Percentage Error and Percentage Uncertainty

Introduction

Saying a result is "a bit out" is not very scientific. How far out? Is that a lot or a little? Two percentages answer these questions. Percentage error compares your result with the accepted value. Percentage uncertainty tells you how much doubt your apparatus introduces. Comparing the two shows whether a discrepancy is simply due to the limits of your equipment or points to a real problem in the method.

Core explanation

Percentage error. When an accepted value is known, the percentage error measures accuracy:

percentage error = ( experimental value − accepted value ÷ accepted value) × 100%

The vertical bars mean "ignore the sign". Sometimes the sign is kept to show whether the result is too high or too low. A small percentage error means an accurate result.

For example, if a student measures the density of copper as 8.60 g/cm³ and the accepted value is 8.96 g/cm³: percentage error = (0.36 ÷ 8.96) × 100% = 4.0%.

Absolute uncertainty. Every instrument reading has an uncertainty. A common convention is:

- For a single reading on an analogue scale (such as a thermometer), the uncertainty is half the smallest division. - For a digital instrument, it is ± one unit in the last digit. - Manufacturers often state the tolerance of glassware directly, for example a 25.0 cm³ pipette ±0.06 cm³.

Where a quantity is found from two readings — such as a burette titre (final − initial) or a mass by difference — both readings contribute, so the uncertainties add. A burette read to ±0.05 cm³ gives a titre uncertainty of ±0.10 cm³.

Courses differ slightly in these conventions, so follow the one your exam board uses.

Percentage uncertainty.

percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%

It shows how significant the uncertainty is compared with the size of the measurement. A ±0.10 cm³ uncertainty in a 25.00 cm³ titre is 0.40%; the same uncertainty in a 5.00 cm³ titre is 2.0%. Larger measurements give smaller percentage uncertainties , which is why chemists design experiments with substantial titres and masses.

Comparing the two. Add the percentage uncertainties of the key measurements to estimate the total percentage uncertainty of the result. Then:

- If the percentage error is smaller than or about equal to the total percentage uncertainty, the difference can be explained by the limitations of the apparatus. - If the percentage error is much larger , there must be other errors — usually systematic errors in the method, such as heat loss or impure reagents.

Formulae

percentage error = ( experimental − accepted ÷ accepted) × 100%

percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%

Step-by-step reasoning

To evaluate a result:

1. Find the difference between the experimental and accepted values. 2. Divide by the accepted value and multiply by 100 to get the percentage error. 3. Work out the absolute uncertainty of each measurement. 4. Convert each to a percentage uncertainty and add them. 5. Compare the percentage error with the total percentage uncertainty and draw a conclusion.

Visual explanation

Draw a number line with the accepted value marked. Around your result, draw an error bar stretching the total uncertainty either side. If the bar reaches the accepted value, the difference is within the apparatus limits; if there is a gap, something else has gone wrong.

Real-world analogy

Being 1 minute late for a 5-minute meeting is a big deal; being 1 minute late on a 3-hour journey hardly matters. The absolute amount is the same, but the percentage tells you how serious it is — exactly the idea behind percentage uncertainty.

Real-world example

Food labelling laws allow the actual contents of a packet to differ from the stated mass only within set tolerances. Manufacturers use percentage figures because a 5 g shortfall is trivial in a 1 kg bag but significant in a 50 g bar.

Why?

Why use percentages rather than absolute differences? A percentage puts the error or uncertainty in proportion to the size of the quantity. This lets you compare measurements of different sizes and different units — a mass, a volume and a temperature change — on the same scale.

Common misconception

"Percentage uncertainty and percentage error are the same thing." Percentage error compares your answer with an accepted value and describes accuracy. Percentage uncertainty comes from the resolution of your apparatus and exists even when no accepted value is known.

Worked example

Question: A 250 cm³ volumetric flask has an uncertainty of ±0.2 cm³ and a balance has a stated uncertainty of ±0.005 g. A student weighs 1.25 g of solid by a single reading. Find each percentage uncertainty and the total.

Reasoning: Flask: (0.2 ÷ 250) × 100% = 0.08%. Mass: (0.005 ÷ 1.25) × 100% = 0.4%. Total = 0.08% + 0.4% = 0.48%, about 0.5%.

Answer: Flask 0.08%, mass 0.4%, total about 0.5%; the mass measurement contributes most of the uncertainty.

Quick check

1. An experiment gives 52 °C where the accepted value is 50 °C. What is the percentage error? Answer: (2 ÷ 50) × 100% = 4%.

Exam focus

Show the formula, the substitution and the answer with a percentage sign. Remember to double the uncertainty for quantities found from two readings, such as a titre. A common evaluation question asks you to suggest how to reduce percentage uncertainty: use a larger quantity or a more precise instrument.

Advanced insight

When quantities are multiplied or divided, their percentage uncertainties add to give the percentage uncertainty in the result. When they are added or subtracted, the absolute uncertainties add instead. More advanced treatments combine independent uncertainties in quadrature (square root of the sum of squares), which gives a slightly smaller, more realistic total.

Summary

Percentage error = experimental − accepted ÷ accepted × 100% and measures accuracy. Percentage uncertainty = absolute uncertainty ÷ measured value × 100% and reflects the limits of the apparatus. Larger measurements reduce percentage uncertainty. If the percentage error exceeds the total percentage uncertainty, other errors, usually systematic, must be present.

Practice questions

1. A student finds the molar mass of a gas to be 42 g/mol; the accepted value is 44 g/mol. Calculate the percentage error. Answer: (2 ÷ 44) × 100% = 4.5%. 2. A burette titre of 20.00 cm³ is measured with each reading ±0.05 cm³. Calculate the percentage uncertainty of the titre. Answer: Total uncertainty = 2 × 0.05 = 0.10 cm³; (0.10 ÷ 20.00) × 100% = 0.50%. 3. Why does using a titre of 40 cm³ instead of 10 cm³ reduce percentage uncertainty? Answer: The absolute uncertainty stays the same, but it is divided by a larger measured value, giving a smaller percentage. 4. A result has a percentage error of 12% and a total percentage uncertainty of 1.5%. What can you conclude? Answer: The error is far larger than the apparatus uncertainty, so there must be other errors, probably systematic errors in the method.