In the Classroom
Precision and Accuracy in Measurements A Tale of Four Graduated Cylinders Richard S. Treptow Department of Chemistry and Physics, Chicago State University, Chicago, IL 60628
Students need to be aware of the fact that even our best scientific measurements unavoidably involve some error. To develop this awareness, general chemistry courses introduce the concepts of precision and accuracy. The distinction between the two terms is illustrated in textbooks through the analogy of a target with arrows more or less clustered around the bull’s eye. After this brief introduction many texts do not develop the subject further. This paper expands upon the topic of precision and accuracy at a level suitable for general chemistry. It serves as a bridge to the more extensive treatments in analytical chemistry texts and the advanced literature on error analysis. It presents a simple model experiment involving graduated cylinders, a discussion of some complications, and suggestions for classroom implementation. Previous articles in the Journal cover such related topics as significant figures (1–5), propagation of error (6–13), experiments and demonstrations of uncertainty (14–21), and a general review (22). Definition of Terms We begin by defining the terms central to the theme of this paper. Measurement: the act or process of measuring. The process may be as simple as reading a thermometer or as complex as all the operations required for a chemical analysis. Precision: reproducibility. A measurement is precise if it is close to other values obtained by repeating the determination using the same procedure. Accuracy: correctness. A measurement is accurate if it is close to the true value. Since the true value of a quantity can never be exactly known, an accepted value is commonly used as the test of accuracy. Error: anything that causes a measurement to differ from the true value. The amount by which the measured value differs from the true value is also called the error. Model Experiment To illustrate the difference between precision and accuracy let us imagine a laboratory exercise in which four graduated cylinders are used to measure the volume of a known quantity of water.1
Four Graduated Cylinders The four graduated cylinders are illustrated in Figure 1. Suppose each has been filled with exactly 3.420 mL of water. Cylinders A and C have 5-mL capacities and graduation marks at each 0.1 mL. They can be read to the hundredths place by estimating the position of the meniscus between the graduations. Cylinders B and D have 25-mL capacities and 992
graduations at each 1 mL. They are readable to the tenths place. A student well trained in the use of graduated cylinders is asked to make five independent readings of the volume in each cylinder. The data obtained would be of the sort listed in Table 1. All the measured values have some uncertainty in the last digit written, since it is an estimate. Table 1 also gives the arithmetic mean of the five values for each cylinder. Let us now judge the performance of each cylinder. The measured values of cylinder A are both precise and accurate. They are precise in that they agree well with each other, and they are accurate in that they are in good agreement with the true value. Cylinder B gives results less precise than those of cylinder A because it is readable only to the tenths place. Its large size makes cylinder B a poor choice for measuring such a small volume. If we make allowance for its imprecision, cylinder B can be regarded as accurate. The results of cylinder C are as precise as those of cylinder A, but they are considerably less accurate. Examination of the cylinder reveals it is inaccurate because of a manufacturing flaw. An extraneous bead of glass is fused at its bottom. This flaw causes all readings to be high. Finally, cylinder D gives results that are both imprecise and inaccurate. It is too large for the task at hand, and it, too, suffers from a glassbead manufacturing flaw.
Expressions of Precision and Accuracy Precision and accuracy can be expressed quantitatively in various ways. The simplest expression of the precision of a series of measurements is its range, defined by the formula range = maximum value – minimum value A small range indicates the measurements are precise. A more statistically meaningful expression of precision is the standard deviation of the series. Its formula is
s=
Σ
xi – x
2
n –1
where the xi are the individual measured values, ¯x is the mean, and n is the number of measured values in the series. Standard deviation measures how closely the individual values are clustered around the mean. If the measurements follow a normal distribution curve, 68% of all values will fall within the interval x¯ ± s. A small standard deviation indicates greater precision. Table 1 lists the range and standard deviation of the measurements for each cylinder. Accuracy is commonly expressed by error, defined by the formula error = measured value – true value Table 1 lists the error of the mean for each cylinder. In the next section the error of the mean will be given the more
Journal of Chemical Education • Vol. 75 No. 8 August 1998 • JChemEd.chem.wisc.edu
In the Classroom
specific term systematic error. The table also lists the percent error for each mean.
Random and Systematic Errors Errors are commonly classified according to how consistently they occur. A systematic or determinate error is one that occurs consistently each time the measurement is repeated. Such an error can be detected by running a calibration standard. The glass bead that causes cylinder C to be inaccurate is a systematic error. It can be removed from the cylinder by grinding, or we can simply adjust for it by subtracting a correction factor from all readings. In general, systematic errors can be corrected by modifying the measuring technique. Regrettably, some inaccuracy will accompany every correction effort. A random or indeterminate error is one that occurs inconsistently each time a measurement is repeated. Such an error causes replicate measurements to differ in spite of our best efforts to repeat them in exactly identical fashion. The imprecision of cylinder B is caused by random errors that arise because of the uncertainty in estimating the digit in the tenths place. Corrective action can sometimes be taken to reduce random errors. For example, a magnifying lens could be attached to cylinder B or better illumination could be provided to improve our viewing of the meniscus. Beyond such instrument improvements, random errors can be reduced by making replicate measurements and averaging the results. If the measurements strictly follow a normal distribution, taking their average will eliminate random errors entirely. Figure 2 illustrates the various terms introduced in this and the previous section. Twelve measured values of a quantity are plotted as dots on a horizontal scale. Their mean, range, and standard deviation are shown. The true value of the quantity is also plotted. Each measured value has a random
error, which is its difference from the mean. The mean has a systematic error, which is its difference from the true value. The sum of the random and systematic error for each measurement is simply its error. In general, random errors cause a set of measurements to be imprecise, and systematic errors cause their mean to be inaccurate. Problems in Classifying Errors The classification of errors as random or systematic is not always as simple as our model experiment might imply. An error can have both a random and a systematic component. Two examples will illustrate this point. The meniscus in a graduated cylinder appears at different heights as the viewing angle changes. To minimize this parallax error the meniscus should be viewed exactly horizontally. As shown in Figure 3, the error will be random if the sighting is randomly too high or too low, but it will be systematic if the sighting is consistently from above. A second example occurs in gravimetric methods of analysis. If the precipitate is not properly dried, its weight will be too high. While the error may be primarily systematic, it will surely have a random component because the moisture content of the precipitate will vary with each determination. We see that random and systematic errors are not fundamentally different. Their distinction lies in the details of the measuring procedure. In spite of these complications, errors can be roughly classified. Examples of those that are primarily random include (i) uncertainty in estimating the position of a meniscus or the needle on a meter, (ii) inhomogeneity in the test sample taken for a chemical analysis, and (iii) fluctuations in the digital readout of an instrument due to power supply irregularities. In contrast, examples of errors that will be mostly systematic are (i) absorption of moisture on the surface of a weighing
Table 1. Data from Graduated Cylinders Illustrated in Figure 1 Cylinder
Figure 1. Graduated cylinders of the model experiment.
