Measurements and uncertainties (2.2.1)
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The error of a measurement is the difference between an individual measurement and the true value of the quantity being measured. There are two types of errors, random and systematic.
Random error results in a random fluctuation of the measured value about the true value over repeated measurements.
Examples of causes of random error include:
- Fluctuations in the external conditions, such as electronic noise in an electrical component.
- Reading the measuring instrument differently each time, such as the level of the line on a thermometer.
Random error is often unavoidable, but its effect can be reduced by using more precise measuring instruments or taking many repeated measurements and averaging them to find the mean value.
Systematic error results in a skewing of the measured data by a given amount related to a flaw in the measurement process. Systematic errors can be reduced by calibrating the measurement apparatus or by comparing the results of different measurement techniques if possible.
Examples of systematic error include:
- Zero error – caused by the measuring instrument not being calibrated correctly, such as a weight scale showing a non-zero reading when no object is placed on it.
- Scale error – when measurements are consistently different from the true value by a certain proportion. For example, a weight scale may measure higher than the true value.
Random errors, where all results are affected by fluctuating amounts, affect the precision of a measurement, which describes the closeness of independent test results made under the same conditions. Precision depends only on the distribution of the random errors about the true value and does not depend on the true value itself.
Systematic errors, however, skew all results by a consistent amount and impact the accuracy of a measurement. Accuracy reflects how closely an individual test result aligns with the true value.
Sometimes, an accepted reference value can be used as the true value; however, the true value is usually unknown and must be measured.
The way that systematic and random errors affect accuracy and precision can be visualised by imagining a person throwing darts at a dartboard.
In the left image, the darts land near the bullseye with a random spread. This is an example of random error. The dart throws are both precise and accurate.
Systematic error in dart throwing in the right image causes a consistent deviation from the intended target, causing darts to consistently land in a particular area off-centre, rather than being randomly scattered. It’s often caused by a flawed technique, such as an improper grip, a poor follow-through, or an incorrect stance, which biases the throws in a predictable direction.
The darts will have a similar spread, but their average position will be displaced from the centre of the bullseye. This is an example of systematic error. The dart throws are precise but not accurate.
Uncertainty is an estimate attached to a measurement which characterises the range of values which should contain the true value.
- When using analogue measurement tools with a graduated scale that can be read, the uncertainty is taken as half of the smallest graduation. For example, a ruler with divisions of has an absolute uncertainty of
- When using a digital apparatus, the uncertainty is equal to the smallest graduation. For example, a one decimal place ammeter has an absolute uncertainty of
Percentage uncertainty is the ratio of the absolute uncertainty to the quantity measured as a percentage, which can be written as:
For example, if a ruler with absolute uncertainty is used to measure a length of the percentage uncertainty is:
When multiple measurements are combined, the uncertainty in the final result will be a combination of the uncertainties in each measurement. So, when measurements are added or subtracted, the absolute uncertainties are summed.
An example of absolute uncertainty is in finding the tensile strain of a metal bar by measuring the change in length before and after the application of force.
If the initial length is and the final length is then the change in length is:
The uncertainty is:
So the change in length is:
When measurements are multiplied or divided, the percentage uncertainties must be added.
An example of percentage uncertainty is the calculated resistance of a resistor. It is found by measuring the current, for an applied voltage, .
The percentage uncertainty in and respectively is:
Ohm’s law can be rearranged to:
This expression shows that the percentage uncertainty in is found by adding the percentage uncertainties in and . Therefore, the percentage uncertainty in is:
Raising to a power is a special case of multiplication. The percentage uncertainty is multiplied by the power to which the value is being raised.
For example, consider measuring the power supplied to a resistor of resistance with current flowing through it. The percentage error in is and the percentage error in is
Electrical power, is given by:
Therefore, the power through the resistor is:
The percentage uncertainty in the power is equal to twice the percentage uncertainty in since it is raised to the power of two, added to the percentage uncertainty in which is:
Therefore, the uncertainty in the power is:
The calculated power is written as:
Measurements can be made more accurate and precise by recording many data points over a range of values.
In this example, the resistance of a resistor can be measured by recording the current as a function of the applied voltage over a range of voltages and using the gradient to determine the resistance .
The red line is the line of best fit through the data. The line of best fit is drawn going roughly through the middle of the data points, mitigating any anomalous results. Ohm’s law can be written as:
The resistance of the resistor, can be found from the gradient of the line of best fit, by:
The gradient of the line of best fit can be found by identifying two points on the line and and using:
For the graph above, this leads to and hence
Each data point, based on a measured value, has an associated uncertainty. Error bars are a way to visually represent this uncertainty. They are drawn stretching above and below the data point by the absolute uncertainty in each direction. If there are no horizontal or vertical error bars for a data point, then the error is negligible in the X axis or Y axis measurement.
Error bars can be used to find anomalous data points. If an error bar does not pass through the line of best fit and is a significant distance away from it, the data point is likely an anomaly.
Error bars can be used to estimate the uncertainty in the gradient of the line of best fit, which can be done by drawing the worst lines of fit. These lines are drawn from the bottom of the first error bar to the top of the last error bar and vice versa.
The difference between the gradient of the line of best fit and each line of worst fit is calculated. The greatest of the two differences is taken as the absolute uncertainty in the gradient.
The gradient of the line of best fit is and the lines of worst fit have gradients and Each of the worst line gradients has a difference of to the line of best fit gradient, so this is the absolute uncertainty and the gradient of the line of best fit is:
The percentage uncertainty is:
Since the resistance only depends on the gradient and no other variables, it has the same percentage uncertainty. Therefore, the absolute uncertainty in the resistance is:
The calculated resistance is written as:
It is useful to know how to determine the difference between experimental values and accepted values to confirm that the experiment was performed correctly. The percentage difference between an experimental and accepted value is equal to:
An example of the difference between experimental values and accepted values is in the true and experimental values of a resistor. If it were known that the true resistance of a resistor was and the measured experimental value was then the percentage difference between the true value and the experimental value would have been: