Introduction

No scientific measurement is perfectly exact. Every measuring instrument has limitations, and every measurement contains a small amount of uncertainty.

Instead of pretending measurements are perfect, scientists estimate the amount by which a measurement could differ from the true value. This estimate is called the measurement uncertainty.

Understanding uncertainty allows scientists to judge the quality and reliability of their data and compare the precision of different experiments.

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What is Uncertainty?

Uncertainty is an estimate of the possible error in a measurement.

It tells us the range within which the true value is likely to lie.

For example:

Length = 25.4 ± 0.1 cm

This means the true length is expected to lie between:

• 25.3 cm
• 25.5 cm

The symbol ± means "plus or minus."

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Absolute Uncertainty

The absolute uncertainty is the uncertainty expressed in the same units as the measurement.

Examples:

Absolute uncertainty tells us the possible error directly.

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Estimating Absolute Uncertainty

For many measuring instruments:

Analogue Instruments

The absolute uncertainty is usually taken as:

± half the smallest scale division

Example:

A ruler marked every 1 mm

Smallest division:

1 mm

Absolute uncertainty:

±0.5 mm

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Digital Instruments

For digital instruments, the uncertainty is usually:

± one smallest displayed unit

Example:

Digital balance displays:

15.62 g

Smallest displayed value:

0.01 g

Absolute uncertainty:

±0.01 g

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Percentage Uncertainty

Sometimes absolute uncertainty alone is not enough.

Scientists often calculate percentage uncertainty to compare different measurements.

Percentage uncertainty tells us how large the uncertainty is relative to the size of the measurement.

The formula is:

\( Percentage \ Uncertainty = \frac{Absolute \ Uncertainty}{Measured \ Value} \cdot 100 \)%

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Example 1

Length measured:

50.0 cm

Absolute uncertainty:

±0.5 cm

Percentage uncertainty:

\( \frac{0.5}{50.0} \cdot100 = 1 \)%

Answer:

1%

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Example 2

Mass measured:

200 g

Absolute uncertainty:

±2 g

Percentage uncertainty:

\( \frac{2}{200} \cdot100 = 1 \)%

Answer:

1%

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Example 3

Time measured:

5.0 s

Absolute uncertainty:

±0.2 s

Percentage uncertainty:

\( \frac{0.2}{5.0} \cdot100 = 4 \)%

Although the absolute uncertainty is small, the percentage uncertainty is much larger.

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Comparing Measurements

Percentage uncertainty allows fair comparisons.

Consider two measurements.

Both have the same absolute uncertainty.

However:

• The first measurement is much more reliable.
• The second has a much larger relative uncertainty.

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Uncertainty and Reliability

Smaller percentage uncertainties usually indicate:

• More reliable measurements
• Greater precision
• Better-quality experimental data

Large percentage uncertainties suggest the experiment may need improvement.

Scientists always try to reduce uncertainty whenever possible.

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Reducing Uncertainty

There are many ways to reduce measurement uncertainty.

These include:

• Using instruments with finer scale divisions.
• Taking repeated measurements.
• Calculating the average value.
• Measuring larger quantities when appropriate.
• Reading scales at eye level to avoid parallax error.
• Calibrating equipment before use.

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Reporting Measurements

Measurements should include both the measured value and its uncertainty.

Examples:

• 15.2 ± 0.1 cm
• 0.845 ± 0.005 kg
• 12.50 ± 0.02 s

This gives other scientists a clear understanding of the quality of the measurement.

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Real-World Applications

Measurement uncertainty is essential in:

• Scientific research
• Medical testing
• Pharmaceutical manufacturing
• Engineering
• Aerospace
• Environmental monitoring
• Construction
• Quality control

Knowing the uncertainty helps scientists decide whether results are trustworthy and whether differences between measurements are meaningful.

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Worked Examples

Example 1

A ruler has millimetre divisions.

What is its absolute uncertainty?

Answer:

±0.5 mm

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Example 2

A balance measures:

120.0 g

Absolute uncertainty:

±0.1 g

Calculate the percentage uncertainty.

Solution:

\( \frac{0.1}{120.0} \cdot100 = 0.083\)%

Answer:

0.083%

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Example 3

A thermometer reads:

25.4 ± 0.5°C

What does this mean?

Answer:

The true temperature is likely between:

24.9°C and 25.9°C.

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Example 4

Which measurement is more reliable?

A.

10.0 ± 0.5 cm

B.

100.0 ± 0.5 cm

Answer:

Measurement B has the smaller percentage uncertainty and is therefore more reliable.

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Example 5

Give two ways to reduce uncertainty.

Answer:

Possible answers include:

• Use more precise measuring equipment.
• Repeat measurements and calculate the average.
• Calibrate measuring instruments.
• Avoid parallax error.

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Did You Know?

Scientists at particle accelerators such as CERN often report measurements with extremely small uncertainties. Even tiny improvements in uncertainty can lead to major discoveries, helping physicists distinguish between ordinary measurement variations and evidence for entirely new particles or physical phenomena.

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Key Terms

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Key Takeaways

• Every scientific measurement has some uncertainty.
• Absolute uncertainty is expressed in the same units as the measurement.
• Percentage uncertainty compares the uncertainty with the size of the measurement.
• Smaller percentage uncertainties generally indicate more reliable measurements.
• Repeating measurements, using more precise instruments, and careful measuring techniques help reduce uncertainty.
• Scientific measurements should always be reported together with an appropriate estimate of their uncertainty.