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OCR for page 97
-- 97 --
:~lubit F
_ ~_
CARVE F~ DING PROBATE EON IN ~= BITE STUDIO
Gordon Bit teahouse
Sedimentation Division, Soil Conservation Service
Washington, D. C.
In 1931, Dryd=t discussed the real =.d;apparent accuracy in per-
centage representation of heavy mineral freo~uencies, Curares Cores present-
ed from which the probable error (in percent of the mineral 's frequency)
could be detested for counts of 25 to 750 grains and for mineral trequen-
cies of 5, 10, 20, 40, 60' and 80 percent. Recently the writer has studied
a river Band containing more than 20 species of heavy minerals, of Itch
most have frequencies of less than 10 percent. Me variations in
frequency of some of these minor constituents appeared s~gnitic~t-. Since
the probable error for most of then could not be extrapolated with sutti-
cient accuracy froan Dryden's curves for 5 and 10 percent frequency, probable
errors were Petered ma curie plotted for frequencies between 0.1 and
20 percent. The pu~poseof the present paresis tonal these curves avail-
ablo for others who may wish to detente the probable errors for minerals
of low frequency. For completeness, curares for 30~ 40, 50, 6O, 70, Id 80
percent frequencies have also been included.
Curves showing the probable error in percent of the m~n~al's
frequency, as do Drydents, are presented in figure 1. Since :t is also
desirable to know the probable error in terms of percent of the total number
of all heavy mineral grains, Runes from which such values flay be c~btai Red
are present~in figure 2. Probable errors from which both sets of curves
can be plotted are presented in table 1.
The basis of all the probable error calc~alatic~ns ~s the accepted
formula used by Dryden, namely, P.E. (in number of grapnel - 0~6745 gnpg,
in which ~ is the nether o:t grains counted, p is the chance that any
grain w111 belong to a certain species, and g is the chance that any
grain will rat belong to that species. Ibe P.:E. (in percent of the
mine:~'s frequency) ~ 0.6745~;/np, The P.E. (in percent of the total
numbs of all heavy mineral grains) ~ 0.67*5 4npq/n. These axe equations
for parabolas arid consequently the curves plot as straight lines on
logarithmic graph paper. Since such presentation has the three-fold advan-
tage of requiring fewer points for plotting,, of permitting quicker and
more accurate construction of the coves, end of sho';rin`$ more data in a
Dryden, A. L . Accuracy in Percentage Representation of Heavy Mineral
Frequencies. Nat. Acad. Sci. Proc. ~vol. 17, pp.; 233-238, 1931.
OCR for page 98
Exhibit F
su ~. _
-- 98
limited space, figures ~ and 2 have been plot ted on a log~it~c base.2
In using the curves in both figure 1 and figure 2, the ordinate
for the total nuder of grains counted (n) is followed to its intersection
with the cure of the required f~equar~cy (or an extrapolated point when
the curare for that frequency is not given). The probable error ~n percent
is read directly from the abscissa scale. For example, in a count of axoO
grairls, if the frequency of t~ornblea~o is 3 percent, the into~rsoct~or~ of
the 400 grain ordinate and the 3 percent frequency curve is 19.2 percent
in figure l. In figure 2 it is 0.58 percent. The chances are equal,
therefore, that the observed frequency of hornblende hill deviate 19~2
percent of 3 percent, or wi11 be 3.0~.58 percent (2.42 to 3.58 percent).
In applying the probable error formula to the determination of
accuracies in percentage representation of mineral f~rec~uenc~es, several
points should be stressed. Firat'. it is assumed that the true frequency
of a mineral is know. AcWally the probable error cannot be computed
from em erimenta, ly determined frequ~ci es . ~' ~ has beers di scussed by
Rein am Petti john.3
Secondly, this applleati~ of the probable error formula gives
only the error ~e to reduction ~n sample ~4 The means by Ah the
size Or sample is reduced does nof matter so long as the sapling is
random. Furors introduced bait non-random sampling during sieved, separate
tion, microsplitti=, ~d counting, or by incorrect identification of
mineral grains, Act be added to the error of reduction in sample size to
give the total error. If laboratory technique is good and about 300
grains are counted, the error of reduction in sample size will probably
approximate the total erro}~.0
2Logarithmic graph paper (2 cycle :: 3 cycle) is made by Eugene Diet zgen
Company and Em!~l and Esser Company.
C
3~umlbein, W. C., and Petti John, F. J. Actual of Sedimentary Petrology,
p. 471. Appleton Century, Now York. ti338e.
This is the "counting errors of statistics. ~ The statistically correct
term is not introduced; because the ted "counting" has been used ~n a
different sense in previous harry mineral papers.
mbein, W. C. and Rasmussen, W. O.
