1.10:
Uncertainty in Measurement: Reading Instruments
Measurements from defined quantities or the direct counting of distinct objects give exact numbers. For example, a dozen bananas is exactly 12 bananas. Similarly, apples in a carton can be directly counted.
However, other measurements are not exact, and have uncertainty associated with them due to limitations in the measurement process.
Consider a runner competing in a 5 kilometer race. The judges record the finish time using a simple analog watch and a digital watch.
The analog watch reads the finish time as between 25 and 30 minutes, so, the judges estimate the time to be 28 minutes. Only the first digit, ‘2’, is certain. The last digit, ‘8’ is an estimate between 25 and 30, and is thus uncertain.
The digital watch, on the other hand, read 26.25 minutes, with full certainty in the first three digits. The finish time could vary between 26.24 and 26.26 minutes, but is certainly not 28 minutes.
In a laboratory setting, a digital balance may display a reading of up to six digits. A reading of 10.1241 grams indicates that the first 5 digits are certain, but the last digit is uncertain. However, all the numbers displayed on a digital instrument are to be reported.
In contrast, when reading analog instruments, the last digit must be estimated. A 50-mL graduated cylinder may have markings every one milliliter. The measurement read is dependent on estimating the volume of liquid at the meniscus, the lowest point on the curved surface of the liquid in the graduated cylinder.
If the meniscus lies between the 42 and 43 markings, the quantity of liquid is more than 42 mL but less than 43 mL. The measurement should be estimated between the two markings; therefore, the volume should be read to the nearest 0.1 mL.
In this case, the meniscus is closer to the 43-mL mark, therefore the number could be reported as 42.8. Another person could estimate the volume as 42.7 or 42.9 as the last digit is uncertain, but the measurement should always be reported to the last uncertain digit.
1.10:
Uncertainty in Measurement: Reading Instruments
Counting is the type of measurement that is free from uncertainty, provided the number of objects being counted does not change during the process. Such measurements result in exact numbers. By counting the eggs in a carton, for instance, one can determine exactly how many eggs are there in the carton. Similarly, the numbers of defined quantities are also exact. For example, 1 foot is exactly 12 inches, 1 inch is exactly 2.54 centimeters, and 1 gram is exactly 0.001 kilograms. Quantities derived from measurements other than counting, however, are uncertain due to practical limitations of the measurement process used.
Significant Figures
Every measurement has some uncertainty, which depends on the device used (and the user’s ability). For instance, the volume of liquid in a graduated cylinder is measured by reading the bottom of the meniscus — the lowest point on the curved surface of the liquid. Suppose the bottom of the meniscus lies between the 15 and 16 markings; it means the liquid volume is certainly greater than 15 mL but less than 16 mL. The meniscus appears to be a bit closer to the 16-mL mark, and so a reasonable estimate of the liquid’s volume would be 16.6 mL. In this measured value, the digits 1 and 6 are certain, but the last digit at the tenths place, 6, is an estimate. Some people might estimate the meniscus position to be equally distant from each of the markings and estimate the tenth-place digit as 5, while others may think it to be even closer to the 16-mL mark and estimate it to be 7. The numerical scale on this graduated cylinder has 1-mL divisions; thus, volumes may be measured to the nearest 0.1 mL. Similarly, a standard electronic balance may read the mass of a quarter as 5.74 g. The digits 5 and 7 are certain, and the 4 indicates that the mass of the quarter is likely between 5.73 and 5.75 grams. The quarter weighs about 5.74 grams, with a nominal uncertainty in the measurement of + 0.01 grams. If the coin is weighed on a more sensitive balance, the mass might be 5.743 g. This means its mass lies between 5.742 and 5.744 grams, an uncertainty of 0.001 grams.
This text is adapted from Openstax, Chemistry 2e, Section 1.5: Measurement Uncertainty, Accuracy, and Precision.