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 |
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.
Example
A ruler has millimeter divisions.
Smallest division:
Estimated uncertainty:
Digital Instruments
For digital instruments, the uncertainty is usually taken as
Example
A digital balance displays
The uncertainty is
Writing Measurements with Uncertainty
The uncertainty is usually written with the same number of decimal places as the measured value.
Examples:
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:
Measured value:
Absolute uncertainty:
Relative Uncertainty
Sometimes it is useful to compare the uncertainty with the measured value.
The relative uncertainty is
Example
Measured length:
Relative uncertainty
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.
Example
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:
Estimated uncertainty:
The measurement is written as
Example 2
A digital thermometer reads
The display changes in steps of
The measurement is recorded as
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.