Beach Sand for Heavy Mineral analysis
The Probable Error in S=pling
, . (Unpubli shed ~
NOTE:- Figs ~ and 2 follow and are considered pages 99 and 100.
Table 1 also follows and is page 101.
OCR for page 99
o
o
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w
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o
LL
llJ
J
a,
o
NUMBER OF HEAVY Ml NERAL GRAINS COUNTED
o
o.° ~
0 0 0 0 0 0 0 °
0 0 0 0 0 0 0 °
~o, °
o
o
~,
f req u e n cy of ony
mineral (In percent of
tota I minerol 9 ro ins counted
1 1 1 1 1 1 1111 ~111111111111111111
flGURE 1. - Curves for delermining probable errors in heavy nnineral
stud~es. The probable errors ore ex p reseed as percent of the heavy
mineral s frequency. ~ Example, with 3.0°/O frequency ond 400 grains
counted, the probable error is 19.2 °/O of 3 °/O. )
o
o
oso om
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`1-°
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OCR for page 100
OCR for page 101
Frequen
cy
Pct.
,
061
.2
.3
~4
.5
.6
.7
~8
~9
1.0
1~2
1,5
2.0 e.e
2.5
3.Q .~e
4.0
5.0 ate
6.0 ~ a.
7~0 .~.
8~0
9.0
10~0
12.0
14.0
17.0
20.0
30.0
40.0
5060
6060
70.0
80~0
~ ·
· a.
. .
· ·
· ·
· ·
· ·
· ·
· ·
..
· ·
-- 101 -- Ash; bit
T}BL]5 1
PALE MOORS FOR V~OUS AN COATS ~ ~ MUNCIE
Probable error of the mineral's
frequency
Probable error of the total
number of all heavy mineral
Brains
Number of grains c`:r~ted
100 400 1, 600 6'400
N~.aber of grains counted
100 400 ~ ,600 6,400
Pct. Pct. Pct. Pct . Pct. Pet. Pct. Pct.
213.2 106.6 53.30 26.65 0.2132 0.1066 0.05330 0~02664
150.7 75~33 37.67 18.83 .3013 .1507 O07534 .03767
123.0 61.48 30.74 15.37 .3689 .1844 .09222 .04611
106.4 53.23 26~61 13.50 .4257 .2129 .1064 .05522
95~15 47.58 23.79 11.89 .4760 .2380 *1190 .05949
86.82 43,41 21.70 10.85 .5209 .2604 .1302 .06511
80.33 40.17 20.08 10.04 .5623 .2812 .1406 .07029
75.11 37.55 18,78 9.389 .6009 ,3004 .1502 .O?511
70.78 35.39 ~ 7.69 8.847 .6370 .3185 .1592 .07962
67.11 53~56 16.78 8~389 .6711 .3356 .1678 .08389
61~20 30.60 15.30 7.650 ,7344 .3672 .1836 .09180
54.66 27.35 13.66 6.832 .8199 .4099 .2050 .1025
47.42 23.71 11.85 5.927 .9443 .4722 .2361 .1180
42~12 21~06 10.53 5~265 1.053 .5265 .2632 .1316
38~35 19.18 9.588 4~794 1.151 .5753 .2876 .1438
313.04 16.52 8.261 4.130 1.322 .6609 .3304 .1652
29.40 14.70 7.350 3~675 1.470 .7350 .5675 .1838
26.62 13.51 6.656 3.328 1.598 .7g88 .3994 .1997
24;.58 12.29 6.146 3~073 1~721 .8605 O4302 .2151
22~87 11~4dS 5.718 2.859 1.830 .9149 .4575 .2287
21~32 10.66 5.329 2.664 1.918 .9592 ,4796 .2398
20.23 10.12 5~059 2.529 2,023 1.012 .5059 .2529
18.27 9.133 4~566 2.283 2,192 1.(~96 .5479 .2740
16~72 8,362 4~181 2.091 2.340 1.170 .5851 .2925
14.90 7.451 3.726 1~863 2~534 1.267 .6234 *3167
13~49 6~745 3.372 1.686 2.698 1.349 .6745 .3572
10 ~ 30 5 ~ 151 2 ~ ~ 76 1 . 288 3 .091 1. 545 . 7 727 . 3864
S.263 4.131 2.066 1~033 3.305 1.~53 .8263 .4131
6.745 3~372 I, 6~36 .8430 3O573 1~686 .8431 .4215
5.50~3 2.754 1.377 .6886 3.305 1~653 .82G3 .4131
4.416 2.208 1.104 .5519 3~091 1~545 .7727 .3864
3.375 1.686 .8431 .4215 2.6g8 1.349 .6745 .3372
Representative terms from entire chapter:
heavy mineral