Measured Mean/ Value/ mL a mL
Precision
Accuracy
Standard Range/ Deviation/ mL mL
Error/ mL
Percent Error
0.06
A
3.42 3.43 3.41 3.44 3.41
3.422
0.03
0.013
0.002
B
3.5 3.3 3.4 3.3 3.4
3.38
0.2
0.084
᎑0.04
C
3.67 3.65 3.64 3.68 3.65
3.658
0.04
0.016
0.238
D
4.2 4.1 4.3 4.3 4.1
4.20
0.2
0.100
0.78
aEach
᎑1.2
6.96
22.8
cylinder contains exactly 3.420 mL.
JChemEd.chem.wisc.edu • Vol. 75 No. 8 August 1998 • Journal of Chemical Education
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In the Classroom
vessel, (ii) the premature color change of an indicator before the equivalence point of a titration, (iii) chemical interference from an extraneous substance in the test sample, and (iv) the slight solubility of a precipitate in a gravimetric analysis.
Knowing the True or Accepted Value The true value must be known if we are to calculate the error in an individual measurement or the systematic error of a mean. The true value is also required for instrument calibration. Various approaches can be used to establish a value acceptable as true for student laboratory experiments. A wide range of test materials with certified chemical or physical properties can be obtained from the National Institute of Science and Technology.2 Unknowns for analysis can also be purchased from commercial sources, or they can be simply prepared by the instructor by mixing reagent grade substances. For experiments whose purpose is to measure a physical constant or some property of a substance it is usually possible to find an accepted value in the literature.
Classroom Recommendations The distinction between precision and accuracy can be used to increase students’ awareness of experimental errors. The terms should be used correctly in laboratory conversations. For example, when we require students to read a buret to the nearest hundredth of a milliliter, our intent is to improve the precision of their work. When we urge them to thoroughly clean a pipet so it will drain properly, the objective is to improve their accuracy. General chemistry experiments often require replicate determinations of some unknown quantity. Such experiments provide an opportunity to reinforce the theme of this paper. In written laboratory reports, students may be asked to: 1. Determine the precision of your results by calculating their range and standard deviation. 2. List the errors that cause imprecision in your data, comment on their relative importance, and suggest ways of reducing them.
If the true or accepted value of the measured quantity is known, the discussion can be extended to include the accuracy of the results and the errors associated with inaccuracy. Finally, a useful problem for generating classroom discussion is: Two students are assigned the task of determining the temperature of a classmate with a fever. Jane uses a single thermometer three times and reports an average. Bill uses three different thermometers once each and reports an average. Discuss the precision, accuracy and errors of each student’s results. Whose result would you believe?
Our goal throughout these discussions is to develop critical thinking skills and a healthy respect for the limitations of scientific measurements. Summary The theme of this paper can be summarized by two concept charts. Figure 4 explains how precision and accuracy differ and how they are expressed quantitatively. Figure 5 illustrates how errors are classified as random or systematic and how the effect of each can be minimized. Notes Figure 2. Illustration of terms for expressing precision, accuracy and error.
1. Although the graduated cylinder exercise is described here as only a thought experiment, an innovative teacher can modify it into an actual experiment or a classroom demonstration. A reliable procedure for filling the cylinders with an exactly known quantity of water would be to calculate the mass required from the density of water at the laboratory temperature, and then weigh out the required mass with an analytical balance. The “manufacturing flaws” required in cylinders C and D can be created by placing a few drops of clear epoxy at the bottom of each. 2. Formerly the National Bureau of Standards. Inquiries can be directed to the National Institute of Standards and Technology, Standard Reference Materials Program, Gaithersburg, MD 20899-0001.
Literature Cited
Figure 3. Random and systematic errors caused by parallax.
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Journal of Chemical Education • Vol. 75 No. 8 August 1998 • JChemEd.chem.wisc.edu
In the Classroom
Figure 4. Concept chart for contrasting precise and accurate measurements.
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Clever, H. L. J. Chem. Educ. 1979, 56, 824. Schwartz, L. M. J. Chem. Educ. 1985, 62, 693. Earl, B. L. J. Chem. Educ. 1988, 65, 186. Graham, D. M. J. Chem. Educ. 1989, 66, 573. Thompson, H. B. J. Chem. Educ. 1991, 68, 400. Thomas, R. J. Chem. Educ. 1991, 68, 409. Sen, B. J. Chem. Educ. 1977, 54, 468. Anderlik, B. J. Chem. Educ. 1980, 57, 591.
Figure 5. Concept chart for contrasting random and systematic errors.
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