Random and systematic errors

A measurement error is the difference between a measured value and the true value of the quantity. Errors can affect repeated readings in different ways: they may cause readings to scatter, or they may shift readings consistently in one direction.

Random errors

Random errors vary unpredictably from one measurement to the next. They cause repeated readings of the same quantity to scatter, with some readings higher and others lower.

Taking repeated measurements and calculating their mean reduces the effect of random errors: variations in different directions tend to partly cancel. The mean is therefore generally a better estimate than a single reading, although random errors are not eliminated entirely.

Systematic errors

A systematic error shifts measurements consistently in the same direction. For example, if a balance reads above zero when it is empty, its readings will be too high by that zero offset. An incorrectly calibrated instrument can also produce a systematic error.

Taking more readings under the same conditions does not remove the shift: the mean will still be affected. Instead, the systematic error must be identified and corrected, or reduced by improving the measurement method.

Measurement quality

A measurement can be close to the true value, agree closely with other measurements, or both. These are different qualities, so they must be described separately.

Accuracy and precision

Accuracy describes how close a measured value is to the true value of the quantity. Precision describes how closely repeated measurements agree with one another.

For example, suppose the true length of an object is $$10.0\ \text{cm}$$. Repeated readings of $$10.4\ \text{cm}$$, $$10.4\ \text{cm}$$ and $$10.5\ \text{cm}$$ are close to one another, so they are precise. They are not accurate, however, because they are not close to the true length.

A systematic error can cause readings to be precise but inaccurate. For example, an instrument that consistently reads too high can produce closely grouped readings that are all above the true value.

Repeatability and reproducibility

Measurements are repeatable when repeated readings agree closely using the same method, equipment and observer under the same conditions. Repeatability describes agreement when the measurement is repeated in the same way.

Measurements are reproducible when they agree closely despite a change in observer, equipment or method. Reproducibility therefore tests agreement across different ways of obtaining the measurement.

Resolution

Resolution is the smallest change in a measured quantity that an instrument can distinguish. An instrument with limited resolution cannot record changes smaller than that amount, so it restricts the detail in the readings.

Finer resolution allows smaller changes to be distinguished, but it does not by itself make a reading accurate. An instrument may display small changes while still giving values that are consistently too high or too low.

Measurement uncertainty and significant figures

A measured or calculated value is not usually known exactly. Its uncertainty expresses the extent to which the value is uncertain. A result can be written as $$x \pm \Delta x$$, where $$x$$ is the reported value and $$\Delta x$$ is its absolute uncertainty.

The absolute uncertainty has the same units as the value. For example, $$12.3 \pm 0.3\ \text{m}$$ expresses an uncertainty of $$0.3\ \text{m}$$ on either side of the reported value.

Relative uncertainty

The same absolute uncertainty can be small or large compared with the value itself. For a nonzero value, fractional uncertainty compares the absolute uncertainty with the magnitude of the value. Percentage uncertainty expresses that fraction as a percentage.

Both relative uncertainties have no units: the units of $$\Delta x$$ and $$x$$ cancel. The magnitude $$|x|$$ is used so that the relative uncertainty is not negative if the reported value is negative. These expressions are not defined when $$x = 0$$.

For $$12.3 \pm 0.3\ \text{m}$$, the fractional uncertainty is approximately $$0.024$$, or $$2.4%$$.

Significant figures and quoted uncertainty

The digits reported for a result should reflect its uncertainty. Extra digits can suggest a level of precision that the result does not support.

An absolute uncertainty is commonly rounded to one significant figure, although two are sometimes used. The value is then rounded to the same decimal place as the quoted uncertainty.

For example, suppose a calculated length is $$12.347\ \text{m}$$ with an absolute uncertainty of $$0.26\ \text{m}$$. Quoting the uncertainty to one significant figure gives $$0.3\ \text{m}$$. Both the uncertainty and the value are then expressed to the nearest tenth of a metre: $$12.3 \pm 0.3\ \text{m}$$.

If the uncertainty is instead kept to two significant figures, it remains $$0.26\ \text{m}$$, and the value is rounded to the nearest hundredth: $$12.35 \pm 0.26\ \text{m}$$.

Combining uncertainties

A result calculated from measurements is uncertain because the measured inputs are uncertain. The rule for combining their uncertainties depends on the operation used in the calculation.

