An uncertainty is not an admission that you did something wrong. It is a statement of how sharply your equipment could ever have answered the question — and comparing it against how far off you actually were is the most useful thing you can do with a set of results.
📚 What you need to know
An uncertainty is the range around a measurement within which the true value is expected to lie.
An error is something in the equipment or technique that makes a reading differ from the true value. They are not the same idea.
Analogue instrument: ± half the smallest division. Digital: ± the last digit shown. Repeated data: ± half the range.
Absolute uncertainty is the ± amount; fractional is that divided by the value; percentage is the fraction × 100.
Adding or subtracting: add the absolute uncertainties.
Multiplying or dividing: add the percentage uncertainties.
Reduce a percentage uncertainty by using finer equipment or by measuring a larger quantity.
Uncertainty is not error
The words get swapped constantly and they describe different things. An uncertainty is a property of the measurement: a burette marked every 0.10 cm3 simply cannot resolve better than about ±0.05 cm3, no matter how careful you are. An error is something that pushed the reading away from the truth — heat lost to the room, an uncalibrated probe, a reading taken from above.
The practical consequence: you can reduce an uncertainty by choosing better apparatus, but you reduce an error by changing the method.
Where the number comes from
That last line covers most of the practical work you will do. A titre, a ΔT and a change in mass are all found by reading an instrument twice, so each carries twice the single-reading uncertainty.
Absolute, fractional, percentage
Three ways of saying the same thing. Take a burette reading of 19.60 cm3 on a scale divided every 0.10 cm3.
The same uncertainty, three ways
absolute = 0.10 ÷ 2 = ±0.05 cm3 fractional = 0.05 ÷ 19.60 = 0.0026 percentage = 0.0026 × 100 = 0.26%
The percentage form is the one that lets you compare. An uncertainty of ±0.05 cm3 is trivial on a 25 cm3 titre and serious on a 2 cm3 one, and only the percentage tells you which situation you are in.
This is exactly why percentage uncertainty falls when you measure more. The ±0.05 cm3 stays the same whatever you do, so making the titre bigger — by using a more dilute titrant, say — puts a bigger number underneath it. The other route is finer equipment, which shrinks the top instead.
Combining uncertainties
The last step matters. A result quoted as 2.45 × 10–3 ± 0.91% is not wrong, but the expected form is an absolute uncertainty in the same units as the result.
Uncertainty against error: the useful comparison
Here is the move that turns a set of numbers into an evaluation. Work out the total percentage uncertainty of your result, then work out the percentage error against the literature value, and compare them.
If the error is far larger than the uncertainty, no amount of careful reading would have saved the result. Something in the method is biasing it, and that is what the evaluation should name.
Uncertainty bars on a graph make the same point visually. Draw each point with a bar of ± its absolute uncertainty; if a straight line can be drawn passing through every bar, the data supports a linear relationship. If it cannot, either the relationship is not linear or something has been underestimated.
The coefficient of determination
Spreadsheets will offer you R2 alongside a trend line. It measures how well the line fits the points: 0 means no predictive value at all, 1 means the line passes through every point exactly, and anything between describes the quality of the fit.
Treat it carefully. A high R2 says the line describes those points well; it does not confirm that a straight line was the right model in the first place, and it says nothing about whether a systematic error shifted them all together.
WORKED EXAMPLE
A titration delivers 24.50 ± 0.10 cm3 of a solution of concentration 0.1000 ± 0.0005 mol dm–3. Calculate the amount in moles and its absolute uncertainty.
Step 1 — the two percentage uncertaintiesvolume: 0.10 ÷ 24.50 × 100 = 0.41%concentration: 0.0005 ÷ 0.1000 × 100 = 0.50%Step 2 — the operation is a multiplicationn = cV, so add the percentages.0.41 + 0.50 = 0.91%Step 3 — the valuen = 0.1000 × 0.02450 = 2.450 × 10⁻³ molStep 4 — convert back to absolute0.91% of 2.450 × 10⁻³ = 0.022 × 10⁻³n = (2.450 ± 0.022) × 10⁻³ molThe titre uncertainty is ±0.10 not ±0.05, because two burette readings were taken.
WORKED EXAMPLE
In a calorimetry experiment a thermometer reading to ±0.5 °C gives an initial temperature of 21.5 °C and a final temperature of 34.0 °C. Calculate the temperature rise, its absolute uncertainty and its percentage uncertainty.
Step 1 — the rise∆T = 34.0 − 21.5 = 12.5 °CStep 2 — a subtraction, so add the absolutes0.5 + 0.5 = ±1.0 °CStep 3 — convert to a percentage1.0 ÷ 12.5 × 100 = 8.0%∆T = 12.5 ± 1.0 °C, or ±8.0%8% is large, and it comes straight from the thermometer. A probe reading to ±0.1 °C would cut it to 1.6% without changing anything else.
WORKED EXAMPLE
Using the temperature rise above, a student obtains an enthalpy of combustion of –520 kJ mol–1 where the literature value is –726 kJ mol–1. The total percentage uncertainty in the experiment is about 10%. Comment on the result.
Step 1 — the percentage error(726 − 520) ÷ 726 × 100 = 28.4%Step 2 — compare it with the uncertainty28.4% is much larger than 10%the uncertainty cannot account for the discrepancyStep 3 — so what is left?A systematic error must be present. The result is far less exothermic than it should be, which is consistent with heat being lost to the surroundings and to the apparatus rather than reaching the water.Notice the logic: the comparison is what licenses you to say “systematic”. Without it, “there must have been heat loss” is only a guess.
💡 Exam tip
State the rule you are using: half a division, the last digit, or half the range.
Double the uncertainty for anything found from two readings — titres, temperature rises, mass changes.
Add absolutes for + and −; add percentages for × and ÷.
Convert the final percentage back into an absolute uncertainty with units.
To reduce percentage uncertainty, say finer equipment or a larger measured quantity.
In evaluations, compare percentage error with total percentage uncertainty before naming an error as systematic.
⚠️ Common mix-up
Using uncertainty and error as synonyms. One is a property of the instrument, the other of the method.
Adding percentages when the operation was a subtraction. Differences take absolute uncertainties.
Quoting a titre as ±0.05 cm3. Two readings were taken, so it is ±0.10.
Leaving the answer as a percentage instead of converting back to an absolute with units.
Reading a high R2 as proof of accuracy. It measures fit, not correctness.
Up next: Graphing Skills — uncertainty bars, gradients and intercepts all live on a graph, and a graph drawn properly answers questions that a table of numbers cannot.
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