“STUDENT’S” METHOD OF COMPUTING PROBABLE ERROR IN AGRICULTURAL EXPERIMENTS.

New Zealand Journal of Agriculture, Volume XXX, Issue 1, 20 January 1925, Page 26

“STUDENT’S” METHOD OF COMPUTING PROBABLE ERROR IN AGRICULTURAL EXPERIMENTS.

F. W. HILGENDORF,

D.Sc.

One of the best ways of conducting those agricultural experiments in which cumulative effect is not a prime consideration is to attack small questions one at a time, and lay out paired trials of the standard method versus the innovation, either in adjacent parts of the same field, or on different fields or farms, or in different years. In this way a great number of replicates can be made at once so as to reduce probable error, and soil variations are largely eliminated, since the two members of each pair of trials are on very similar soil. ; The differences only are computed, the actual yields of the two members of the pairs being'of nodirect interest. For the past two years at Lincoln we have made large use of Beaven’s halfdrill strip method for this purpose, and find it conducive to accuracy of results combined with rapidity of handling. The middle coulter of the drill is blocked up, and half the drill is filled with each, of the two varieties or manures. The drill is then driven wheel on wheel-mark up and down the field, and the result is that one gets pairs of plots sown for as long as one keeps drilling. We find from twenty to thirty pairs usually sufficient. . For computing the probable error , of [the difference between paired trials “ Student’s ” method is very suitable, because it is based on an estimation of the differences, which is what the experimenter is interested in, and because it takes cognizance of the correlation that exists between the members of each pair. The method is adapted to any kind of paired experiments that can be devised, and so has been used for the manurial trials recorded elsewhere in this issue-of the Journal, any two trials made on one farm in one year being regarded as a pair. The calculation is as follows : Find the difference between each pair of plots ; enter each difference with its appropriate arithmetical sign. Find the mean difference M, with its arithmetical sign ; find the difference d of each difference from the mean difference (their total = O), and finally square each d. Then, Vll,d 2 M if n is the number of variates, the standard deviation a — \/ and Z = — v - n <r The odds can then be found from the table appended. It will be noted that the factor -67 used for turning standard deviation into probable error is not introduced. It has no particular advantage, and must not be used in conjunction'with the table here quoted.

V2S’OOQ g— = 2-04 •68 ■ - ■ ■ Z = ■ = -33 2’04 and odds are by attached table about 4 to 1 that the difference is significant. Skeleton Table of Odds for Z Values estimated by “-Student’s” Method. (n number of variates.)

Some Cases of Practical Certainly. Z= -7 'n= 12 odds = 48-5 Z= 5 n= 20 odds 46-4 Z -45 n= 25 odds = 46-6 A much fuller table is given by H. H. Love, Jour. Am. Soc. Agronomy, vol. xvi, No. 1, 1924, p. 68.

' Year or Farm. 2 cwt. Guano. 1 J cwt. Super. Difference. d — Difference from Mean. d 2 . A - .. IO-2I 9-61 —o-6o -1-28 1-638 B . . 26-82 3l-8o + 4-98 + 4-3O 18-490 C ’ 9’43 8-42 . — I-OI — 1-69 2-856 D 27-11 ■ 27-22 4- o-11 -o-57 •325 E 9-27 10-41 + 1-14 4-0-46 •212 F 24-21 23-67 -o-54 — 1-22 1-488 Total 107-05 hi-i.3 o-oo 25-009 Difference 4-08 4-08 Mean = M 4-0-68

Example. To find difference and the odds in favour of its significance between two manures applied to turnips on each of six farms:-

z. . n = 5. n = 7. n — 10. n — 12. n = 15. • n = 17. n = 20. n = 25. n = 30. 0-2 1-82 2-12 2-55 2-84 3-29 3-60 4-08 4-94 5-90 o-4 3'27 4.48 6-67 8-45 xi-8 14-5 19-5 3i-4 • 49-3 o-6 5-75 9-42 18-0 26-8 47’3. ' 68-4 117 277 666 o-8 9-82 19-5 49’3 88-3 207 356 832 3.332 9,999+ i-o 16-2 39-2 132 293 908 1,999 9,999+ 9,999 1-2 25-9 75-9 344 908 » 3,332 9,999+ 1-4 35'8 119 666 3,332 9,999 + i-6 600 255 i,999 . 9,999 + i-8 86-7 434 4,999 2'0 122 713 9,999 + 2'2 68 1,249 9,999 + 2- 4 232 1,999 9,999 + 1