BMS1031 - Uncertainities booklet

Introduction to Uncertainties

• Definition: In any scientific field, we work with measurements that inherently include uncertainties.

• Importance: Understanding uncertainties is crucial for evaluating scientific claims and conducting effective experiments.

• Practical Applications: In some experiments, the focus will be on understanding phenomena rather than precision, while others will involve actual measurement comparisons to theoretical values.

• Learning Goal: Students will learn how to estimate uncertainties and understand their significance in experiments.

Example of Uncertainties in Measurement

A1: Circle Measurement Example

• Task: A primary school student measures the circumference and diameter of circles to determine the ratio of circumference to diameter.

• Data Table:

  • Circle A: Circumference = 98 mm; Diameter = 30.00 mm; Ratio = 3.27

  • Circle B: Circumference = 127 mm; Diameter = 40.00 mm; Ratio = 3.17

• Expected Result: The ratio should equal π (approximately 3.14).

• Analysis of Measurements:

  • Measurements are approximate due to the limitations of the measuring tool (tape measure).

  • Circumferences can have uncertainties, expressed as c ± u(c) = 127 ± 0.5 mm, where u(c) is the estimated uncertainty due to the smallest measurement division.

Basic Rules for Estimating Uncertainties

A2: General Guidelines

• Different types of measurements have different sources and methods of estimating uncertainty.

• A clear distinction between 'uncertainty' (scientific term) and 'error' (often misinterpreted as a mistake).

A2.1: Repeated Measurements

• Rule 1: Calculating uncertainty for repeated measurements:𝒖(𝒙) = ± 𝟏/𝟐 (range of most measurements) (A.2)

  • Rationale: Multiple measurements give better average values and improve confidence.

  • Example: Time for liquid to flow through a plug was measured multiple times getting results between 11.9 s and 13.8 s.

    • Selected range for uncertainty calculation is from 11.9 s to 12.6 s, leading to 𝑢(𝑡) = ±0.4 s.

A2.2: Outliers

• Definition: Outliers are measurements significantly different from others.

• Rule 2: Investigate and repeat measurements before discarding any outlier.

  • Example: If time recorded as 13.8 s appears unusually high, check for issues with the measurement method.

A2.3: Significant Figures

• The number of digits shown should correlate with the uncertainty's significance.

• Rule 3: When uncertainty is shown to one significant figure, round the measured value to the same decimal place.

  • Example: From 𝑥 = 62.82 ± 0.31 m, round final value to 𝑥 = 62.8 ± 0.3 m.

A2.4: Reading Values

• Rule 4: Uncertainty can be half the smallest division on the instrument scale or half of the last digit shown on a digital device.

A2.5: Fractional Uncertainty

• Rule 5: Fractional uncertainty = 𝒖(𝑇) / 𝑇. This is important for calculating absolute uncertainty based on a known measurement.

  • Example: If a thermometer has a fractional uncertainty of 1%, then measuring 30°C results in an absolute uncertainty of 0.3°C.

A2.6: Combining Individual Uncertainties

• Rule 6: For multiple sources of uncertainty, the overall uncertainty is based on the largest individual uncertainty.

  • Example: If measuring body temperature, calculate uncertainties from different sources and use the largest one for final reporting.

Combining Uncertainties - Overview

A3: When Using Multiple Measurements

• The resultant uncertainties from different measurements can be combined to calculate total uncertainty in derived values.

A3.1: Multiplying and Dividing Measurements

• Rule 7a: For calculated quantities obtained through multiplication or division, add the fractional uncertainties:𝒖(𝑫) / 𝑫 = 𝒖(𝒙) / 𝒙 + 𝒖(𝒚) / 𝒚 (A.11)

  • Density example derived from mass and volume shows how to calculate total uncertainty based on individual measurements.

A3.2: Adding and Subtracting Measurements

• Rule 7b: For totals or differences, add the uncertainties:𝑢(𝑆) = 𝑢(𝑥) + 𝑢(𝑦) (A.12).

Improving Accuracy

• Strategies to enhance measurement accuracy include:

  • Allow adequate time for instruments to stabilize.

  • Regularly calibrate instruments to aligned standards.

  • Be mindful of zero offset and systematic errors that may bias results.

References for Uncertainties and Experimental Work

• Kirkup L and Frenkel RB (2006) An Introduction to Uncertainty in Measurement (Cambridge: University Press).

• Bureau Internationale de Poides et Measures (BIPM) resources for uncertainty expression guidelines.