Measurement Uncertainty | Learn Physics with Erkan Koymen

Definition

No measurement is perfectly exact.

Every measurement contains a degree of uncertainty because measuring instruments, observers, and environmental conditions all have limitations.

Measurement uncertainty is an estimate of the range within which the actual value of a measured quantity is expected to lie.

Why Do Measurements Have Uncertainty?

Every measuring process has limitations. Some common sources of uncertainty include:

Limited resolution of measuring instruments
Human judgment when reading analog scales
Small variations in experimental conditions
Random fluctuations during measurement

Even when the same quantity is measured repeatedly under identical conditions, the measured values may not be exactly the same.

Instrument Resolution

The smallest value that an instrument can measure is called its resolution (or least count).

The resolution determines the smallest change that can be detected and therefore affects the uncertainty of the measurement.

A higher-resolution instrument generally allows a smaller measurement uncertainty.

Examples

Instrument	Resolution
Ruler	1 mm
Vernier caliper	0.02 mm
Micrometer	0.01 mm
Digital balance	0.01 g

Table 1.8.1 — Instrument Resolution. Examples of the resolution of common measuring instruments.

Estimating Measurement Uncertainty

The uncertainty depends on the type of measuring instrument.

Analog Instruments

For most analog instruments, the measurement uncertainty is commonly estimated as half of the smallest scale division.

\pm \frac{\text{smallest scale division}}{2}

Example

A ruler has millimeter divisions.

Smallest division:

1\,\mathrm{mm}

Estimated uncertainty:

\pm 0.5\,\mathrm{mm}

Digital Instruments

For digital instruments, the uncertainty is usually taken as

\pm 1\ \text{least significant digit}

Example

A digital balance displays

52.37\,\mathrm{g}

The uncertainty is

\pm 0.01\,\mathrm{g}

Writing Measurements with Uncertainty

The uncertainty is usually written with the same number of decimal places as the measured value.

Examples:

\begin{aligned} (12.4 \pm 0.1)\,\mathrm{cm} \\ (5.62 \pm 0.01)\,\mathrm{g} \\ (2.38 \pm 0.05)\,\mathrm{s} \end{aligned}

This notation indicates that the actual value is expected to lie within the stated interval.

Absolute Uncertainty

The uncertainty written with the measurement is called the absolute uncertainty.

Example:

(8.5 \pm 0.2)\,\mathrm{cm}

Measured value:

8.5\,\mathrm{cm}

Absolute uncertainty:

\pm 0.2\,\mathrm{cm}

Relative Uncertainty

Sometimes it is useful to compare the uncertainty with the measured value.

The relative uncertainty is

\text{Relative uncertainty}=\frac{\text{Absolute uncertainty}}{\text{Measured Value}}

Example

Measured length:

50.0 \pm 0.5\,\mathrm{cm}

Relative uncertainty

=\frac{0.5}{50.0}=0.010

Relative uncertainty has no unit because it is the ratio of two quantities with the same unit.

Percentage Uncertainty

Relative uncertainty is often expressed as a percentage.

\text{Percentage uncertainty}=\frac{\text{Absolute uncertainty}}{\text{Measured Value}}\times 100\%

Example

\frac{0.5}{50.0}\times 100=1.0\%

Percentage uncertainty makes it easier to compare the quality of different measurements.

Example 1

A ruler with a smallest scale division of 1 mm is used to measure the length of a pencil.

Measured value:

15.8\,\mathrm{cm}

Estimated uncertainty:

\pm 0.05\,\mathrm{cm}

The measurement is written as

(15.80 \pm 0.05)\,\mathrm{cm}

Example 2

A digital thermometer reads

24.6\,^\circ\mathrm{C}

The display changes in steps of

0.1\,^\circ\mathrm{C}

The measurement is recorded as

(24.6 \pm 0.1)\,^\circ\mathrm{C}

Key Points

Every measurement has uncertainty.
Measurement uncertainty does not mean the measurement is incorrect.
Instrument resolution limits the precision of a measurement.
Analog and digital instruments use different methods for estimating uncertainty.
Measurements should be reported together with their uncertainty.
Relative and percentage uncertainty make it easier to compare the precision of different measurements.

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