FITTING A CHI -square CURVE TO AN OBSERVI:D FREQUENCY DISTRIBUTION By w. T Federer BU-14-M Jan. 17, 1951

Transcription

1 FTTNG A CH -square CURVE TO AN OBSERV:D FREQUENCY DSTRBUTON By w. T Federer BU-4-M Jan. 7, 95 Textbooks n statstcs (for example, Johnson, Statstcal Methods n Research; Love, Applcaton of Statstcal Methods to Agrcultural Research; and Smth and Duncan, Samplng Statstcs and Applcatons) are not clear as to the correct procedure for fttng a ch-square curve to an observed frequency dstrbuton of sample varances. Therefore, the purpose of ths note s to set out the procedure n detal wth examples of 4~ sample varances (Table ) obtaned by members of class n Advanced Statstcal Methods,.3, and of sample varances obtaned by Johnson (Chapter ). TABlE. 44 Sample Varances Obtaned from Samples of Sze from the Populaton of Pg Gans, Table.3., Snedecor. ' lo6.o o o o.l o o loo Totalg 3,4oo.8 X - = 93.o6llll Ordnates for a ch-square curve wth n degrees of freedom are obtaned by substtutng values of X n the formula

2 and evaluatng the result. n (\n) () Each of the 44 sample varances h~s 9 degrees of freedom assocated wth t. Therefore, t s desred to obtan ordnates for a ch-square curve wth n equal to 9 The values of the ordnates are presented n Table for selected values of ch-square. TAB~. Com!!;taton: of ordnates for A curve wth degrees of freedom. Value of X. - ':, x (f-) e.oo (x ) e - = ordnate * o.o.o o.6o o.367s79 o.ol5g o.ol9s.4.46 o.3oll94 o.o '34.33 o.o oco o.o a5S o.o8o85 o.o87 6.o 59.9 o.ol: o 97.~93 o.o3ol97 o.lo4 s.o 44S.l5 o.ol836 o.o S7.oo o.oo9.93 o.o 36.3 o.oo673s o.oso o.oo487 o.o6: o.o o.ool5o3 o.o45 4.o 67. o.ooo9 o.o356 6.o 63S4.o o.ooo335 o.oo9 8.o 47~.3. o.oool3 o.on6 o.o o.oooo45 o.oo6 * 6 5(.ff) (-l)l () () () () T. ( ) =./ (7)(5 )(3)()!F. = = Usng class ntervals of the frequency dstrbuton of the 44 sample varances s gven n Table 3. --

3 TABLE 3. Frequency dstrbuton of 44 sample varances. x = class center f freguency x = class center f freguency x = class center f freguency Total frequency L f. = ~ 44 The X values multpled by ~ = e34ol3 and the correspondng orc:.nates for a frequency of 44 wth a class nterval of are gven n Table 4. TABLE 4. X values and ordnates for the frequency dstrbuton of Table 3. Values of 39.6 x ordnate values of 39.6 x ordnate Sl x Sl x x n for X curve x n for X curve o lo3.4o ' ~.os e5J o o6.8o o.85 For ths example t wll be assumed that the true varance s not gven and that the mean of the varances, 93eo6, s a sutable estmate of the true varance, d The procedure usually followed n practce wll be to take the mean of the sample varances equal to d n Table 4 the factor 39.6 s multpled by the ordnates of a X curve for n = 9 degrees of freedom. t s not necessary to have the X values correspond to class centers or class endponts. The resultng curve through the -3-

4 new ordnates wll then have the same area between the curve and the abscssa as s ncluded n the hstogram of the frequency dstrbuton of varances. Ths factor s obtaned as follows: (L r )(Class nterval)(n = d.r.) ll'(l )(9) = ~ = ~9 6 true varance The abscssa for varances, s, corresponds to the abscssa for X multpled by ns d /n snce X = ~ for sums of squares of normally dstrbuted varates. Therefore, snce an expanson factor s used for the X values along the abscssa, t s necessary to use the recprocal of the factor for the values of the ordnate. The addtonal factor, frequency tmes class nterval, s necessary n order to change the ordnate values from a curve wth unt frequency to one wth a total frequency, ~ r, and a gven class nterval. Mathe~~tcally the Jacoban of the transformaton from X to s ordnates nvolves the factor n/d thus f(x ) dx =!! f(x ) ds. d or the ordnates for the s curve are obtaned as ~ tmes the ordnates of the X curve. The factor L r (class nterval) merely changes the unt area to a total area bass. f the class nterval were, the results gven n Table 5 would be obtan~ TABLE 5. Frequency dstrbuton and ordnates for curve wth a class nterval of. X. = class f. :;! Value of d * x ordnl. l. centers freguenc;y: ". -n X ate for X, cuty:e o o , o o l l QE.8.7 L r = 44-4-

