Introductory Chemistry: A Foundation - Zumdahl S.S., DeCoste D.J. 2019

Measurements and Calculations
Uncertainty in Measurement


· To understand how uncertainty in a measurement arises.

· To learn to indicate a measurement’s uncertainty by using significant figures.

When you measure the amount of something by counting, the measurement is exact. For example, if you asked your friend to buy four apples from the store and she came back with three or five apples, you would be surprised. However, measurements are not always exact. For example, whenever a measurement is made with a device such as a ruler or a graduated cylinder, an estimate is required. We can illustrate this by measuring the pin shown in Fig. 2.5(a). We can see from the ruler that the pin is a little longer than cm and a little shorter than cm. Because there are no graduations on the ruler between and , we must estimate the pin’s length between and cm. We do this by imagining that the distance between and is broken into equal divisions [Fig. 2.5(b)] and estimating to which division the end of the pin reaches. The end of the pin appears to come about halfway between and , which corresponds to of our imaginary divisions. So we estimate the pin’s length as cm. The result of our measurement is that the pin is approximately cm in length, but we had to rely on a visual estimate, so it might actually be or cm.

Uncertainty in Measurement

Andrew Lambert Photography/Science Source

A student performing a titration in the laboratory.

Figure 2.5.A set of two illustrations shows, labeled a and b. The first illustration shows a small portion of a ruler. A pin is placed such that the head of the nail is exactly below the front end of the ruler. A vertical dashed line at the end of the pin is marked between 2.8 and 2.9 centimeters on the ruler. In the second illustration, the point at which the dashed line touches the ruler is zoomed to show 2.85 centimeters.

Measuring a pin.

Because the last number is based on a visual estimate, it may be different when another person makes the same measurement. For example, if five different people measured the pin, the results might be


Result of Measurement











Note that the first two digits in each measurement are the same regardless of who made the measurement; these are called the certain numbers of the measurement. However, the third digit is estimated and can vary; it is called an uncertain number. When one is making a measurement, the custom is to record all of the certain numbers plus the first uncertain number. It would not make any sense to try to measure the pin to the third decimal place (thousandths of a centimeter) because this ruler requires an estimate of even the second decimal place (hundredths of a centimeter).

It is very important to realize that a measurement always has some degree of uncertainty. The uncertainty of a measurement depends on the measuring device. For example, if the ruler in Fig. 2.5 had marks indicating hundredths of a centimeter, the uncertainty in the measurement of the pin would occur in the thousandths place rather than the hundredths place, but some uncertainty would still exist.

The numbers recorded in a measurement (all the certain numbers plus the first uncertain number) are called significant figures . The number of significant figures for a given measurement is determined by the inherent uncertainty of the measuring device. For example, the ruler used to measure the pin can give results only to hundredths of a centimeter. Thus, when we record the significant figures for a measurement, we automatically give information about the uncertainty in a measurement. The uncertainty in the last number (the estimated number) is usually assumed to be unless otherwise indicated. For example, the measurement kilograms can be interpreted as kilograms, where the symbol means plus or minus. That is, it could be or .