1.9: Uncertainty in Measurement: Accuracy and Precision

Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to evaluate both the precision and the accuracy of their results. Measurements are said to be precise if they yield very similar results when repeated in the same manner. A measurement is considered accurate if it yields a result that is very close to the true or the accepted value. Precise values agree with each other; accurate values agree with a true value.

Suppose a quality control chemist at a pharmaceutical company is tasked with checking the accuracy and precision of three different machines, meant to dispense 500 mL of cough syrup into storage bottles. The chemist proceeds to use each machine to fill five bottles and then carefully determines the actual volume dispensed, as reported in Table 1.

Considering these results, the chemist reported that dispenser #1 is precise but not accurate. All the values from dispense #1 are close to each other, but none of the values are close to the target value of 500 mL. Results for dispenser #2 showed improved accuracy (values are close to 500 mL) but worse precision (not close to one another). Finally, the chemist reported that dispenser #3 is working well, and it is dispensing cough syrup both accurately (all volumes are within 0.2 mL of the target volume) and precisely (volumes differ from each other by no more than 0.2 mL).

Highly accurate measurements tend to be precise, too. However, highly precise measurements may not necessarily be accurate. For example, an improperly calibrated thermometer or a faulty weighing balance may give precise readings that are inaccurate.

Random and Systematic Errors

Scientists always try their best to record their measurements with the utmost accuracy and precision. However, sometimes errors do occur. These errors may be random or systematic.

Random errors are observed due to the inconsistency or fluctuation in the measurement process or variations in the quantity itself that is being measured. Such errors fluctuate from too high or too low from the true value in repeated measurements. Consider a scientist measuring the length of an earthworm using a ruler. Random error in this measuring process might be the result of the inconsistent method in which the scientist reads the scales, or if the earthworm isn’t still and its body movements might pose difficulty in taking correct length measurements. Random error cannot be avoided; however, it can be averaged out with repeated trials.

Systematic errors arise from a persistent issue and result in a consistent discrepancy in measurement. These errors tend to be consistently either too high or too low from the true value. These are predictable and are mostly instrumental in nature. For instance, an improperly calibrated weighing balance may consistently weigh objects heavier than their true value. However, unlike random error, systemic errors cannot average out with repeated measurements.

This text is adapted from Openstax, Chemistry 2e, Section 1.5: Measurement Uncertainty, Accuracy, and Precision.

Scientists make repeated measurements of a quantity during experimentation to ensure that their results are accurate and precise.

The accuracy of a measurement is the degree of closeness of the results to the true or accepted value.

Consider two students, A and B, who repeatedly weighed a gold bar known to have a true mass of 10 grams. Student A reported three values － 9.5 grams, 10 grams, and 10.5 grams, while student B reported masses of 8.5 grams, 8.6 grams, and 8.5 grams. Student A reported values closer to the true mass of the bar compared to student B. Thus measurements by student A were, therefore, more “accurate”.

Precision, on the other hand, is the measure of how closely the results agree with each other, or how reproducible they are. A measurement is said to be precise if it gives highly similar results when repeated under the same conditions. For instance, the values for the mass of the gold bar reported by student B were very similar to one another, as compared to student A. That is “precision”.

Accuracy and precision are two distinct qualities of measurement which are independent of each other. Thus, a particular set of measurements can be either accurate, or precise, or neither, or both.

The measurements for the mass of the gold bar by student A were more accurate, close to the true value of 10 grams, but not precise, as they were not close to each other. On the contrary, the measurements by student B were precise, but not accurate.

Highly accurate values tend to be precise too. Like a weighing balance showing true or close to true masses for all the objects, repeatedly. However, highly precise measurements may not necessarily be accurate; if the same balance is improperly calibrated, it may give precise but inaccurate readings. This may lead to scientific errors. 

Errors in the measurement process is a common problem. Such errors may fall into two categories － random and systematic. 

Random errors are the result of inconsistency in the measuring process or variations in the quantity being measured. These result in fluctuations, too high or too low, around the true value. Consider a scientist measuring the length of an earthworm using a caliper. Inconsistency of the scientist to read the scales correctly, or continuous body movement of the earthworm during the measurement, may result in incorrect length measurements. Random error cannot be avoided, however, it can be averaged out with repeated trials.

Systematic errors are the results of a persistent issue and lead to a consistent discrepancy in measurement. These errors tend to be either all too high or all too low compared to the true value. For instance, weights being measured using an improperly calibrated weighing balance. These are predictable and mostly instrument-related. However, unlike random error, it cannot be averaged out with repeated measurement.