Understanding Accuracy and Precision in Analytical Measurements, Exercises of Laboratory Practices and Management

Download Understanding Accuracy and Precision in Analytical Measurements and more Exercises Laboratory Practices and Management in PDF only on Docsity! Accuracy, Precision and Analytical Measurements What are accuracy and precision? Accuracy is how close a measurement is to its desired or theoretical value. For example, if we need to dispense 25.0 mL of dilute HCl, then dispensing 24.9 mL is more accurate then dispensing 25.7 mL. Accuracy usually is reported as a percent error % error = actual value − expected value expected value × 100 which, for the two examples cited above, are 24.9−25.0 25.0 × 100 = −0.4% and 25.7−25.0 25.0 × 100 = +2.8%. Note that an error that affects accuracy is either positive or negative. Precision is the reproducibility of a set of measurements. Three identically prepared solutions with pH values of 6.76, 6.73, and 6.78, for example, are more precise than a duplicate set with pH values of 6.76, 6.54, and 6.92. Precision usually is reported as a standard deviation, s, which we define as s = √∑ (xi − x̄)2 n − 1 where x̄ is the average, or mean result, and xi is one of the n different results. If you closely examine this equation you will see that a standard deviation essentially is the “average” deviation of the individual measurements from their mean value.1 Note, as well, that squaring the term in the numerator guarantees that the standard deviation is always positive. As an example, the mean pH for the measurements 6.76, 6.73, and 6.78 is 6.76 + 6.73 + 6.78 3 = 6.757 and the standard deviation is s = √ (6.76 − 6.757)2 + (6.73 − 6.757)2 + (6.78 − 6.757)2 3 − 1 = 0.0252 Alternatively, we can express the standard deviation as the percent relative standard deviation, sr, or rsd sr = s x̄ × 100 For the example above, the relative standard deviation is 0.0252 6.757 × 100 = 0.373%.2 1Note that the standard deviation is not a true average because we divide the numerator by n − 1 instead of by n. The reason for this is not important to us at this time. 2Although you can calculate the mean and the standard deviation by hand, it is inconvenient to do so for a large data set. All scientific calculators and spreadsheets have the ability to calculate the mean and the standard deviation; learn how to do so with your calculator and/or favorite spreadsheet. 1 Figure 1: Clusters of five rifle shots illustrating the difference between accuracy and precision. Is a pH of 6.76 both more accurate and more precise than a pH of 6.8? Good question. It is tempting to say that a pH of 6.76 is more accurate than a pH of 6.8 because it contains more significant figures, but this is not necessarily correct. In fact, if the instrument used to measure the pH is not calibrated, then neither pH reading is accurate. Regardless of its accuracy, we can say that a pH of 6.76 is known more precisely to us than a pH of 6.8 because the absolute uncertainty for the first measurement is ±0.01 while that for second is ±0.1.3 If the pH meter is calibrated properly, then a more precise measurement can lead to a smaller percentage error and, consequently, to better accuracy. If a measurement is accurate, must it also be precise? Interestingly, the answer to this question is no. As we see in Figure 1, there are four possible combinations of accuracy and precision. The target at the far left shows both accuracy and precision as the shots are clustered together (they are precise) in the target’s center-most ring (they are accurate). The next example shows results that are precise, due to a tight clustering of the shots, but inaccurate because they are at the target’s outer edge instead of its center. The third example is considered accurate because the five shots cluster around the target’s center, but they are not precise because the individual shots are quite far apart from each other. The final example shows a dispersion of shots that is both inaccurate and imprecise. Note that the average for a set of measurements may be accurate even if the individual measurements deviate significantly from the desired or theoretical value. What factors affect accuracy and precision? Three main factors affect the accuracy and the precision of a measurement: the quality of the equipment we use to make the measurement, our ability to calibrate the equipment, and our skill using the equipment. These factors are considered further in this section. We cannot make an accurate measurement if our equipment is not calibrated properly. To calibrate equipment we analyze a system where the response is known to us and either adjust the equipment to give that response or determine the mathematical relationship between the measured result and its known value. The two examples of accurate target shooting in Figure 1—the target on the far left and the target second from the right—require that you calibrate the rifle’s scope so that an accurate result is possible. In addition, the potential accuracy of any individual measurement is greater with better quality equipment or instrumentation; the better the scope, the closer each shot is to the target’s center. Precision, on the other hand, is influenced by both the quality of the equipment and the skill of the person using it. The importance of the user’s skill is obvious; when shooting a rifle, for example, you must have a steady hand to achieve a tight, precise pattern of shots. Although less obvious, the quality of the equipment 3We also can think about this in terms of relative uncertainty where 6.76 has a relative uncertainty of 1 part in 676, or 0.148%, and where 6.8 has a relative uncertainty of 1 part in 68, or 1.5%. 2