5 TABLE 5 (contnued) (L f.)(class nterval)(n) * ()(9) = --: t~ varance n Chapter of "Statstcal Methods n Research," Johnson presents a ftted ch-square curve, Fgure, but gves no explanaton for constructng the fgure. The sample varances (Johnson, Table 8) were obtaned from the populaton of pg gans (Snedecor) from random samples of sze 5. Johnson uses d = obtaned from the populaton of pg gans. Ths procedure s not ahrays possble, snce d wll be unknown for the majorty of populatons from whch observed frequency dstrbutons are optaned. Ordnates for a ch-square curve wth 4 degrees of freedom are computed n Table 6. TABLE 6. Ordnates for a ch-square curve wth 4 degrees of freedom. Value of x e ~x /.5 X e-x =--~rdnate * e9o4837.o : o89.s o o o.o49787.o o58 8.o.836.o o5o..oo6738.ol68..oo4o87...oo479.oo74 6..ooo335.3 o.o.oooo45.ooo -5-

6 TABlE 6 (contnued) = 4 =.5 Assumng that Johnson used a class nterval of, the ordnates n the above table must be multpled by (4 )(loo) = 8 =(class nterval)(n)(l:f) varance = o Fttng a curve through these values gves a fgure whch agrees wth Fgure n Johnson's book. Also, the frequency dstrbuton of the quanttes, X = sum of sguaros d, could be compared wth the ch-square curve but qute often the sample varances s wll bo avalable nstead of the sums of squares. Therefore, the X values must be multpled by o/n n order to correspond to tho values of tho sample varances. Tho comparson of an observed frequency dstrbuton of sample varances wth tho theoretcal ch-square dstrbuton s exemplfed n Table of Johnson's book. He dvdes tho observed and theoretcal frequency dstrbutons nto 9 classes and compares them vl th a ch-square goodness of ft test. Tho resultng ch-square value has 9- = 8 degrees of freedom. f Johnson had used o equal to tho mean of tho sample varances, 85.9, tho resultng goodness of ft X value HOUlcl have!:!! degrees. freedom ~nstoac; Of _ snce o s estmated from tho data. n tho event that Johnson had used tho mean of tho sample moans, 9.83 (Table 3), nstead of tho populaton moan, 3o, and tho average of the sample varances, 85.9, cu vded by 5 as o& nstead of o& = /5 =, tho rex X sultng ch-square for comparng the observed wth tho expected froquoncos, Table 4, would have had - 3 = 7 degrees of freedom nstead of 9 snce tho observed number must equal the expected number and two constants, m = 9.83 and d& = 7.84, were obtaned from the data. X Another nterestng pont to note s that the expanson factor for the -6-

7 ordnates of the ch-square curve, N(class nterval) d-;/n s of the same form as the expanson factor for fttng the normal curve to an observed frequency dstrbuton, ' N(class nterval) sample standard devaton = s -?-

8 .. -!.,.st FGURE. Ch-square curves for 4 and 9 degrees of freedom f(x )...lot.re).o6.o4j.o -4= === x~

9 .. t-'- 5t J - 5t >a~ "' 8(H <D'-'> g.~ <DO' &~ lot 5+,-. FGUR!.!:. Hstogram and ftted curve for data of Table 3! ' -! ' L- Jl-f-'~L-\ oo ~ s = d = n

10 3 t- '! 5 +! l + T 5 t ;;- >., >< g~ m Q) ('(\ ;- ~.{'\ ' ' Q) w t:~ ~ 5 t,,-. ~ n! ~ ~ j\! ~ ' ~ k \J... J.. FGURE 3. Hstograra and ftted curve for data of Table ' '---~ _l_, ~--l.'--r---= +~=J ' t =, o 5 3oo 35o 4oo ---;-t-- 45o - s _- o X =.343 X ----?> n