Uncertainty in Measurement

Introduction

If you measure the length of a pencil as 14.3 cm, is it exactly 14.3 cm? Almost certainly not. It might be 14.28 cm or 14.33 cm; your ruler simply cannot tell the difference. Every measurement, however careful, has some doubt attached to it. Scientists call this doubt uncertainty . Stating the uncertainty is not an admission of failure — it is honest reporting that tells other people how much they can trust the number and whether two results really differ.

Core explanation

Why uncertainty is unavoidable. Three things limit every measurement:

- The instrument. Its resolution sets a limit on how finely the value can be read, and its calibration may not be perfect. - The observer. Judging the position of a meniscus, the moment a colour changes or when to stop a stopwatch involves human judgement. - The thing being measured. Temperatures drift, liquids evaporate and objects are not perfectly regular in shape.

Even with a perfect instrument, the last digit is always rounded, so there is always a small range of possible true values.

Writing uncertainty. A measurement is written as

value ± uncertainty, unit

For example, 14.3 ± 0.1 cm means the true length is expected to lie between 14.2 cm and 14.4 cm. The uncertainty has the same unit as the value, which is why it is called the absolute uncertainty .

Estimating the uncertainty of a single reading. A common rule is:

- For an analogue scale read at one point, the uncertainty is about half the smallest division. A thermometer marked every 1 °C gives readings of about ± 0.5 °C. - For a digital display, the uncertainty is at least one unit in the last digit. A balance reading 5.27 g gives about ± 0.01 g.

Measurements that use two readings. Many quantities are found as a difference between two readings — a length measured from one mark to another, or a burette volume found as final reading minus initial reading. Each reading has its own uncertainty, so the uncertainties add. A burette read to ± 0.05 cm³ at both ends gives a titre with an uncertainty of ± 0.10 cm³.

Matching digits to uncertainty. The value and its uncertainty should end at the same decimal place. Writing 14.327 ± 0.1 cm makes no sense: if you are unsure about the first decimal place, the second and third mean nothing. It should be 14.3 ± 0.1 cm.

Comparing results. Uncertainty lets us decide whether two results truly differ. If one group finds 25.2 ± 0.1 cm³ and another 25.3 ± 0.1 cm³, their ranges overlap, so the results agree within uncertainty.

Step-by-step reasoning

1. Take the reading to the full resolution of the instrument. 2. Decide the uncertainty: half a division for an analogue reading, one unit in the last digit for a digital display. 3. If the quantity comes from two readings, add their uncertainties. 4. Round the value so it ends at the same decimal place as the uncertainty. 5. Write value ± uncertainty with the unit.

Visual explanation

Draw a number line with the measured value as a dot. Around the dot draw a bar stretching the uncertainty in each direction. The true value lies somewhere along that bar. Results whose bars overlap agree; results whose bars do not overlap genuinely differ.

Real-world analogy

A weather forecast that says "18 to 22 °C tomorrow" is giving a value with an uncertainty: about 20 ± 2 °C. It is more useful than a single number, because it tells you how much the forecast could be off.

Real-world example

Food packaging in the UK carries an "e" mark, meaning the average contents meet the stated quantity and only a small permitted shortfall is allowed in individual packs. Manufacturers must understand the uncertainty in their filling machines to meet these rules without wasting product.

Why?

Why do uncertainties add when a value is found from two readings? Each reading could be off by up to its own uncertainty, and in the worst case the errors act in opposite directions, so the difference could be off by the sum of the two.

Common misconception

"Uncertainty means I made a mistake." Uncertainty is not the same as a mistake. A mistake, such as misreading a scale, can be avoided; uncertainty is a natural limit that exists in every measurement, even when everything is done correctly.

Worked example

Question: A burette's initial reading is 0.50 cm³ and its final reading is 24.35 cm³. Each reading has an uncertainty of ± 0.05 cm³. Find the titre and its uncertainty.

Reasoning: Titre = 24.35 − 0.50 = 23.85 cm³. Two readings are combined, so the uncertainties add: 0.05 + 0.05 = 0.10 cm³.

Answer: 23.85 ± 0.10 cm³.

Quick check

1. A thermometer is marked every 1 °C. What is the uncertainty of a single reading? Answer: About ± 0.5 °C, half the smallest division.

Exam focus

Know the conventions: half a division for a single analogue reading, one unit in the last digit for a digital display, and add uncertainties when two readings are subtracted. Make sure the value and uncertainty end at the same decimal place.

Advanced insight

Professional metrologists distinguish between uncertainties estimated from repeated readings (Type A) and those estimated from other knowledge, such as calibration certificates (Type B). When independent uncertainties are combined properly, they are often added "in quadrature" — the square root of the sum of squares — which gives a smaller total than simple addition. The simple adding rule used here is a safe upper estimate.

Summary

No measurement is perfectly exact: instruments, observers and samples all introduce doubt. Report results as value ± uncertainty, with the same unit and the same number of decimal places. Use half a division for an analogue reading and one unit in the last digit for a digital display. When two readings are combined by subtraction, add their uncertainties.

Practice questions

1. Give three sources of uncertainty in a measurement. Answer: The limited resolution or calibration of the instrument, the judgement of the observer, and variation in the object or sample being measured. 2. A ruler marked in millimetres is used to measure a length from the 0 mark to the 86 mm mark, each reading having an uncertainty of ± 0.5 mm. What is the length and its uncertainty? Answer: 86 ± 1 mm, because the two reading uncertainties of 0.5 mm add to 1 mm. 3. Rewrite 7.4382 ± 0.05 g correctly. Answer: 7.44 ± 0.05 g, so the value ends at the same decimal place as the uncertainty. 4. Two results are 12.4 ± 0.2 °C and 12.7 ± 0.2 °C. Do they agree within uncertainty? Explain. Answer: Yes; the ranges 12.2–12.6 °C and 12.5–12.9 °C overlap.