969-972.
- - , 1965b, Transformation of weight- and number-
frequencies for phi-normal size distributions: Jour.
Sed. Petrology, v. 35, p. 973-975.
- - , 1968, Thin-section size analysis and the moment
problem: Sedimentology, v. 10, p. 147-151.
SEVON,W. D., 1968, First and second degree regression
correlation of the size analysis statistical parameters
of Trask, lnman, Folk and Ward, and Friedman: Jonr.
Sed. Petrology, v. 38, p. 238-240.
STRECOK,A. J., 1968, On the calculation of the inverse
of the error function: Mathematics of Computation,
v. 22, p. 144-158.
SWAN, D., J. J. CLAGUE,AND J. L. LUTERNAUER,1976,
Some problems in the use of grain size statistics:
Geol. Surv. Canada Paper 76-1C, p. 273-275.
THOMAS,R. L., A. L. W. KEMP,ANDC. F. M. LEWIS,
1972, Distribution, composition and characteristics of
the surficial sediments of Lake Ontario: Jour. Sed.
Petrology, v. 42, p. 66-84.
APPENDIX
List of Symbols
a moment about the mean of a distribution
c cumulative weight frequency percent
A difference in percentiles calculated
using linear and probability (gaussian)
interpolations between known points on
a distribution
E error term introduced by using linear
interpolation in calculating graphic sta-
tistical parameter
f fractional number frequency

17. 878 DA VIS SWAN, JOHN J. CLA GUE AND JOHN L. LUTERNA UER
g normal probability function (defined in
text)
K kurtosis of a grain-size distribution
K* transformed kurtosis (defined in text)
f standardized moment used in generation
of hypothetical grain-size distribution
m particle mass
ix mean grain size (phi units)
n number of particles of a given size
N total number of particles in a sample
6 -log 2 diameter (mm)
p particle density (2.65 gcm 3 assumed)
~r standard deviation of a grain-size dis-
tribution
Sk skewness of a grain-size distribution
w fractional weight frequency
W total weight of sample
z normal deviate (z score) generated from
a normal population
z' non-normal deviate generated for a
specified distribution from z
w,n subscripts used to denote parameters
defined in terms of weight frequency
and number frequency, respectively
V i e w p u b l i c a t i o n s t a t s