### 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.*