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-- 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.
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.
o o c~ z w - 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 *o.° `1-° `~o o \o9.oo ',.o ~o 6.0 5o A0 3.0 z.5 ~,0 \~5 \.o oo.e 01 o,6 05 0~A 0-3 03 0-\ O -3~33
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
