On the Confidence Limits on the Product of Two Uncertain Numbers Sir: The problem of how to estimate the mean ( 2 ) and confidence interval ( ~ 1 ~ of 2 ) the product of two uncertain 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 and S , = s y / ( 4 f i y ) ] are errors o f f and 4 [SI = s,/(f 6,) 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. I t 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
Table I. S u m m a r y of Precision Studies
Syringe size, pl
Liquid
Isopropanol Isopropanol Gasoline (Premium) Gasoline (Premium) Mineral spirits Mineral spirits
100 10
100 10 100 10
Arithmetic mean of experimental density, g/I.
0.784 0.768 0.708 0.714 0.777 0.770
Table 11. S u m m a r y of Accuracy Studies Liquid Std dev,
C/b
0.209 0.728 0.507 0.242 0.386 0.541
Isopropanol Isopropanol Gasoline Gasoline Mineral spirits Mineral spirits
Syringe size, rl
Referee value, g/l.
Experimental density, g/l.
Error,
100 10
0.777 0.777 0.719 0,719 0.774 0.774
0.784 0.768 0.708 0,714 0.777 0.770
0.91 1.14 1.51
100
10 100
10
~‘5
0.60
0.49 0.55
Apparatus. Standard 10-11 and 100-11 syringes (Unimetrics Universal Corp.) were employed. The syringes were used without special cleaning. The results were compared with data obtained by the standard method for determining densities, using 50-ml volumes of the sample in a closed container. The weighings were made on a semi-microbalance having a sensitivity of 0.000050 gram. Sample sizes greater than 50 microliters may be readily weighed on a balance with a sensitivity of 0.0001 gram. Reagents. “Spec-pure” and reagent grade chemicals were used as received. Samples of commercial “premium” grade gasoline and mineral spirits were used to represent complex liquids. Procedure. Calibration of t h e Syringe. Although micro -syringes are guaranteed by their manufacturer to be better than 1% accurate, it is recommended that syringes used for density measurements be calibrated prior to use. In this way, any volume errors resulting from erroneous syringe markings can be compensated for and a correction applied to subsequent data. Ordinary distilled or deionized water, a common laboratory liquid, is ideally suited for calibration purposes. The procedure for making the measurements is the same as described for the sample in the following paragraph. The true values for the densities of water at given temperatures are readily available from the technical literature and handbooks. Syringes employed in the following measurements were found to be well within the standard deviation for the overall measurements. Therefore, no corrections were applied to the data. Analysis of the Sample. A sample of the liquid was drawn into the clean, dry syringe. All bubbles were removed by inverting the syringe and tapping it gently against the other hand. The plunger was then set at the appropriate volume mark, and the tip of the needle was cleaned with a paper tissue. The weight of the syringe plus contents was then determined and recorded. The sample was then expelled from the syringe, the tip was again cleaned, and the syringe was reweighed. The density was calculated from the difference in the two weights divided by the volume change taken.
minutes. T h e nominal temperature of the laboratory was 22 “C. Accuracy. Table I1 shows t h a t the relative errors encountered in the examples given were 1.5%or less. T h e majority of the analyses were found to be in error of less than 1%. Limitations of Syringes. T h e accuracy of syringes employed in the determination was reported t o be better than 1%of the volume delivered. Minute bubbles of entrapped air in t h e sample may be additive in the overall error found. Care must be taken t o ensure that the syringes contain samples with no detectable air entrainment. Limitations of Balance Sensitivity. A weighing error of 50 pg on 10.0 pl of sample that has a density of 0.75 gram per milliliter would result in a relative error of 0.7%. When possible, t h e density of a sample should preferably be made on 100 microliters of the liquid sample. T h e data in Table I1 show t h a t 10 microliter samples will give accurate two place results. Effects of Viscosity. Highly viscous samples such as heavy crude oils cannot be determined by this technique because the sample cannot be readily drawn or expelled into a 10- or 100-microliter syringe. Calibrated syringes having capacities of 1000 pl are equipped with larger gauge needles and therefore may be employed to determine the densities of some moderately viscous materials. E f f e c t s of V a p o r Pressure. T h e experimental results showed t h a t no significant weight loss occurred within the first five minutes of weighing of any of the liquids tested. Open syringes can be readily used for most common laboratory and commercial liquids. Some modifications of the procedure would be necessary for liquids having high vapor pressures.
R E S U L T S A N D DISCUSSION
CONCLUSION
Temperature compensations may be necessary because of “apparent” discrepancies between t h e observed density values and those reported in the literature. “Referee” determinations were employed t o show t h a t good correlation is possible if the experimental was obtained a t the same temperature as t h e “referee” value. Reproducibility. Table I shows that the densities of isopropanol, mineral spirits, and a premium grade gas are readily determined with good reproducibility. Special or meticulous care in t h e handling of the syringes was not taken and the total time per analysis was less than two
Small scale density determinations are possible on a variety of liquids. T h e results are accurate to a t least two significant figures, using 1 0 - ~ 1syringes. Larger syringes enhance the precision and accuracy of the results. Densities of samples having high vapor pressures, which are abnormally viscous, or which are corrosive t o the syringe, cannot be conveniently measured by the proposed technique.
EXPERIMENTAL
RECEIVEDfor review January 7, 1974. Accepted May 3, 1974. This study was supported by t h e Roy C. Ingersoll Research Center, Borg-Warner Corporation.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
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