Uncertainty

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All readings (values taken from an instrument) should have an absolute uncertainty attached to them. For example, d = 3.7 cm ± 0.2 cm  is taken to mean that the value of d is likely to lie between 3.5 cm and 3.9 cm.

Likewise, all results (values calculated or otherwise derived from readings) should have absolute uncertainties attached to them. To calculate these, it is necessary to use the concept of fractional uncertainty.

There also exists percentage uncertainty (= fractional uncertainty ´ 100%)

Assigning uncertainties to readings

If all else fails (e.g. in situations where you have only one reading or repeat readings have been identical), declare the size of the smallest scale division as the absolute uncertainty. The size of the smallest division is sometimes called the precision of the instrument.

If you have taken a small number of repeats - up to three, say - you can subtract the smallest from the largest to find the range, and then declare the uncertainty as half the range.

If you have a larger number of readings, you can find the standard deviation, and divide it by the square root of the number of readings. This is the best method.

Fractional uncertainties

Fractional uncertainty is defined by

The first thing to do is to make fractional uncertainty the subject and hence convert all the absolute uncertainties of the readings into fractional uncertainties. So taking the example above, we see that the fractional uncertainty is (0.2 cm / 3.7 cm) = 0.054. Notice that the units have cancelled. (It is possible to express this as a percentage uncertainty of 5.4%, but there is rarely any advantage in doing so.)

We write the fractional uncertainty as dd/d (or, better, as

The vertical layout is harder to deal with in html, and so this note will use the horizontal layout)

Combining uncertainties

The basic equation is

Suppose d in the example above is the diameter of a tube of which we wish to find the cross-sectional area. We first find the radius

r = ½ d = 1.85 cm

In this equation, d appears just once on the right-hand side, so the fractional uncertainty in r is equal to the fractional uncertainty in d. So dr/r = 0.054. The constant ½ in the formula plays no part in calculating the fractional uncertainty. (Note that similar criteria apply in other situations: e.g. the fractional uncertainty in the period of a pendulum is the same as the fractional uncertainty in the original reading of the time for 20 swings; the fractional uncertainty in the thickness of one sheet of paper is the same as the fractional uncertainty in the measured thickness of the whole stack, etc). This appears to be quite a difficult point to grasp. The main thing to appreciate is that, in this example, the diameter is a measurement and the radius is a calculate result. The absolute uncertainty of a measurement is derived from the set of measurements that were made (range/2, st dev, etc), whereas the absolute uncertainty of a calculated result can only be determined by the method above.

For the area we have

A = p r ² = 10.75 cm²

Since r appears twice on the right-hand side, we must include it twice in the overall fractional uncertainty equation

dA/A = dr/r + dr/r =  0.108

Notice, once again, that p, being a constant, plays no part in this calculation. Finally, we calculate the absolute uncertainty of A from the defining equation

dA = A ´ dA/A  = 10.75 cm²  ´ 0.108 = 1.16 cm²

so that the final result is

A = 10.7 cm²  ± 1.2 cm²

Notice that we have dropped the second decimal point. If we are uncertain even about the number of square centimetres, it would be absurd to worry about square millimetres!

Sums and differences

The above rule for combining uncertainties only works if the results equation involves just multiplication and division (including powers and roots). Consider, for example, the problem of finding the density of a stone by weighing it and then immersing it in a measuring cylinder of water. Here are the readings

m = 38 g ± 1 g          V₁ = 50 cm³ ± 1 cm³           V₂ = 67 cm³ ± 1 cm³

The obvious formula to use is

r = m / (V₂ - V₁)

but this has a difference on the bottom line, which renders the uncertainty equation unusable. Here is how we get round the problem.

Define V_D as the volume difference (V₂ - V₁), which has the obvious value 17 cm³. But, if luck went against us, with V₁ = 49 cm³ and V₂ = 68 cm³, it could be as high as 19 cm³; and it could equally well be as low as 15 cm³. So now we have

V_D = 17 cm³ ± 2 cm³, leading to  dV_D/ V_D = 2/17 = 0.117

The fractional uncertainty in m is, of course,  dm/m  = 1/38 = 0.026. As the density equation is now r = m / V_D, with no subtractions or additions, we may use the usual procedure of adding fractional uncertainties to arrive at

dr/r = dm/m + dV_D/V_D =  0.026 + 0.117 = 0.1443

The density calculation itself gives r  = 38 g / 17 cm³ = 2.235 g/cm³, so the absolute uncertainty in density is

dr = r ´ dr/r  = 2.235 g/cm³ ´ 0.1443 = 0.322 g/cm³

Thus we declare a final result of  = 2.2 g/cm³ ± 0.3 g/cm³. Notice that, because we are uncertain even about the second significant figure, there is no point in quoting anything less significant. So we would give our final conclusion as

r  = 2.2 g/cm³ ±  0.3 g/cm³

Notice also that, because it is a volume difference that appears in the equation, the volume uncertainties have a much greater effect than one might at first have suspected. Any improvements to this experiment need to focus on reducing the volume uncertainties rather than on finding a more accurate balance!

In summary, as a rule of thumb, double the uncertainty whenever you have a difference of values to deal with.

Graphs

Before plotting a graph, you should form a column to contain the absolute uncertainties of the quantities to be plotted. Sometimes the plot will be of readings, in which case this is very straightforward. At other times the plot will be of calculated values, in which case the procedures outlined above will need to be followed for each value to be plotted. Excel can make short work of this.

The points should be plotted as + signs: the vertical and horizontal lines of each + should be made of such lengths as indicate the absolute uncertainties for that pair of results/readings. These horizontal and vertical lines are known as uncertainty bars (sometimes, erroneously, as 'error bars'). There is nothing magic about the point of intersection of the +. All that matters is that the best fit line/curve should pass through (or as near to as feasible) the uncertainty bars.

It often happens that one wants to find the gradient or y-intercept of the graph. These instructions apply equally well to either requirement

• Draw the steepest graph you think the plots will support and find its gradient in the usual way. Let's call it M₁, (with a y-intercept of C₁).
• Draw the shallowest graph you think the plots will support and find its gradient in the usual way. Suppose it is M₂, (with a y-intercept of C₂).
• Find the mean of M₁ and M₂ - let's call it M_M - and half the range - let's call it dM_M. The fractional uncertainty of the gradient is now dM_M/M_M , and this may be used in calculations further down the line. The mean C_M and fractional uncertainty dC_M/C_M of the y-intercept may be dealt with in exactly the same way.

This technique is particularly useful in situations where the original readings constitute a table with no repeats.