When conducting experiments in chemistry, every measurement taken has a degree of imperfection, meaning the exact true value is unattainable. This limitation is known as measurement uncertainty, which describes the range of values within which the actual quantity is expected to reliably lie. Calculating this range is fundamental because it reflects the quality and reliability of any experimental result derived from those initial readings. By quantifying uncertainty, scientists can properly compare their findings with theoretical models or results obtained by other laboratories, ensuring that conclusions drawn from chemical data are scientifically justifiable and robust.
Assigning Initial Uncertainty to Measurements
Before mathematical operations, the initial uncertainty inherent in each piece of raw data must be determined. This initial value depends entirely on the type and precision of the measuring instrument used. Instruments are categorized as either analog or digital, and each follows a different convention for assigning this error.
Analog instruments, such as a glass burette or a standard ruler, require the user to visually estimate a reading between marked lines. For these devices, the accepted uncertainty is estimated as half of the smallest marked increment on the scale. For example, if a burette has lines marked every \(0.1\) milliliters, the reading uncertainty would be \(\pm 0.05\) milliliters due to the need for visual estimation.
Digital instruments, like an electronic balance or a pH meter, provide a direct numerical readout. The uncertainty for these devices is taken as the smallest increment the instrument can display or is provided by the manufacturer’s specifications. A digital balance that reads to \(0.01\) grams, for instance, has an uncertainty of \(\pm 0.01\) grams in its measurement.
Propagating Uncertainty Through Addition and Subtraction
Once individual uncertainties are assigned, the next step involves calculating how these errors combine when measurements are added or subtracted. This procedure is known as uncertainty propagation, and it determines the overall error in the final calculated value. When combining two independent measurements, their absolute uncertainties do not simply add linearly, as this simpler method would often overestimate the total error.
The standard statistical approach uses the root sum of squares (RSS) method to determine the overall absolute uncertainty. This method accounts for the fact that individual errors are equally likely to cancel each other out as they are to compound, providing a more realistic estimate. The total uncertainty is found by squaring each individual absolute uncertainty, summing the squares, and then taking the square root of that final sum.
Consider the calculation of mass lost, determined by subtracting a final mass (\(M_{final}\)) from an initial mass (\(M_{initial}\)). If the initial mass is \(5.00 \pm 0.01\) grams and the final mass is \(3.50 \pm 0.01\) grams, the total uncertainty is calculated by applying the RSS formula: \(\sqrt{(0.01^2 + 0.01^2)}\), resulting in a propagated uncertainty of approximately \(\pm 0.014\) grams.
Propagating Uncertainty Through Multiplication and Division
The method for propagating uncertainty changes significantly when measurements are combined using multiplication or division, such as in the calculation of density from mass and volume. Unlike addition and subtraction, where absolute uncertainties are combined, these operations require the use of relative uncertainties. Relative uncertainty is the absolute uncertainty divided by the measured value itself.
To find the final uncertainty in a calculation involving products or quotients, first calculate the relative uncertainty for each individual measurement. For example, a mass of \(10.0 \pm 0.1\) grams has a relative uncertainty of \(0.1/10.0\), or \(0.01\). A volume of \(5.0 \pm 0.1\) milliliters has a relative uncertainty of \(0.1/5.0\), or \(0.02\).
The relative uncertainties are then combined using the root sum of squares method. This means squaring the relative uncertainties, summing them, and taking the square root of the total to find the overall relative uncertainty of the final result. If density is calculated by dividing the mass by the volume, the total relative uncertainty is \(\sqrt{(0.01)^2 + (0.02)^2}\), which equals approximately \(0.022\).
This final relative uncertainty must then be converted back into an absolute uncertainty for reporting the final value. This conversion is achieved by multiplying the final calculated value by the overall relative uncertainty. If the calculated density is \(2.0\) grams per milliliter, the absolute uncertainty is \(2.0 \times 0.022\), yielding an absolute uncertainty of \(\pm 0.044\) grams per milliliter.
Reporting the Final Value and Its Uncertainty
The final stage is presenting the calculated value and its associated uncertainty in a standardized format that reflects the precision of the experiment. Specific conventions dictate how both the calculated value and the propagated uncertainty must be rounded. The governing rule is that the absolute uncertainty should be rounded to only one or, at most, two significant figures.
This rounding dictates the precision of the final calculated result. The final value must be rounded so that its last significant digit aligns with the decimal place of the rounded uncertainty. For example, if a calculation yields a value of \(15.453\) and an uncertainty of \(\pm 0.0345\), the uncertainty is first rounded to \(\pm 0.03\). The final result is then correctly reported as \(15.45 \pm 0.03\).