The exactness of every measurement can be described in terms of accuracy and precision. **Accuracy** is how close a measurement is to a value that is widely accepted to be the true value. For example, if an 11-cm length of rope is measured with an instrument that reports the value as 12 cm, the measurement is not very accurate. The **true value** is the value that would be obtained in an ideal measurement. **Precision** is the degree to which a measurement can be reproduced. If that same 11-cm rope is measured three times, and the three readings are 10 cm, 11 cm, and 12 cm, the measurements are not very precise.

**significant figures**, which are the number of digits in a measurement that are certain, plus one digit that contains some uncertainty. The number of significant figures of a measurement provides information about the amount of precision the measurement has. In the example with the rope, the measurement is precise to within 1 cm.

Values that contain leading and trailing zeroes have a specific format when significant figures are specified. Any placeholder zeros that come immediately before or after the decimal point are not considered to be significant figures. Any zeros that come at the end of a measurement that includes a decimal value are significant. Therefore, the measurement 0.01200 m has four significant figures. If a whole number ends in zero, as in the measurement 1,000 m, placing an otherwise unnecessary decimal point at the end of the number indicates that the measurement has four significant figures: 1,000. m. But what if that measurement has less than four significant figures? In that case, the solution is to use **scientific notation**, which is a method of expressing numbers as a coefficient greater than or equal to 1 but less than 10, multiplied by 10 raised to the appropriate power. For example, $1.0\times{10}^3\;\rm{ m}$ has two significant figures, and $1.00\times{10}^3\;\rm{ m}$ has three. Both are equal to 1,000 meters, but the certainty of the measurements differs, reflecting the precision of the device used to take the measurement.

### Error in Measurements

Significant figures provide information about the precision of a measurement. The precision of a measurement depends on the tool used to make the measurement. Consider the example of an 11-cm rope. If the length of the rope were measured with a meterstick marked in 1-cm increments, it is possible to mentally divide the space between markings and estimate the true length to within a fraction of a centimeter, perhaps to within 0.5 cm. In this case, the measurement should be reported as 11.0 ± 0.5 cm. Including the margin of error, ± 0.5 cm, makes it clear that the measurement is precise to within 0.5 cm. In other words, the measurement might be 10.5 cm, 11.0 cm, or 11.5 cm. If it is possible to estimate the measurement to within 0.1 cm, the measurement should be reported as 11.0 ± 0.1 cm.

On the other hand, if the rope were measured with a meterstick marked off in 1-mm increments, the user would be able to estimate the rope length down to a fraction of a millimeter, and a proper reading would require an additional significant figure: $110.5\pm0.5\;\rm{ mm}=11.05\pm0.05\;\rm{ cm}$. This difference in precision is called the precision of the instrument, and any error introduced by the precision of instruments is independent of the user of the instrument.

A **systematic error** is a consistent error in measurements that leads to a precise but inaccurate measurement. This type of error is analogous to darts tightly clustered around a spot that is not the bullseye. For example, a systematic error is the type that would result from a thermometer that is calibrated incorrectly so that it always provides a reading lower than the true temperature. An error caused by an unknown or unpredictable source, such as an experimenter making a mistake, is called a **random error**.

### Common Laboratory Tools

#### Common Laboratory Glassware

**meniscus**. If the molecules of the liquid are more strongly attracted to one another than to the glass, which is the case with mercury, the meniscus is convex, like an upside-down U. Laboratory glassware is calibrated so that an accurate reading of the volume occurs at the center of the meniscus (the lowest point if the meniscus is concave or the highest point if it is convex). When making a measurement, it is also important to be sure that the meniscus is held at eye level so that the angle of viewing does not skew the reading.

#### Reading the Meniscus

**balance**, which measures mass. Many types of balances are available with a wide range of precisions. A triple-beam platform balance typically has a precision of ±0.1 g, and an expensive digital balance can have a precision of ±0.0001 g.

### Calculations with Significant Figures

Note that when a calculator is used to find the volume, the result is 438.7041. Because the answer is limited to three significant figures, it must be rounded. The general rule is to increase the last digit by 1 if the first digit dropped is 5 or greater and to keep the last digit the same if the first digit dropped is less than 5. For example, if a result should have only three significant figures, 438.7 rounds to 439, and 438.4 would round to 438.