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.