Uncertainties in AS/A GCE Physics 1. Introduction No physical quantity can ever be measured with 100% accuracy; no measuring instrument is perfect. There will always be some intrinsic variation in measurements, as there are natural/inherent limitations in the apparatus used. Even digital instruments have an uncertainty, although the (apparent) precision of the display may hide this.

No real experiment will yield an answer that is guaranteed to be correct. Instead, we will end up with a range of possible answers, which we hope will include the "true" value. The centre of this range is taken as the best answer and half the size of the range is the uncertainty. Example 1: Imagine measurements (of a length) give results lying between 79.0 cm and 79.4 cm. Then the answer should be quoted as 79.2 0.2 cm. The value " 0.2 cm" is the uncertainty in the final answer of 79.2 cm.

2. Mistakes

A mistake can be caused by misreading scales or by faulty equipment. Two of the most common causes of mistakes are faulty transcription (a failure to write down a measurement straight away and thus mis-remembering the number), or the careless use of a calculator. All mistakes can and should be eliminated, by a painstaking approach and by taking check readings. A mistake is clearly wrong: an error in either the chosen experimental procedure or in the way the experiment is carried out. The term "error" will not be used below.

3. Errors and uncertainties You may find these words used in different ways by different people - possibly even interchangeably. The term error implies that something has gone wrong. Uncertainties, however, cannot be avoided. An uncertainty is not an error (nor is it a mistake). When designing and carrying out experiments, you should aim to minimise uncertainties. You may be asked to carry out a numerical estimate of uncertainties, in particular the practical examinations.

4. Expressing uncertainty The uncertainty estimates the range either side of the mean value, within which the true answer is likely to lie. (A statistical estimate of probabilities can be made but that is not required in Advanced Supplementary or Advanced GCE Physics.) The uncertainty will normally be expressed as either a percentage or as an absolute value. Occasionally, factional uncertainties are used. Only an absolute uncertainty has a unit; the other two are dimensionless, being ratios.

Example 2: Imagine a metre rule is used to make a single determination of the width of a bench and gives 462 1mm. The absolute uncertainty is l mm, whereas the fractional uncertainty is 1/462 = 0.0021(6) and the percentage uncertainty is (1/462) x 100 = 0.21(6)%. Remember that the absolute uncertainty does require a unit.

5. Random and systematic uncertainties

Random uncertainties can be caused by: Human misjudgements of various kinds (eg incorrectly estimating exactly where the

image produced by a lens is in focus) Limitations in the equipment employed (eg the natural lack of precision of the

instrument) A lack of reproduceability in the actual measurement (eg a real variation in the e.m.f.

of the electrical mains). To expand on the first example, the observer will have to judge the position of the pointer of an analogue instrument. If the pointer lies between two divisions of the scale, a judgement will have to be made as to its position: by what fraction is it above the lower number? This estimate will only be exactly correct by chance. The recorded number could be above or below the true value.

Thus, random uncertainties can shift the answer above or below the "true" value. Repeat readings will show a spread and taking additional readings should tend to reduce the random uncertainty but can never eliminate it. A precise experiment has a small random uncertainty and vice versa. Systematic uncertainties shift measurements in one direction, away from the "true" value. Zero errors will create this son of uncertainty, as will incorrectly calibrated instruments. Examples include:

A thermometer calibrated in impure water (and thus with 0cC and 100C wrongly marked)

A stopwatch consistently running fast (or slow) Using a micrometer screw gauge without allowance for zero error.

Check readings will not reveal any information on systematic uncertainties. Where feasible, the whole experiment should be repeated using a completely independent method to assess the reliability of the approach(es). Imagine two students measure the resistance of nichrome wire by the ammeter/voltmeter method. One student uses a current of 10 mA whereas the other one uses a current of 1 A. The wire heats up appreciably in the second case, increasing the resistance, and this will still be so no matter how many check readings are taken. It is only by the use of an, independent set-up, for example using a low current digital ohmmeter, that the systematic uncertainty can be discovered.

An accurate experiment has a small systematic uncertainty whereas an inaccurate one will have larger systematic uncertainties.

It is unlikely that candidates will be asked in an examination to assess the numerical size of any systematic uncertainties, apart from zero error checks of some instruments such as micrometer screw gauges.

