The quality of measurements is very important in science and engineering because it affects how reliable the results are. There are a number of factors that affect the quality of measurements, like accuracy, precision, uncertainty and error. Use this resource to learn about these concepts.
Accuracy and precision
Accuracy shows how close a measurement is to the true value, while precision shows how consistent the measurements are.
Consider a target board used for archery.
Accuracy refers to how close the arrows are to the bullseye. If the arrows are consistently hitting near the bullseye, they are accurate.
Precision refers to how close the arrows are to each other. If all arrows land in a small cluster, even if they're not near the bullseye, they are precise.
Ideally, you want both accuracy and precision, meaning the arrows are tightly grouped around the bullseye. Accuracy and precision, by RMIT, licensed under CC BY-NC 4.0
Uncertainty
We measure physical quantities using devices called instruments. A ruler measures length, a thermometer measures temperature, a scale measures weight, a spring balance measures force and a stopwatch measures time.
Each of these instruments has uncertainty, meaning they can't give exact measurements. This uncertainty comes from things like the instrument's limits or changes in the environment. For example:
A ruler marked with millimetres cannot measure smaller details like fractions of a millimetre.
High humidity and moisture in the environment can affect a scale's sensors, leading to inaccurate weight measurements.
Temperature affects the pressure reading on a barometer.
An analogue stopwatch can only be read to the nearest tenth of a second. It is not able to measure finer time intervals accurately.
Knowing about these uncertainties helps to make sure that your measurements are accurate and reliable.
Reporting measurement uncertainty
Measurement uncertainty can be expressed as the absolute uncertainty of the mean. This is the uncertainty in the average value obtained from multiple measurements and gives an indication of how much the average is expected to vary.
Absolute uncertainty is denoted by \(\Delta\bar{x}\) and is calculated using the maximum and minimum measurement values:
\[\textrm{Absolute uncertainty}\,(\Delta\bar{x}) = \pm \frac{x_{\textrm{max}}-x_{\textrm{min}}}{2}\]
From this, the percentage uncertainty can be calculated. This represents the uncertainty as a proportion of the measurement average.
\[\textrm{Percentage uncertainty} = \frac{\textrm{Absolute uncertainty}}{\textrm{Measurement average}} \times 100\]
Example – calculating uncertainty
Suppose you measure the length of a table five times and get the values \(200.1\textrm{ cm}\), \(200.3\textrm{ cm}\), \(199.9\textrm{ cm}\), \(200.2\textrm{ cm}\) and \(200.0\textrm{ cm}\). The average measurement is \(200.1\textrm{ cm}\). Find the absolute and percentage uncertainty of the measurement.
Using the formula, you would calculate the absolute uncertainty to be:
\[\begin{align*} \Delta\bar{x} & = \frac{x_{\textrm{max}}-x_{\textrm{min}}}{2}\\
& = \frac{200.3-199.9}{2}\\
& = 0.2\textrm{ cm}
\end{align*}\]
We would therefore state the measurement as \(200.1\pm0.2\textrm{ cm}\).
So, the measurement would be \(200.1\textrm{ cm}\pm0.1\%\). This means that the measurement deviates from the average of \(200.1\textrm{ cm}\) by \(0.2\textrm{ cm}\) or \(0.1\%\). That is, there is a \(0.2\textrm{ cm}\) or \(0.1\%\) uncertainty in the result.
Error
Error refers to the difference between the measured value and the true value. Errors are classified as systematic or random.
Systematic error
A systematic error is caused by consistent problems with the measurement instruments or the methods used. Common systematic errors are:
Parallax error, which occurs when reading a measurement from an angle, rather than directly in line with the scale Parallax error, by RMIT, licensed under CC BY-NC 4.0
Calibration error, which happens when an instrument hasn't been adjusted to the correct standard, leading to consistently high or low readings
Zero error, which occurs when an instrument, like a balance, does not start from a true zero, affecting all measurements taken with that instrument.
To reduce systematic errors, you should make sure your instruments are calibrated, zeroed (or "tared") before use, and that you are using the correct techniques.
Random error
Random errors happen due to unpredictable changes during the experiment. Examples include:
Environmental fluctuations, like random changes in temperature, humidity, voltage, wind, etc, that can affect readings
Human reaction time, which can introduce small variations when you have to manually start and stop a timing device.
We can limit the effect of random errors by taking repeat measurements and averaging them, improving the way we measure things (like using an automated tool rather than relying on your own reaction speed), and controlling environmental factors where we can (e.g. maintaining constant temperature).
Reporting measurement error
The error in a measurement can be expressed as absolute error. This is calculated using:
\[\textrm{Absolute error} = \left| \textrm{Measured value}-\textrm{True value} \right| \]
This gives an indication of the accuracy of the result; the lower the absolute error, the more accurate the result, i.e. the closer it is to the true value.
Error can also be represented by percentage error, which represents the accuracy as a proportion of the true value.
\[\textrm{Percentage error} = \left| \frac{\textrm{Measured value}-\textrm{True value}}{\textrm{True value}} \right| \times100 \]
Example – calculating error
If the true value of a length measurement is \(15.0\textrm{ cm}\) and the measured value is \(14.6\textrm{ cm}\), calculate the absolute and percentage error.
\[\begin{align*} \textrm{Absolute error} & = \left| \textrm{Measured value}-\textrm{True value} \right| \\
& = \left| 14.6-15.0 \right| \\
& = 0.4\textrm{ cm}
\end{align*}\]
Exercise – calculating error
A student measures the mass of an object as \(48\textrm{ g}\), but the actual mass is \(50\textrm{ g}\). Calculate the percentage error of the measurement.
If a volume is measured as \(250\textrm{ mL}\) but the true volume is \(245\textrm{ mL}\), what is the absolute error?
A student measures the density of a liquid as \(1.02\textrm{ g cm}^{-3}\), while the true density is \(1.00\textrm{ g cm}^{3}\). Calculate the percentage error.
\(4\%\)
\(5\textrm{ mL}\)
\(2\%\)
Mistakes
Unlike uncertainty and error, which are natural in the measurement process, mistakes are usually accidental slip-ups that can be fixed with careful checking and practice.
Mistakes include:
writing down the wrong value, e.g. the scale reads \(72\) but you write down \(27\) by mistake
using the wrong instrument, e.g. picking the wrong pipette and adding \(1\textrm{ mL}\) instead of \(10\textrm{ mL}\)
misusing the calculator, e.g. pressing the \(\div\) button instead of \(\times\).
