Measurement Uncertainty & Limits of Error – CSEC Physics Practical Guide

In physics, every measurement comes with a degree of uncertainty. Understanding how to estimate, express, and work with this uncertainty is essential for conducting experiments and reporting reliable results. This guide covers the key concepts of precision, accuracy, limits of error, and significant figures as outlined in the CSEC Physics syllabus.

Key Idea: No measurement is ever exact. The “true value” of a quantity is always unknown. What we report is an estimate with a range of possible values.

1. Precision vs. Accuracy

Accuracy refers to how close a measured value is to the true or accepted value.
Precision refers to how close repeated measurements are to each other, and is determined by the instrument’s smallest division (its sensitivity).

Diagram illustrating accuracy vs precision

Figure 1: Visualizing accuracy and precision. High precision means measurements are clustered; high accuracy means they are centered on the true value.

2. Uncertainty (Error) in Measurements

Uncertainty (or error) is the interval within which the true value is expected to lie. It arises from:

Instrument limitations (precision of the device)
Parallax errors (reading a scale from the wrong angle)
Reaction time errors (e.g., starting/stopping a stopwatch)
Environmental factors (e.g., temperature, drafts)

2.1 Determining Uncertainty for Common Instruments

Instrument	Smallest Division	Uncertainty (Generally)
Metre rule (mm marks)	1 mm	±0.5 mm
Vernier caliper	0.1 mm	±0.1 mm
Micrometer screw gauge	0.01 mm	±0.01 mm
Digital balance (0.1 g)	0.1 g	±0.05 g
Stopwatch (0.1 s)	0.1 s	±0.1 s (plus reaction error)

Rule of Thumb: For analogue scales, the uncertainty is usually taken as half the smallest division. For digital instruments, it is ± the last digit shown.

3. Expressing Measurements with Limits of Error

A measurement should be stated as: measured value ± uncertainty with appropriate units.

Example: A length measured with a metre rule (mm divisions) is recorded as 6.4 cm.
The smallest division is 0.1 cm (1 mm), so uncertainty = ±0.05 cm.
The result should be reported as: 6.40 cm ± 0.05 cm or (6.40 ± 0.05) cm.
This means the true length lies between 6.35 cm and 6.45 cm.

4. Significant Figures

Significant figures (s.f.) indicate the precision of a measurement. The last digit recorded is uncertain.

All non-zero digits are significant.
Zeros between non-zero digits are significant.
Leading zeros are not significant.
Trailing zeros after a decimal point are significant.

Examples:
0.0023 has 2 s.f. (leading zeros not counted)
120.40 has 5 s.f. (trailing zero after decimal counts)
1.230 × 10³ has 4 s.f.

4.1 Combining Measurements – Rule of Thumb

When adding, subtracting, multiplying, or dividing, the result should have no more significant figures than the least precise measurement used.

Example: Area of rectangle = length × breadth = 6.02 cm × 4.15 cm = 24.983 cm² (calculator).
Both length and breadth have 3 s.f., so area = 25.0 cm² (3 s.f.).

5. Spreading the Error

Measuring many items together reduces the relative uncertainty per item.

Example: Measuring the thickness of one page of a book directly is impossible with a mm ruler. Instead, measure the total thickness of 100 pages (e.g., 14.2 mm ± 0.5 mm). The thickness of one page = (14.2 ± 0.5) / 100 = 0.142 mm ± 0.005 mm. The uncertainty per page is much smaller.

This principle is used in timing pendulum swings (time 10 swings, then divide by 10) and measuring small masses or volumes.

6. Practical Tips for CSEC Physics Experiments

Always record the precision of your instrument in your method.
State measurements with their uncertainty in your results table.
Use appropriate significant figures in calculated results.
Draw error bars on graphs where possible.
Discuss sources of error and how they could be minimized in your conclusion.

CSEC Exam Practice

1. A student measures the length of a wire five times: 12.1 cm, 12.2 cm, 12.1 cm, 12.0 cm, 12.2 cm. The metre rule used has mm divisions. What is the best estimate of the length with its uncertainty?

Mean = (12.1+12.2+12.1+12.0+12.2)/5 = 12.12 cm ≈ 12.1 cm (to 1 decimal place).
Uncertainty = ±0.05 cm (half of smallest division 0.1 cm).
Answer: (12.10 ± 0.05) cm.

2. A digital ammeter reads 1.35 A. What is the uncertainty in the reading?

For digital instruments, uncertainty = ± the last digit.
Last digit is 0.01 A, so uncertainty = ±0.01 A.
Answer: ±0.01 A.

3. The sides of a rectangle are measured as 5.2 cm and 3.7 cm, each to the nearest mm. Calculate the area with the correct number of significant figures.

Area = 5.2 cm × 3.7 cm = 19.24 cm² (calculator).
Both sides have 2 s.f., so area should have 2 s.f.
Answer: 19 cm² (2 s.f.).

4. Why is it better to time 20 oscillations of a pendulum rather than just one when determining the period?

Timing more oscillations spreads the reaction error over a longer time interval, reducing the percentage uncertainty in the period. The ±0.2 s reaction error becomes less significant when divided by 20.

5. A cylinder has diameter 2.45 cm ± 0.01 cm and height 10.0 cm ± 0.1 cm. Calculate the volume and its approximate percentage uncertainty.

Radius = 1.225 cm ± 0.005 cm.
Volume V = πr²h = π × (1.225)² × 10.0 ≈ 47.1 cm³.
% uncertainty in r = (0.005/1.225)×100% ≈ 0.41%.
% uncertainty in h = (0.1/10.0)×100% = 1%.
Since r is squared, its contribution doubles: 2×0.41% = 0.82%.
Total % uncertainty ≈ 0.82% + 1% = 1.82% ≈ 2%.
Answer: Volume ≈ 47 cm³ ± 2%.

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