6. Estimating the uncertainty in the measurement of a single parameter Method 1 The uncertainty cannot be smaller than half the smallest scale division, although the convention that the random uncertainty should be quoted as plus or minus the smallest scale division is becoming increasingly widespread. For example, a micrometer screw gauge can be read to - 0.01 mm, whereas analogue Vernier callipers can be read to 0.2 mm (or perhaps 0.1 mm). Method 2

By consulting the manufacturer's handbook. This will normally tell us that the overall uncertainty is (much) larger than the precision to which an instrument can be read. For example, a digital voltmeter may be able to be read to 0.01 V in 20 V, yet the manufacturer's handbook may quote an accuracy of 0.5% (ie 0.1V at 20 V).

7. The advantages of taking check data

If we have multiple readings of a single parameter, then we can assess if a mistake has been made. An anomalous measurement should always be recorded but discounted when averaging. However, you must cross out the anomalous measurement or state the reason for discounting it. If measurements are simply ignored, without comment. Examiners will wonder why. That could lead to a loss of marks.

Further, the spread of these data gives a good indication of the random uncertainty. There are statistical methods for calculating the random uncertainty but these are not required in AS/A GCE Physics. Do not use them in an examination. They consume time and may cost you marks (as you are unlikely to show the fall method, writing down intermediate values displayed on your calculator.)

The following quick method gives a reasonable estimate. Subtract the smallest datum from the largest one, ignoring any anomalous results. Then assess the uncertainty as half the range.

Example 3: Six students measure die resistance of a lamp. Their answers in . are: 609; 666; 639; 661; 654; 628.

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Mean resistance; 643 Smallest resistance 609 Largest - smallest: 57 Largest resistance 666 Dividing by 2 29

Resistance = 643 29 ft (Better expressed, slightly pedantically, as 0.64 0.03 k)

Example 4: Three pairs of perpendicular measurements are made of the diameter of a cylinder. Results, in mm, are: 21.8; 21.6; 22.1; 26.1; 21.9; 22.1

The datum 26.1mm is very different from the others and can be discarded as a mistake (perhaps a transcription error?). Thus present your results, in mm, as 21.8; 21.6; 22.1; 26.1; 21.9; 22.1

The average of the "good" five is 21.9 mm. The difference between the largest and smallest measurements is 0.5 mm Hence, the uncertainty is 0.5/2 = 0.2(5) mm. The cylinder has a diameter of 21.9 0.3 mm.

To apply this method, at least three data are required, preferably more. In practical examinations, there is rarely enough time to take large numbers of check data. Normally a total of three readings is sufficient. If the data is very variable (e.g. measuring the time

interval when a sphere rolls down an incline), the Examiners may expect more readings to be taken, but never more than five for an individual measurement. 8. Answers Ensure that the final answer and its absolute uncertainty are quoted to the same number of decimal places. It would be rare to quote the uncertainty to more than one significant figure. Don't forget units.

9. Estimating uncertainties in selected common measurements

9.1 Periodic time A common task during experiments is to measure the period of some oscillatory system. Modern (digital) stop clocks and stopwatches have a display reading to 0.01s but that does not mean that your uncertainty is 0,01 s. Both the starting and the stopping of the timer involve reaction time uncertainties, which can be reduced in two ways:

a) by anticipation: (Count -3. -2, -1, 0, 1,2... 19, 20, starting the timer on "0" and stopping it on "20") and

b) by using a central mark from which to time.

A good way of measuring your uncertainty, which will vary from day to day, is to make at least three measurements for one set-up and then apply the method as illustrated in examples 3 and 4. However, you must not claim too high a precision.

Example 5: For a particular length of a pendulum, twenty oscillations are timed, in s, at. 1473; 14.69; 14.75.

The spread of readings (largest - smallest) is 0.06 s. Thus it might be thought that the uncertainty in 20 T is 0.03 s. Examiners would be concerned about this as it is too small. Allowing for reaction time uncertainties, a realistic estimate of the uncertainty would be at least: 0.1 s. This result is explored further in section 9.7, Example 6: Imagine however a different student timed the twenty oscillations, in s, as follows: 14.51; 15.13; 14.82 The spread of readings here is 0.62 s. Thus the uncertainty in 20 T is at least 0.3 s.

=>20T= 14,8 0.3 s.

9.2 Digital meters Without consulting the manufacturer's handbook, you cannot tell if the final (least significant) digit has been rounded up or down. The uncertainty must be at least (1 in the least significant digit). This may well imply a false precision; the true accuracy can be found from the manufacturer's handbook. One exception: digital ohmmeters. The accuracy of this sort of meter depends upon fee accuracy of the calibration of the internal source of e.m.f. and may well change with time. An estimate of (1 in the least significant digit) will be a severe underestimate of the resultant uncertainty.

