To lessen the uncertainty in their measurements, scientists typically take repeated observations of a given measurement (known as replicate measurements, or replicates). Experimental error cannot always be removed, but it can be characterized and minimized by averaging across several measurements. The kind of uncertainty in a measurement can be described by looking at the accuracy and precision of the value.
Accuracy of Measured Values
The accuracy of a measurement shows how near the measurement is to the "true" value. In some cases (such as when you measure a sample whose concentration is unknown), this "true" value is unknown, so you will estimate your accuracy from other measurements (a titration, for instance).
Expressing Accuracy
When the "true" value of a measurement is known, the accuracy of a measurement can be expressed through its percent error:
% error = [(measured value − true value) / true value] × 100%
Percent error may be positive (the measured value is above the "true" value) or negative (the measured value is below the "true" value). In general, a larger % error, whether positive or negative, indicates a less accurate measurement.
How much error is acceptable in your reported value depends on what kind of measurement it is. In many first-year lab experiments, an accuracy of ~5% may be expected given the experimental design and the equipment used. A similar measurement in a second-year analytical lab may be expected to have an accuracy of less than 0.5%.
Precision of Measured Values
The precision of a measurement describes how reproducible the value is: if you took this measurement again, how similar would the two measurements be? One way to think about precision is as the spread or deviation of the measurements around their central (average) value. If we return to the "bullseye" example above:
The hits on Targets 1 and 2 are relatively precise: the "hits" are close together, and the dart thrower lands on nearly the same spot each time. Targets 3 and 4 show results that are less precise: the hits are more scattered and cover a wider range of values on the target.
Even though the term precise is used in various ways in everyday English, in chemistry precision, or precise data, refers specifically to the spread and repeatability of measurements, since accuracy is described as a separate concept.
Data can be accurate AND precise at the same time (as in Target 1 above), precise but not accurate (as in Target 2 above), accurate but not precise (as in Target 3 above), or neither accurate NOR precise (as in Target 4 above).
Expressing Precision
The precision of a measurement can be expressed numerically in several ways: by using significant figures, by calculating the standard deviation of a data set, or by applying an error limit (or "error range") to indicate the variability in the data.
Just as with accuracy, what counts as an acceptable value for the precision of your data depends on the kind of measurement being taken. For instance, a typical pH meter will only provide 2 or 3 significant figures in a result (i.e. pH = 2.02 ± 0.01), whereas using a volumetric flask correctly can yield 4 or 5 significant figures in the final answer (e.g. 100.0 ± 0.1 mL).
Significant Figures
Every numerical value carries an implied level of precision that depends on the number of digits recorded. For instance, you might measure a beaker's mass as 124.758 g, or purchase a USB charging cable listed as 2.0 m.
According to the conventions for reporting significant figures in our numbers, we would take these reported measurements to be repeatable to within ±1 of the last digit reported. Thus, if I weighed the beaker again, I would expect a reading between 124.757 and 124.759 g, or if I measured the USB cable, a value between 1.9 and 2.1 m.
A value that can be (correctly) written with more significant figures is more precise. More significant figures in a value indicate that its variation is small relative to its overall magnitude (for example, the 2.0 m USB cable could measure anywhere from 190 cm to 210 cm. A 2.00 m cable would imply the cable should fall within 199–201 cm, a more precise value, since there is less "spread" across the possible measurements).
Standard Deviation
The spread or scatter of experimental data around its average value can be described by the standard deviation (also known as statistical precision), s. Standard deviation can be computed for any data set containing 3 or more data points:
s = √[ Σᵢ (Mᵢ − M̄)² / (n − 1) ]
Where:
- Mᵢ refers to a single measurement or observed value
- M̄ is the mean (the arithmetic average of all observed values)
- n stands for the total count of measurements
- Σᵢ stands for "sum of": here it means that all the available values of the following should be summed
- (Mᵢ − M̄)² is the squared difference between each measurement and the mean
A higher standard deviation means that the measurement set has lower precision (the measurements show a greater "spread").
