11.1 Uncertainties and errors in measurements and results

Absolute uncertainty

Absolute uncertainty = (percentage uncertainty)/100% x value

Quoting the result of addition/subtraction

If a calculation involves just adding or subtracting numbers, the final answer should be quoted to the same number of decimal places as the piece of original data that has the fewest decimal places.

Estimating random uncertainties in digital instruments

In general, the uncertainty of a measurement made on a digital instrument should be quoted as ± the smallest division.

Quantitative data

Quantitative data are numerical data, which are obtained from measurements.

Propagation of uncertainties in addition/subtraction

When quantities with uncertainties are added or subtracted, the absolute uncertainties are added.

Accuracy

Accuracy refers to how close a measurement is to the actual value of a particular quantity.

Estimating random uncertainties in analogue instruments

As a rule of thumb, the uncertainty of a measurement is half the smallest division to which you are reading.

Quoting the result of multiplication/division

When carrying out calculations involving multiplication and/or division, the general rule is that the final answer should be quoted to the number of significant figures of the piece of data with the fewest significant figures.

Rounding values in calculations

When carrying out multi-stage calculations, it is important to avoid rounding errors that could, after several stages, introduce large inaccuracies into the calculation. As a general rule, all numbers should be carried through in a calculation and rounding should only happen when an answer to a particular part of a question is required.

Systematic errors

1) A systematic error is an error introduced into an experiment by the apparatus or the procedure. Systematic errors result in a loss of accuracy, i.e. the measured value is further away from the true value. 2) Systematic errors are always in the same direction. 3) The effect of a systematic error cannot be reduced by repeating the readings. 4) Systematic errors can be reduced by changing the way the experiment is carried out.

Precision

1) Precision relates to the reproducibility of results. If a series of readings is taken with high precision, it indicates that the repeated values are all very close together and close to the mean (average) value. 2) For a single reading, a more precise value is a value to more significant figures.

Random uncertainties

1) Uncertainty in a measurement due to the limitations of the measuring apparatus and other uncontrollable variables that are inevitable in any experiment. 2) The effects of random uncertainties should mean that the measurements taken will be distributed either side of the mean, i.e. fluctuations will be in both directions. 3) The effect of random uncertainties can be reduced by repeating the measurements more often, but random uncertainties can never be completely eliminated.

Percentage error

1) We can use the percentage error to compare the experimental value with the accepted literature value. 2) Percentage error = (experimental value - accepted value)/accepted value * 100%

Percentage error and random uncertainties

1) When compared to the literature value, if the percentage error is smaller than that due to random errors then the experiment has worked well as a way of obtaining a value for this particular quantity. 2) If the percentage error is larger than the percentage uncertainty due to random errors then this suggests that there are systematic errors in the experimental procedure.

Propagation of uncertainties in multiplication/division

1) When multiplying or dividing quantities with uncertainties, the percentage uncertainties should be added. 2) When multiplying or dividing a quantity with an uncertainty by a pure number, the absolute uncertainty is multiplied/divided by that number so that the percentage uncertainty stays the same.

Percentage uncertainty

Percentage uncertainty = (absolute uncertainty)/value x 100% A percentage uncertainty has no units (%).

Nature of science: Making quantitative measurements with replicates to ensure reliability

Precision, accuracy, systematic, and random errors must be interpreted through replication. Scientists often repeat experiments to reduce random uncertainties but the effects of systematic errors can be more difficult to spot.

Qualitative data

Qualitative data are non-numerical data. These would normally be observations made during an experiment.

Quoting values with uncertainties

The uncertainty is usually quoted to one significant figure, and the measurement should be stated so that the uncertainty is in the last significant figure - no figures should be quoted after the uncertainty.