Analytical Chemistry News & Features: The Weakest Link Exercise

August 1, 1999

Frank A. Settle
Michael Pleva
Washington and Lee University

Imagine that you have been assigned the task of determining the concentrations of heavy metals in a heterogeneous sample such as a railcar-load of coal or incinerator waste. A powerful, computer-controlled, inductively coupled plasma-optical emission spectrometer (ICP-OES) is available for the measurement. However, obtaining useful data also means taking representative samples and preparing them for measurement while preserving their integrity (1).

Chemical analysis involves three major operations--sampling, sample preparation, and measurement. This educational exercise is designed to emphasize the importance of sampling. Although powerful instrumental techniques such as ICP-OES tend to focus attention on measurement, the other two operations are also critical to the quality of the data obtained from the analysis. The quality of the data can be no better than the least precise operation in the method; thus, while the precision of the measurement may be high, the overall data quality may be much lower because of imprecise sampling or sample preparation techniques.

The goal of the current exercise is to identify the weakest or least precise operation in an analysis and evaluate the magnitudes of the variance for each of the three operations. Variance, which is the square of the standard deviation (s²), is used because, unlike standard deviations, variances are additive. Thus, the variance of a method is the sum of the variances of its component operations: Variance (total) = Variance (sampling) + Variance (sample preparation) + Variance (measurement). Once the values of the variances for the component operations are known, the weakest link can be identified, and efforts to improve the overall precision of the method can be undertaken.

Determining the sodium content of salted snack foods provides a relatively simple and engaging opportunity to introduce the concept of the weakest link. Options for the measurement link include gravimetry, flame emission spectrometry, atomic absorption spectrometry, volumetric titration (2), and potentiometry. Each method has its assets and liabilities. Students work in teams of two or three, depending on the level of the student and the laboratory environment, and they are either given specific procedures for the determination or asked to develop their own.

The snacks are sampled directly from the bag or box, and the sodium chloride is extracted from the samples into an aqueous solution. The design of the experiment is intended to isolate the variance for each step. The students also compare the results from two of the measurement techniques to see whether they differ statistically. Figure 1 shows the overall strategy for this exercise (3), which is designed for k samples (for this example, k = 4).

In the lab

The issue of obtaining a representative sample is significant. In heterogeneous materials, the variance associated with the sampling component is expected to be the largest source of uncertainty in an analytical method. The procedures for sampling solid materials include considerations of the minimum mass, particle size, and sample splitting (4-7). A possible variation of this exercise would be to study the effects of systematic versus "grab" sampling by running the entire experiment twice. The students used only the grab sample technique.

A three-level nested design based on duplicates (Figure 1) is used to estimate the variances caused by sampling, sample preparation, and measurement (8). Other sets of variables, such as between laboratory, within laboratory, and between analysts, can be investigated using the same design. In this experiment, S_1-4 represents the four samples (k = 4) taken randomly from a bag of Fritos. After grinding and crushing, each of the four test samples are divided into two approximately equal portions, labeled a and b, and the mass of each of the eight (2k) portions is obtained.

Figure 1. Experimental design.

Each portion is extracted with water, filtered, and diluted in volumetric flasks of the appropriate volume to give samples V_1a, V_1b, V_2a, etc. This volume, based on the percentage of sodium stated on the snack food's label, should give a reasonable concentration of sodium for the measurement and provide the quantity necessary for replicate measurements. The sodium percentages are represented as A1a, A1b, A2a, A2b, B1a, B1b, etc. A data set from silver nitrate titrations (Table 1) illustrates the procedure for recording and processing the 16 measurements. The sodium content in a subset of the samples (Table 2) also was determined by flame emission spectrometry.

Table 1. Titration data.

Table 2. Comparison of measurement techniques.

Determining variances

The variance associated with level III (measurement by titration) is determined first by using values from the 16 titrations, Table 1, and

with 2k degrees of freedom (8 df). Column d3 in Table 1 contains the differences between the averages of the duplicate titrations of samples V_ia and V_ib.

When the variance associated with the measurement is known, the variance for level II can be determined using

with 4 df. This value of s_II² contains both the sample preparation and the measurement effects. In fact,

The "2" in the denominator of the s_meas² term takes into account the fact that two titration values were used to generate one value for an average of the percent sodium. The differences of these averages are used to determine the variance associated with sample preparation.

Finally, the variance for level I is determined by replacing each pair of the eight averages in Table 1 with four averages. The variances from all component operations--sampling, sample preparation, and measurement--are nested in s_I²

with 3 df. The "2" and "4" in the denominators of the second and third terms of Equation 4 take into account the number of values required to calculate one value in Table 1's right-hand average column. Incidentally, the best estimate of the percentage of sodium in the sample is the average of the 4 final averages, which corresponds to the overall average of the 16 separate titrations in level III. Note that the methods for calculating variances mean that the sum of the component variances will not be equal to the variance of the averages in Table 1's right-hand column.

