# Accuracy, Precision, and Uncertainty

Accuracy and precision are characteristics that describe the exactness of a measurement, or closeness of the true value of the measured quantity.

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.

It is because all measurements contain some inherent error that scientists must take multiple readings in order to get as close as possible to the true and exact value of a measurement. By taking multiple readings of a single measurement, scientists can take an average of all the readings, which reduces the effect of the error contributed by a lack of precision. For example, the average of the three readings of the 11-cm rope is the true length of the rope.
$\text{average}=\;\frac{10\;\rm{cm}+11\;\rm{cm}+12\;\rm{cm}}3=11\;\rm{cm}$
Because all measurements contain uncertainty, scientific measurements are reported with a certain number of 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.

Significant figures do not provide any information about the accuracy of a measurement. Remember that a measurement is accurate if it is close to the true value and that a measurement is precise if it is close to other measurements of the same thing. One way to remember the difference in precision and accuracy is to think about a dartboard with several darts thrown at it. If all darts hit the dartboard near one another, they have been thrown with precision, but if the cluster of darts is far from the bullseye, they have not been thrown with accuracy. Conversely, if the darts are spread out on the board but largely centered around the bullseye, the average location of the hits is close to the bullseye and is accurate, but the individual darts have been thrown with low precision.

### Error in Measurements

Measurements contain inherent error that must be reported.

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

Different laboratory tools have different levels of precision.
In a general chemistry lab, several different kinds of glassware are used for measuring and containing liquids used in experiments. Specific types of glassware are designed to measure or dispense precise volumes of liquid, including graduated cylinders, burets, pipettes, and volumetric flasks. Other types of glassware are used primarily to hold liquids and are not used for measuring volumes. These include a beaker and an Erlenmeyer flask. Although these types of glassware typically have graduated volume markings, the markings are approximate.

#### Common Laboratory Glassware

The surface of a liquid contained in a thin glass cylinder is not flat. Instead, liquids form either concave or convex curves at the surface due to forces within the liquid and forces between the liquid and the glass. The surface of the liquid forms a U shape if the molecules of the liquid are more strongly attracted to the glass than to one another. This curved surface of a liquid in a thin tube is called the 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.

Another common tool in a chemistry lab is a 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

The number of significant figures to report depends on the type of calculation being performed.
When calculations are performed with measurements, the resulting calculation should neither gain nor lose precision compared with the experimental measurements. This can be achieved by following two rules. First, when measurements are added or subtracted, the result should have the same number of decimal places as the measurement with the fewest decimal places. For example, consider the following sum.
\begin{aligned}202.1&\rm\;{mL}&\text{Least precise number; only one digit after the decimal}\\\;50.214&\rm\;{mL}&\\\;2.01&\;\rm{ mL}&\\\overline{254.324}&\rm\;{mL}\end{aligned}
After rounding, the answer is 254.3 mL. Multiplication and division are handled differently. When the calculation uses multiplication or division, the result should have the same number of significant figures as the least precise measurement in the calculation. This is the value with the fewest significant figures. For example, consider the following calculation of a volume.
$\begin{gathered}\begin{gathered}10.25\;\rm{cm}\\4\;\text{significant}\;\\\text{figures}\end{gathered}&\times&\begin{gathered}3.24\;\rm{cm}\\3\;\text{significant}\;\\\text{figures}\end{gathered}&\times&\begin{gathered}13.21\;\rm{cm}\\4\;\text{significant}\;\\\text{figures}\end{gathered}&=&\begin{gathered}439\;\rm{cm}^3\\3\;\text{significant}\;\\\text{figures}\end{gathered}\end{gathered}$
The answer can only have three significant figures because the second measurement only has three.

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.