An absolute uncertainty is expressed in the same units as the measured value. A fractional uncertainty compares the absolute uncertainty with the magnitude of the value:

Multiply the fractional uncertainty by $$100$$ to obtain the percentage uncertainty.

Addition and subtraction

When measured quantities are added or subtracted, add their absolute uncertainties. Use addition for the uncertainties even when the measured values are subtracted.

For example, subtracting $$(3.0 \pm 0.1),\mathrm{m}$$ from $$(12.0 \pm 0.2),\mathrm{m}$$ gives $$(9.0 \pm 0.3),\mathrm{m}$$: the values are subtracted, but the absolute uncertainties $$0.2,\mathrm{m}$$ and $$0.1,\mathrm{m}$$ are added.

Multiplication, division and powers

When measured quantities are multiplied or divided, add their fractional uncertainties. Percentage uncertainties can be added instead, because they express the same fractions as percentages. Convert the combined fractional or percentage uncertainty to an absolute uncertainty if the result needs to be written in the form $$\text{value} \pm \text{uncertainty}$$.

When a measured quantity is raised to a power, multiply its fractional or percentage uncertainty by the magnitude of the power. For example, squaring a quantity doubles its percentage uncertainty. Raising it to the power $$-2$$ also doubles its percentage uncertainty.

Combining uncertainties through several stages

Suppose $$x = (2.0 \pm 0.1),\mathrm{m}$$ and $$t = (4.0 \pm 0.2),\mathrm{s}$$. For the calculated quantity $$Q = x^2/t$$, the central value is:

Both $$x$$ and $$t$$ have a percentage uncertainty of $$5%$$. Squaring $$x$$ doubles its percentage uncertainty to $$10%$$. Dividing by $$t$$ means adding its $$5%$$ uncertainty, giving $$Q$$ a combined percentage uncertainty of $$15%$$.

The absolute uncertainty is therefore $$15%$$ of $$1.0,\mathrm{m^2,s^{-1}}$$, or $$0.15,\mathrm{m^2,s^{-1}}$$. The result can be written as $$Q = (1.00 \pm 0.15),\mathrm{m^2,s^{-1}}$$.

Applying the appropriate rule at each stage gives a conservative estimate: adding uncertainty contributions allows for them to combine rather than assuming that they cancel.

Graphical uncertainties

A measurement plotted on a graph is not necessarily exact. An error bar shows the range indicated by the uncertainty in that measurement. It extends from the plotted value to the lower and upper limits of the range.

A vertical error bar shows uncertainty in the quantity on the vertical axis; a horizontal error bar shows uncertainty in the quantity on the horizontal axis. For example, a vertical value of $$5.0 \pm 0.2$$ has an error bar extending from $$4.8$$ to $$5.2$$. A point can have horizontal error bars, vertical error bars, both, or neither. A graph can also contain some points with error bars and others without them.

Graphical uncertainty in a gradient

For data that follow a straight-line trend, draw a best-fit line through the overall pattern. It need not pass through every point. Its gradient is the best estimate from the graph.

Next, draw the steepest and shallowest reasonable lines consistent with the data and any error bars shown. These lines represent alternative fits, not lines joining only the highest or lowest points. Where error bars are shown, use their ranges to judge whether a line is reasonable; consider the pattern of all the data rather than a single point.

Calculate the gradient of each line using two well-separated points on that line. The extreme gradients give a range of reasonable values. A common estimate of the uncertainty is half the width of that range:

Here $$m$$ represents gradient. The best-fit gradient can then be reported with this estimated uncertainty.

Graphical uncertainty in an intercept

The vertical intercept is the value of the vertical-axis quantity where the line reaches $$x=0$$. Read it from the best-fit line, extending the line if necessary. Then consider reasonable alternative lines that give the largest and smallest intercepts. These need not be the same lines that give the steepest and shallowest gradients.

A common estimate of the intercept uncertainty is half the range of these extreme intercepts:

Here $$c$$ represents the vertical intercept. If the best-fit value is not near the middle of an extreme range, the half-range may not cover both extremes. Comparing the best-fit value with each extreme shows this; the larger difference can be used as a conservative uncertainty.

When error bars are absent

The scatter of points around the trend can guide the choice of reasonable alternative lines. However, scatter does not give a uniquely defined uncertainty for each individual point. A missing error bar should not be taken to mean that the measurement has zero uncertainty.