Introduction

Scientists often calculate new quantities from measurements. For example, you might measure the length and width of a table to calculate its area, or measure distance and time to calculate speed.

Since every measurement contains some uncertainty, the calculated result must also contain uncertainty.

The process of determining how measurement uncertainties affect a calculated result is called error propagation (or uncertainty propagation).

Understanding error propagation helps scientists determine how reliable their calculated results are.

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Why Does Uncertainty Propagate?

Imagine measuring the sides of a rectangle.

Length:

20.0 ± 0.1 cm

Width:

10.0 ± 0.1 cm

Neither measurement is exact.

Therefore, the calculated area cannot be exact either.

Whenever measurements are combined mathematically, their uncertainties combine as well.

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Different Calculations Combine Uncertainties Differently

The way uncertainties combine depends on the type of mathematical calculation.

These simple rules work well for most school-level laboratory calculations.

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Addition

When adding measurements:

Add the absolute uncertainties.

Example

Length A:

12.0 ± 0.2 cm

Length B:

8.0 ± 0.1 cm

Total length:

20.0 cm

Total uncertainty:

±(0.2 + 0.1)

= ±0.3 cm

Final answer:

20.0 ± 0.3 cm

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Subtraction

Subtract the values, but add the absolute uncertainties.

Example

Initial volume:

50.0 ± 0.2 mL

Final volume:

20.0 ± 0.2 mL

Difference:

30.0 mL

Uncertainty:

±(0.2 + 0.2)

= ±0.4 mL

Final answer:

30.0 ± 0.4 mL

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Multiplication

For multiplication, use percentage uncertainties.

Step 1

Calculate the percentage uncertainty of each measurement.

Step 2

Add the percentage uncertainties.

Step 3

Calculate the uncertainty of the final answer if required.

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Example

Length:

20.0 ± 0.2 cm

Percentage uncertainty:

\( \frac{0.2}{20.0} \cdot100 = 1 \)%

Width:

10.0 ± 0.1 cm

Percentage uncertainty:

\( \frac{0.1}{10.0} \cdot100 = 1 \)%

Area:

20.0 × 10.0 = 200 cm²

Combined percentage uncertainty:

1% + 1% = 2%

Final answer:

200 ± 4 cm²

since:

2% of 200 = 4

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Division

Division follows exactly the same rule as multiplication.

Add the percentage uncertainties.

Example

Distance:

100.0 ± 1.0 m

Percentage uncertainty:

1%

Time:

20.0 ± 0.2 s

Percentage uncertainty:

1%

Speed:

100 ÷ 20 = 5.0 m/s

Combined percentage uncertainty:

2%

Final answer:

5.0 ± 0.1 m/s

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Summary of Error Propagation Rules

This simple table is one of the most useful references in experimental science.

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Estimating Uncertainty in Derived Quantities

Many physical quantities are calculated from measurements.

Examples include:

Each calculated quantity inherits uncertainty from the measurements used to calculate it.

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Reliability of Calculated Results

A derived quantity with a small percentage uncertainty is generally considered more reliable than one with a large percentage uncertainty.

For example:

The first result is much more reliable.

Scientists often compare percentage uncertainties when evaluating experimental methods.

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Reporting Final Answers

Always report:

• the calculated value
• its uncertainty
• the correct units

Examples:

• 3.25 ± 0.05 m/s
• 200 ± 4 cm²
• 1.82 ± 0.03 g/cm³

Avoid reporting more decimal places than are justified by the uncertainty.

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Real-World Applications

Error propagation is important in:

• Engineering
• Scientific research
• Medicine
• Manufacturing
• Environmental monitoring
• Space exploration
• Physics experiments
• Chemistry laboratories

Whenever measurements are used in calculations, uncertainty must also be considered.

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Worked Examples

Example 1

A rod measures:

15.0 ± 0.2 cm

Another rod measures:

8.0 ± 0.1 cm

Find the total length.

Solution:

Length:

15.0 + 8.0 = 23.0 cm

Uncertainty:

0.2 + 0.1 = 0.3 cm

Answer:

23.0 ± 0.3 cm

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Example 2

A rectangle measures:

12.0 ± 0.2 cm

by

5.0 ± 0.1 cm

Area:

60 cm²

Percentage uncertainties:

1.7%

2%

Combined:

3.7%

Absolute uncertainty:

3.7% of 60 ≈ 2 cm²

Answer:

60 ± 2 cm²

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Example 3

A car travels:

240 ± 2 km

in

4.0 ± 0.1 h

Calculate the speed.

Speed:

240 ÷ 4 = 60 km/h

Percentage uncertainties:

Distance:

0.83%

Time:

2.5%

Combined:

3.33%

Absolute uncertainty:

3.33% of 60 ≈ 2 km/h

Answer:

60 ± 2 km/h

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Example 4

Which result is more reliable?

A.

40 ± 1

B.

40 ± 5

Answer:

Result A has the smaller percentage uncertainty and is therefore more reliable.

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Example 5

Why should uncertainty be reported with calculated values?

Answer:

Because calculated quantities inherit uncertainty from the original measurements, and reporting the uncertainty shows how reliable the result is.

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Did You Know?

Space missions depend on careful uncertainty analysis. Engineers calculate how uncertainties in fuel mass, engine performance, and navigation measurements combine over time. Even tiny uncertainties can grow during a journey of millions of kilometres, so understanding error propagation is essential for guiding spacecraft accurately to their destinations.

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Key Terms

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Key Takeaways

• Calculated quantities inherit uncertainty from the measurements used to calculate them.
• For addition and subtraction, add the absolute uncertainties.
• For multiplication and division, add the percentage uncertainties.
• Smaller percentage uncertainties generally indicate more reliable results.
• Always report calculated values together with their uncertainties and units.
• Understanding error propagation helps scientists judge the quality and reliability of experimental results.