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Chapter 4 Evaluating Analytical Data 97

of which are the binomial distribution and the normal distribution. is paired or unpaired. For unpaired data it is also necessary to
Knowing the type of distribution allows us to determine the proba- decide if the standard deviations can be pooled. A decision
bility of obtaining results within a specified range. For a normal about whether to retain an outlying value can be made using
distribution this range is best expressed as a confidence interval. Dixon’s Q-test.
A statistical analysis allows us to determine whether our Finally, we have seen that the detection limit is a statistical
results are significantly different from known values, or from statement about the smallest amount of analyte that can be de-
values obtained by other analysts, by other methods of analy- tected with confidence. A detection limit is not exact because its
sis, or for other samples. A t-test is used to compare mean val- value depends on how willing we are to falsely report the analyte’s
ues, and an F-test to compare precisions. Comparisons between presence or absence in a sample. When reporting a detection
two sets of data require an initial evaluation of whether the data limit, you should clearly indicate how you arrived at its value.

4J Suggested EXPERIMENTS

Experiments Cunningham, C. C.; Brown, G. R.; St Pierre, L. E. In this experiment students measure the length of a pestle
The following experiments may be used to introduce the statistical analysis of data in the analytical chemistry
laboratory. Each experiment is annotated with a brief description of the data collected and the type of statistical
analysis used in evaluating the data.

“Evaluation of Experimental Data,” J. Chem. Educ. 1981, 58,
using a wooden meter stick, a stainless-steel ruler, and a
vernier caliper. The data collected in this experiment provide
509–511.
Experiments
an opportunity to discuss significant figures and sources of
In this experiment students determine the density of glass
marbles and the radius of the bore of a glass capillary tube.
Density is determined by measuring a marble’s mass and
Paselk, R. A. “An Experiment for Introducing Statistics to
volume, the latter by measuring a marble’s diameter and error. Statistical analysis includes the Q-test, t-test, and F-test.
Students of Analytical and Clinical Chemistry,” J. Chem.
assuming a spherical shape. Results are compared with those Educ. 1985, 62, 536.
expected for a normal distribution. The radius of a glass
capillary tube is determined using Poiseuille’s equation by Students use a commercial diluter to prepare five sets of
measuring the volume flow rate of water as a function of the dilutions of a stock dye solution (each set contains ten
hydrostatic head. In both experiments the experimentally replicates) using two different diluters. Results are compared
obtained standard deviation is compared with that estimated using t-tests and F-tests.
by a propagation of uncertainty. Richardson, T. H. “Reproducible Bad Data for Instruction in
Statistical Methods,” J. Chem. Educ. 1991, 68, 310–311.
Gordus, A. A. “Statistical Evaluation of Class Data for Two
Buret Readings,” J. Chem. Educ. 1987, 64, 376–377. This experiment uses the change in the mass of a U.S. penny
to create data sets with outliers. Students are given a sample
The volumes of water in two burets are read, and the
difference between the volumes are calculated. Students of ten pennies, nine of which are from one population. The
analyze the data by drawing histograms for each of the three Q-test is used to verify that the outlier can be rejected. Class
volumes, comparing results with those predicted for a normal data from each of the two populations of pennies are pooled
distribution. and compared with results predicted for a normal
distribution.
Harvey, D. T. “Statistical Evaluation of Acid/Base
Indicators,” J. Chem. Educ. 1991, 68, 329–331. Sheeran, D. “Copper Content in Synthetic Copper
Carbonate: A Statistical Comparison of Experimental and
In this experiment students standardize a solution of HCl by Expected Results,” J. Chem. Educ. 1998, 75, 453–456.
titration using several different indicators to signal the
titration’s end point. A statistical analysis of the data using t- In this experiment students synthesize basic copper(II)
tests and F-tests allows students to compare results obtained carbonate and determine the %w/w Cu by reducing the
using the same indicator, with results obtained using different copper to Cu. A statistical analysis of the results shows that
indicators. The results of this experiment can be used later the synthesis does not produce CuCO 3, the compound that
when discussing the selection of appropriate indicators. many predict to be the product (although it does not exist).
Results are shown to be consistent with a hemihydrate of
O’Reilley, J. E. “The Length of a Pestle,” J. Chem. Educ. 1986, malachite, Cu 2 (OH) 2 (CO 3 ) • 1/2H 2 O, or azurite,
63, 894–896. Cu 3 (OH) 2 (CO 3 ) 2 .
—Continued