bliddleport, S . Y.). This recycling system permits a continuous gas-liquid estraction which ensures high recovery of volatiles. Because the system is closed, and the pressure is atmospheric, sample loss is a minimum. The sweep gas (air, S1,O?, etc.), time of si\-eeping, and temperature of sample depend on sample and study in question. Upon completion of collection, the U-tube can be disconnected and the collected sample injected via a system of valves
Design Patterns
LITERATURE CITED
(1) Farrington, P. S., Pecsok, R. L., LIeelier, R. L., Olson, T. J., ANAL. CHEN. 31,1512 (1959). ( 2 ) Snwar, W, W,,Sawyer, F. ll., Beltran. E. G.. Facerson. 1. S.. Ihid.. K. w.,Sawyer, F. M., Fletcher, IT., Fagerson, I. S., J . D a t r y Sci., 43, 839 (1950) (abstract). ( 4 ) Rhoades, J. W.,Food Research 23,
(3) *aiar,‘
for Slopes and Intercepts
C. T. Shewell,’ Research and Development Division, Humble Oil & Refining Co., Baytown, Tex.
many instances in chemisT try and physics where two or more determinations of a property, y, are HERE ARE
made at different levels of an independent variable, 5, such as concentration or temperature, and a straight line is to be drawn through these points to determine a slope, m, or a n intercept, b. This basic piocedure. of course, assumes that a straight line will adequately represent the data over some known region of values of the variable, 5. Preliminary experiments or theory are helpful in deciding irhether a straight line may be used (perhaps after some transformation of the data), and over what permissible range of the independent variable. Although there are in fact errois in the independent variable as well as in the dependent variable, and methods are available for taking account of these errors (S),it is usual to assume that the values of the dependent variable are knoirn with certainty. The choice of location of the experimental points within the allowed field can appreciably alter the accuracy of the determination of the slope or intercept rrithout changing the total number of points. Broadly speaking, the most accurate determination of the intercept is obtained by repeating the measurement S - 1 times a t the loivest permissible value of the independent variable and then obtaining one measurement at the highest permissible value of the independent variable; for the slope, the most accurate determination results from S / 2 repeated tests a t the loiyest permissible value and S 2 at the highest permissible value of the independent variable. A measure of the efficiency of the
design chosen is the variance of the slope, V ( m ) , or the variance of the intercept, V(b),or both. As might be expected, these are both functions of the experimenter’s ability to reproduce his observations a t a given level of the independent variable. This ability may vary widely a t the two ends of the permissible range, being a function of the level of x used. A transformation to logarithms will usually eliminate this undesirable feature. A measure of this ability is the variance, a*, of replicate determinations. The variance is the square of the standard deviation, cr, and may be calculated from the formula
where N is the total number of observations, Zy is their sum, Zy2 is the sum of Table I.
Effects of Design on Accuracy
Values of Independent Variable, z 1,1>? 1,, 1. , 3
1,3,5 l,5,5 3,3,5 3,5,5
1,1,1,1,5 1,1,1,5,5 1,1,5,5,6 1,1,3,5,5 1,2,3,4,5
1,1,5,5 0.1,0.1,0.5,0.5 5,5,25,25
Variances /u2 Intercept, Slope, V (b ) / d V (m ) / d = P = * 0.376 1.376 0.844 0.094 1.458 0 125 i.594 o.o9i 5.3’75 0.375 (.375 0.375 1.000 0.333 0.583 0.083 0.813 0.062 1.150 0,100 1.583 0.083 0.078 0.453 0.552
0.802
0.762
1.100 0.813 0.813 0.813
0.052
0.052 0.062 1.100
their squares, and j is their average. Then for any design V(b) = p a 2 and V ( m ) = q d where p and q depend on the location choices made, on N , and on nothing else. The smaller these variances are, the more precise the determination of the slope or intercept. Table I gives the values for p and q for a number of simple choice patterns t o illustrate the above statements. The values in the table were computed from the equations (6) :
obtained by inverting the matrix of the normal equations of the straight line. Additional pattern variances to cover any desired case may be computed from these equations for p and q. The equations for p and q shorn that p is independent of scaling while q varies inversely with the square of the scaling factor. This is illustrated in the last three sets of values in Table I. The best choice pattern may be adapted t o the cryoscopic or ebullioscopic methods for the determination of molecular weights, where single point methods do not give the desired accuracy, or for other situations which abound in the literature (1, 2, 4 ) . LITERATURE CITED
(1) Felten, E. J., Fankuchen, Isidore, Steieman. Joseuh. ASAL. CHEJI. 31. 1 7 6 (19g9). ’ (2) Gilbert, P. T., Jr., Ibid., 31, 110 (19a9). (3) Kelley, B. K., .l’ature 184, 1086 A
f1F)bF)).
0.0625
6.250
0.0025
Deceased. VOL. 32, NO. 11, OCTOBER 1960
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