When the values for s_samp², s_extract², and smeas² have been determined, students ask themselves, "Are these values statistically different?" The answer is found by calculating the F ratio of any two variances. By convention, the larger variance is divided by the smaller and then compared with the appropriate tabular F value. The degrees of freedom of each variance are used to determine the appropriate value from the table. If the calculated F ratio exceeds the tabular F value, then the null hypothesis (no difference between the two variances at the stated confidence level) is not valid, and we can say that the variances are statistically different at the specified confidence level. After checking all differences in this manner, the process with the largest variance is the least precise and thus is the weakest link in the analysis.

And the weakest link is . . .

Table 1 contains the titration data from the experimental design outlined in Figure 1. From the statistical procedure described in the preceding section,

with 8, 4, and 3 df, respectively. Applying the F test (7) at a 99% confidence level for statistical differences, s_extract²/s_meas² = 63.9, and critical F is 8.81 (one tailed 4,8 df). At a 99% confidence level, there is a difference between the variance associated with measurement and sample preparation, or s_samp²/s_meas² = 259.0, and critical F is 9.60 (one tailed 3,8 df). Thus, at a 99% confidence interval, a statistical difference exists between the variances for measurement and sampling.

The magnitudes of the variances indicate that the sampling is the weakest link. If sampling can be improved, then attention can be devoted to the extraction process. The data indicate that percentages reported at 95% confidence limits for sodium (0.408 ± 0.056% by titration and 0.452 ± 0.033% by flame emission) were lower than the value on the label of the Fritos bag (0.57%). This is not surprising in view of the large variances associated with sampling and extraction. Factors contributing to these differences are the distribution of the salt within the sample, the method of sampling, and the efficiency of the extraction process.

Table 2 lists the data for determining sodium in the eight samples by titration and atomic emission. The emission results represent the average of duplicate measurements. The two measurement techniques can be compared using the paired t test (5), in which the null hypothesis says that no difference between the two means (i.e., _d = 0) exists. The mean of the differences (x_d) is 0.044, and the standard deviation of the differences (s_d) is 0.027. Because _d = 0,

(8)

From the data, the calculated value for t is 4.58. The critical t value at 99% confidence is 3.50 for 7 df. Therefore, the null hypothesis is rejected, and the results from the two measurements are statistically different at the 99% confidence level. This result indicates a systematic error in one of the measurement techniques; however, it is not possible to determine which technique from the existing data.

This exercise familiarizes students with the components of chemical analysis and emphasizes the importance of sampling. The experimental design and statistical procedures allow them to evaluate the impact of each step on the overall quality of the information. The comparison of the different measurement techniques also reinforces the use of statistical methods. Finally, students are forced to think about factors that caused unexpected results and ways to improve analytical methods. These experiences are important for both future analysts and those who will use information from chemical analyses.

We would like to acknowledge the six students who labored long and hard to make this a meaningful exercise--David Cooper, Nathaniel Dunn, Marium Holland, Nicole Johnson, Anne McElhaney, and Leonard Rorrer.

Frank A. Settle was a professor at Virginia Military Institute and a program director at the National Science Foundation before assuming his present position as a professor at Washington and Lee University. Michael Pleva is department chair and a professor at Washington and Lee University. He teaches analytical chemistry and a course on chaos. Address correspondence to Settle at Dept. of Chemistry, Washington and Lee University, Lexington, VA 24450 (settlef@wlu.edu).

References

(1) Storer, D. A. Sample Preparation for Chemical Analysis; Terrific Science Press, Miami University Press: Middletown, OH, 1998.

(2) Harris, D. Quantitative Chemical Analysis, 5th ed.; W. H. Freeman Co.: New York, 1998; 165-66.

(3) Taylor, J. Statistical Techniques for Data Analysis; Lewis Publishers, Inc.: Chelsea, MI, 1990; 50-56.

(4) Schwedt, G. The Essential Guide to Analytical Chemistry; John Wiley and Sons: New York, 1997; 18-21.

(5) Dean, J. Analytical Chemistry Handbook; McGraw-Hill, Inc.: New York, 1995; 1.1-1.4.

(6) Benedetti-Pichler, A. In Physical Methods for Chemical Analysis; Berl, W., Ed.; Academic Press: New York, 1956; Vol. 3; 183-217.

(7) Laitinen, H. A.; Harris, W. E. Chemical Analysis; 2nd ed.; McGraw-Hill: New York, 1975; Chapter 27.

(8) Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry, 2nd ed.; Ellis Horwood: New York, 1992; Chapter 3.