On the Confidence Limits on the Product of Two Uncertain Numbers Sir: The problem of how to estimate the mean ( 2 ) and 2 ) the product of two uncertain confidence interval ( ~ 1 ~ of numbers (2 = 29) frequently arises in the treatment of chemical data. A typical example is the determination of the amount of a chemical compound in a test solution by means of titration; this concentration will be determined by the volume of titrating solution used times the concentration of the titrant. This problem was recently discussed by Olcott ( I ) for the case when f and 9 can be assumed to be normally distributed. The treatment discussed by him was, however, complicated and, in addition, statistically incorrect. In particular, Olcott's method leads to serious underestimation of the confidence interval when the relative standard errors of f and 9 ( S , and S , see below) are unequal. A short comment about simple and realistic treatments suitable for practical use is therefore warranted. The poblem has been treated in the statistical literature (2-4) and chemical literature (4-6) in sufficient detail, so only a summary will be given. When the relative standard errors (see below) o f f and 9 are small, say less than lo%, the product is well estimated by Equation 2 and the variance of the product by Equation 3.
O n leave t o the D e p a r t m e n t of Statistics, U n i v e r s i t y of W i s consin, Madison, Wis. 53706, 1973-74. (1) R. J. Olcott, Anal. Chern., 45, 1737 (1973). (2) L. A. Goodman, J. Amer. Stat. Ass., 55, 708 (1960). (3) F. Yates. "Sampling Methods for Censuses and Surveys," 2nd ed., Charles Griffin & Co, London, 1953, p 198. (4) A. Hald, "Statistical Theory with Engineering Applications." Wiley, New York, N.Y., 1952, pp 118, 246. (5) P. D. Lark, B. R. Craven and R. C. L. Bosworth, "The Handling of Chernical Data." Pergamon Press, London, 1968, p 129. (6) 0. L. Davies and P. L. Goldsmith, "Statistical Methods in Research and Production," 4th ed., Oliver & Boyd, Edinburgh, 1972, pp 54, 62.
sT2 =
y 2 s . 2 / n x+ X2s,2/ny
(3 1
If we, in addition, assume that f and 9 are approximately normally distributed, which is very reasonable for most continuous measurements made in chemistry (see below), z will also be approximately normally distributed. Confidence limits of can then be estimated in the ordinary way be means of t-values: z1,z2
=
z f fl.&s,-
(4 1
where the number of degrees of freedom of the t-distribution (significance C Y ) is n, n, - 2 if the relative standard errors o f f and 4 [SI = s,/(f 6,) and S , = s y / ( 4 f i y ) ] are approximately equal. Otherwise, this number ( f ) is estimated from
+
l/f =
{ ( s ~ ' / ( s+~S' , 2 ) } 2 / ( n , - 1) + ((s$/(s_,2+ s,')}2/(ny
-
1)
(5)
For most practical cases, the formulas given above are sufficient. It should be noted that, for small sample sizes (smaller than 20) distribution models are highly approximate. This means, among other things, that since then I: and 9 in Equation 2 are only approximately normally distributed, there is nothing in theory that says that z in Equation 2 will be less normally distributed than E and 9. To conclude, the use of Equations 1 to 5 for the estimation of the product and its confidence interval is usually sufficient for chemical applications. More refined treatments are warranted only in the case that statistical tests show that or f and 9 are distributed significantly different from normal distributions. Such tests are possible to perform only when the samples are large, and seldom show significant results even in those cases. Svante Wold' Institute of Chemistry Umei University S90187 Umei, Sweden
RECEIVEDfor review January 28, 1974. Accepted June 3, 1974.
I AIDS FOR ANALYTICAL CHEMISTS Rapid Method for Determining Densities of Liquids Using Micro Syringes James E. Burroughs and Charles P. Goodrich Borg- Warner Corporation, Roy C. Ingersoll Research Center, Des Plaines, Ill. 600 18
Calibrated micro syringes are most useful because of their ability to deliver small volumes of liquids with extremely high precision and accuracy. Normally, such syringes are rated to deliver with an accuracy of fl% of the volume delivered. Volatility characteristics of liquids are also minimized because of the small cross-sectional area of the needle. In this study, this characteristic has allowed the routine use of micro syringes for determining densities of certain types of liquid. 1614
This technique is of great value where the volume of a liquid sample is limited. Even though the use of limited range hydrometers may be simpler and the employment of pycnometers will give four place answers, the data will show that the use of micro syringes will allow the determination of densities on samples of limited volumes with precision and accuracy. Also, the technique is simple and rapid and is most suited for both plant and routine laboratory use.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
