Uncertainties
An uncertainty is an estimate of the accuracy of a measurement.
It is largely aimed at putting bounds on the amount of random error that may be present in the reading.
It is sometimes referred to as the "probable error" in a measurement.
An uncertainty is expressed as a +/- value with a unit (or a % symbol). This is very important. Without a unit, the value is meaningless.
E.g. if a length is measured as 50cm +/-4cm then this means that we think the measurement is 50cm, but we could reasonably be out by 4cm in either direction i.e. the length value probably lies between 46cm and 54cm.
If we just said that the length has an uncertainty of +/-4, this is unhelpful. Does this mean +/-4mm, +/-4cm, or +/-4m?
If the Examiner asks you to quote an uncertainty in a measurement, they will expect the +/- element (but will sometimes let this pass). They will never, ever, excuse lack of a unit and this will cost you a mark.
Estimating uncertainties
This will occur at least once on the ISA test paper, so learn this well.
If you have repeated readings, then the uncertainty is half the range of the repeated results.
E.g. for values of length of 10.4cm and 10.8cm, the range is 0.4cm (the difference in the readings) and so the uncertainty is 0.2cm. The average value, our best estimate of the actual length, is 10.6cm. We would say that the length is 10.6cm +/-0.2cm.
At A level, this rule remains if you have more than two readings - find the largest and smallest of the repeat values, determine the range and halve it.
[Strictly, you should start doing standard deviation calculations and the extra readings will then reduce the uncertainty, but this is mathematically more advanced and so not taught at this level.]
IF the repeated readings are identical, or you haven't taken repeated readings (e.g. it was your independent variable) then the uncertainty is taken to be the precision of the measuring equipment. NOT half the precision, but one full scale division. Please get this right as it seems to be a common mistake.
Percentage uncertainties
Another question almost guaranteed to occur in the ISA test paper so learn this properly, too.
Percentage uncertainty = uncertainty/mean value x 100%
Please note that you divide by the mean value.
A typical ISA test paper question asks you to determine the percentage uncertainty in your largest value of (for example) length. The "largest" part of this is merely directing you to the relevant row in your table. Find the uncertainty in your repeated length values and then divide by the mean length, not the biggest length value that you measured at any point.
If you measure a length of 20cm+/-2cm and another length of 50cm +/-2cm, which has been measured "best"? The actual uncertainties are the same but the percentage uncertainties are different.
The first length can be written as 20cm +/-10%; the second as 50cm +/-4%.
The second measurement is "better" (and arguably more accurate) as it has a lower percentage uncertainty.
In any experiment, you should try to reduce the percentage uncertainty in your values. This may involve reducing the actual uncertainty, or measuring a larger value.
The second factor is sometimes easier to do. If you are dropping an object to find the time of its fall, drop it over a large distance rather than a small one. If using a stopwatch, the uncertainty due to your reaction times will stay the same, but dividing it by a larger number will reduce the percentage uncertainty.
Note that measuring a larger value does not "reduce the uncertainty in the measurements". It reduces the percentage uncertainty. This distinction is important if you want to gain any marks for your written explanations.
It is usual to write a percentage uncertainty to only 1 (or maybe 2) significant figures. It is, after all, only an estimate! A percentage uncertainty of +/-2.785% is a very specific uncertainty - claiming that it is +/-2.785% rather than +/-2.787% is a very detailed claim. An Examiner will penalise this.
Combining uncertainties
This is likely to crop up in both the AS and at A2 test papers, with A2 being more challenging.
In a calculation, each piece of data may have its own uncertainty. You assume at A level that the uncertainties simply reinforce.
Rule 1: If adding or subtracting values (uncommon in most calculations) then add the actual uncertainties.
E.g. If X = 10 +/-2 and Y = 20 +/-1 then
X + Y = 30 +/-3 and
Y - X = 10 +/-3
Please don't subtract the uncertainties when subtracting values. If X and Y were both +/-2 then you would claim that subtracting two values with random errors produces a result with no error at all!
Rule 2: If multiplying or dividing values, add the percentage uncertainties.
E.g. If X = 10 +/-2 then X = 10 +/-20%. If Y = 20 +/-1 then Y = 20 +/-5%. This makes
XY = 200 +/-25% i.e. XY = 200 +/- 50
Y/X = 2 +/-25% i.e. Y/X = 2 +/-0.5
Note that the answers for the calculations can be expressed as an answer with a percentage uncertainty, or as an answer where the percentage has been converted back into an actual uncertainty. The latter version is more likely to be demanded at A2, but you do need to read questions carefully.
A specific case of rule 2 is when you raise a measurement to a power (e.g. square or cube it).
E.g. If Y = 20 +/-5% then Y squared = Y x Y = 400 +/-10%. In a similar way, Y cubed = 8000 +/-15%.
Square rooting a value requires more thought, but works along the same theme. You halve the percentage uncertainty as square rooting is raising a quantity to the power of a half.
E.g. If Z = 64 +/-8% then the square root of Z is 8 +/-4%.
[Why? Because if you then multiply the square root of Z by itself, the two values of 4% add back up to the 8% uncertainty in the original value of Z.]