9.3 Analogue measurements and meters Quote the uncertainty in a single measurement as at least (half the smallest scale division). Two exceptions:

Edexcel Practical Examiners require -10 C=> +110 C liquid-in-glass thermometers to be read to better than 1 C. Interpolate the scale. Room temperature might perhaps be read as 21.5 C which can seem like a precision of 0.1 C.

Many (school) Vernier callipers have thick rulings on the scale and can only be set to 0.2 mm.

9.4 Uncertainties when using a metre rule If using a laboratory metre rule, you should be able to read that rule to 0.5 mm. A length measurement involves two readings: positioning one end of the object against "zero"' (or some other suitable scale division) and then taking the scale reading of the other end. As each is - 0.5 mm, the length of the object has been determined to - limn. The centre of mass of that same rule can be found by balancing it on a suitable fulcrum. However, that single measurement is precise only to ~ 1 mm at best.

9.5 Standard masses

So called standard masses are often less accurate than one mieht hope. A batch of 100 g masses from a commercial source were found to vary in the range 97 > 103 g. It would be reasonable to quote any individual mass as 100 3 g.

In the practical Examination, assume that the value stated on the mass is accurate to r 1 in the last significant figure; i.e. "100 g" means "100 1 g".

Should it be relevant, the precision of this sort of apparatus, supplied m the AS/A GCE practical Examinations will be told to you.

9.6 Radioactive counting experiments. Different rules apply to statistical processes of this type. A treatment of these is beyond the scope of this book.

9.7 Two common faults

Consider again the data of example 5. For a particular length of a pendulum, twenty oscillations are timed, in s: 14.73; 14.69; 14.75. => The average value of 20T is 14.72 s => The average period is 0.736 s

A common mistake made by candidates in a practical Examination is to assume that the precision of an instrument is the uncertainty in the measurement. Referring to the data above, it would be -wrong to quote the period as 0.74 0.01 s.

The uncertainty in a single measurement is sometimes and wrongly taken to be the same as that in a multiple measurement. Again referring to the data of example 5, a student might decide that: 20T= 14.7(2) 0.1 s

However, it would then be wrong to quote T = 0.74 0.1 s.

In particular, the percentage uncertainty in 20 periods is exactly the same as the percentage uncertainty in a single period.

10. Compounding uncertainties

If data are to be added or subtracted, add (never subtract) the absolute uncertainty; don't forget units.

If data are to be multiplied or divided, add (never subtract) the percentage uncertainty.

If a parameter is raised to z power, multiply the percentage uncertainty by that power. For example, if the uncertainty in the radius of a circle is y%, then the uncertainty in area is 2y%, as area y2 .

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Example 7: A cylinder has a radius of 1.60 0.01 cm and a height of 11.5 0.1 cm. The volume is % x (1.60)2 x 11.5 = 92.488 cm3. The percentage uncertainty in the radius is (0.01/1.60) x 100 = 0.62%. The percentage uncertainty in the area is 1.2(5)%. The percentage uncertainty in the height is (0.1/11.5 0) x 100 = 0.87%. => percentage uncertainty in the volume - 1.25 + 0.87 = 2.1%. Thus the absolute uncertainty in the volume = 0.021 x 92.488 = 1.96 cm3. In this case, we have found the volume to be 92 2 cm3.

If calculating an intricate function (eg lg (x), sin (y), etc), it is best to work out the largest and smallest values and not try to estimate uncertainties directly. For example, imagine x is measured to z, yet lg (x2) is to be calculated. Then the largest value of x = x1 = x + z; then calculate the logarithm of (x1)2:. Similarly, the minimum value of x = x2: = x - z; then calculate the logarithm of (x2)2. If a graph is to be plotted, the two values x1 and x2 constitute the ends of the uncertainty bar. (Some authors refer to uncertainty bar as "error bar").

Example 8: Applying this method to the data of example 7: The largest feasible radius is 1.61 cm. The largest feasible height is 11,6 cm3.

Largest feasible volume is pi x (1.61)2: x 11.6 = 94.46 cm3. The uncertainty is (94.46 - 92.49) = 2 cm3 (Quoting to a sensible precision.)

This method can be used on all occasions and is preferred except where Examiners require you to use a different procedure. It may give marginally different answers to other approaches - but insignificantly so. The new Specification (starting from September 2000) puts less emphasis on compounding uncertainties than was formerly the case.

11. Graphical Methods

Most AS/A GCE Physics experiment lead to a graph. If calculating an answer, plot the points on the graph, allowing for uncertainties. Then draw two lines: - the steepest feasible line - the shallowest feasible line

Both lines should encompass most, not necessarily all, of the points. The steepest feasible line will graze the top of (most of) the uncertainty bar of the highest points and the bottom of the uncertainty bar of the lowest points. Then calculate the slope of each line. The mean gives the value of the most probable ("best") slope, whilst half the difference gives the uncertainty in that slope.

This approach is not expected in the AS/A GCE Physics examination. Edexcel normally expect only the best fit line to be drawn. Should it be required, the Examiners will guide you through it explicitly.

Note that logarithmic (both lg and In) graphs, from September 2000, feature solely in Advanced GCE but not in the Advanced Supplementary GCE Specification.

12. The role of Uncertainties in testing hypotheses

If testing a hypothesis, plot the points on the graph, allowing for uncertainties. Draw in the best-fit smooth line (which may or may not be straight). If the line fits the experimental points within the limits of the uncertainties, then the hypothesis can be deemed valid over the range of data tested. If a point is significantly far off the line, then there are three feasible reasons: The estimated limits of uncertainty are over optimistic A mistake might have occurred (ie the point is anomalous) The hypothesis being tested is invalid. Example 9: It is believed that a vehicle accelerates uniformly from rest. Data for speed, v, and time, t are as follows: t/s 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 v/'ms-1 4.2 10.4 14,1 18.6 24.2 28.8 33.9 42.1 48.8

The hypothesis is that v t; ie v = kt. A straight line through the origin of slope k is predicted.

Plot the data (Try it yourself.). The first seven points all lie within about 1.5% of the best-fit line, so we can take the experimental uncertainty to be ~ 2%.

The eighth point is about 3 ms-1 above the best-fit line. Its "distance" from the line is thus (3/39) x 100-7.7%.

The ninth point is about 5 ms-1 above the best-fit line. Its "distance" from the line is thus (5/44) x 100= 11%.

As both 7.7% and 11% are considerably greater than 2%, it seems likely that the hypothesis becomes invalid some time at or around 7 s after starting.

Note: A good experimenter would have noticed that something "interesting" was occurring at t ~7 s and would have taken additional data at t/s = 6.5, 7.5, 8.5, etc.

Where something interesting or unexpected is occurring, candidates will get credit for taking additional readings in that region. To quote another example, if investigating the electrical characteristics of a diode, some students might only take data at 0V, IV, 2V, etc. They might "miss" the interesting region, close to the origin, where the diode cuts on at 0.2 V or 0.6V (depending on construction).

13. Some rules (in summary)

An uncertainty is not an error (nor is it a mistake). All experiments have some uncertain! associated with the conclusion.

Random uncertainties can be either positive or negative. Their effects are reduced by the check readings. Systematic uncertainties shift the result in one direction only.

Accuracy and precision have different meanings.

Always take check readings, to reduce the effects of random uncertainties and to help spot mistakes. Note down all the data you capture, even if anomalous values are subsequently not processed. ^

It is often sensible to evaluate the largest and mean (or the largest and smallest) values of complicated answer, rather than trying to estimate uncertainties directly.

Calculators 1. Don't be tempted to use advanced statistical functions. 2. Do quote intermediate answers. 3. Ensure that your calculator meets the current regulations of the Awarding Body.

Answers Ensure that the final answer and its absolute uncertainty are quoted to the same number of decimal places, with the uncertainty quoted to one significant figure Don't forget units, Don1: fuss about small uncertainties or waste time on meaningless precision.

Example 10: The density of a block of plastic is found to be 1.4872 g cm-3. However the uncertainty is subsequently calculated as 0.01782 g cm"3. Rounding this uncertainty to one significant figure gives 0.02 g cm'-3. As the answer is uncertain m the second decimal place, the third and fourth decimal places are meaningless.

Thus the density of the plastic is best quoted as (1.49 0.02) g cm"3. For Edexcel examinations in AS/A GCE Physics, do not waste time calculating uncertainties numerically, unless the question explicitly requires you so to do. However, you must always have an appreciation of the precision to which you are taking your measurements and thus the number of significant figures that you can quote in your final answer