Measurement Uncertainty and Errors
Overview
No physical measurement is perfectly exact. Every reading is limited by:
- instrument resolution
- observer judgement
- experimental method
- environmental effects
Therefore, measured values should be reported with uncertainty and interpreted with an understanding of error sources.
This page expands the uncertainty section of Measurement.
Why It Matters
Uncertainty and error analysis tell you how reliable a measurement is and whether an experimental result deserves confidence.
Definition
Measurement uncertainty is the estimated range within which the true value is likely to lie.
Key Representations
Error vs Uncertainty
Error
Error is the difference between a measured value and the true (accepted) value.
Example:
True value of
Measured value:
Error:
Uncertainty
Uncertainty gives the estimated range within which the true value is likely to lie.
Example:
Likely range:
Uncertainty is usually more useful than quoting error, since the true value is often unknown.
Absolute Uncertainty
Absolute uncertainty is written in the same unit as the measurement.
Examples:
Instrument-Based Estimate
For analogue instruments:
- uncertainty is often taken as half the smallest scale division
Example:
Metre rule marked every
Estimated uncertainty:
For digital meters:
- uncertainty often taken as least significant digit unless stated otherwise
Figure: For a single reading from an analogue instrument, uncertainty is often estimated as half the smallest scale division.
Fractional Uncertainty
Example:
Then:
Percentage Uncertainty
Using the previous example:
Why Percentage Uncertainty Matters
It allows comparison of quality of different measurements.
Compare:
- →
- →
Although both have absolute uncertainty, the first is much better relatively.
Figure: Random errors cause scatter about the true trend, while systematic errors shift the best-fit line away from the true line.
Systematic Errors
Systematic errors shift readings consistently in one direction.
They reduce accuracy.
Examples
- zero error on vernier calipers
- stopwatch consistently slow
- poor calibration
- heat loss in calorimeter
- friction ignored in theory
Features
- repeated readings may agree closely
- averaging does not remove it
- corrected by improving method or calibration
Random Errors
Random errors cause scatter in repeated readings.
They reduce precision.
Examples
- reaction time variation
- fluctuating surroundings
- slight reading judgement differences
- unstable power supply
Features
- values spread above and below mean
- reduced by repeated measurements
- averaging helps
Precision vs Accuracy
Precision
How close repeated readings are to one another.
High precision means:
- small spread
- low random uncertainty
Accuracy
How close reading (or mean value) is to true value.
High accuracy means:
- low systematic error
Typical Cases
High Precision, Low Accuracy
Readings:
Very close together, but if true value is , then inaccurate.
Low Precision, Good Average Accuracy
Readings:
Spread is larger, but average may be close to true value.
Repeated Measurements
Repeated readings improve reliability.
Mean Value
Range Estimate
Sometimes uncertainty estimated from spread:
Used when repeated values vary significantly.
Example: Pendulum Timing
Instead of timing one oscillation:
- reaction time large relative to measurement
Better method:
- time 20 oscillations
- divide by 20
This reduces percentage uncertainty.
Recording Measurements Correctly
Write measured value and uncertainty consistently.
Correct
Incorrect
(decimal places inconsistent)
Rule for Decimal Places
The measured value should be rounded to the same decimal place as the uncertainty.
Examples:
Correct
Correct
Incorrect
Practical Examples
Example 1: Measuring Wire Diameter
Using metre rule:
resolution poor
Using micrometer screw gauge:
better precision
Hence choose the better instrument.
Example 2: Measuring Time of Motion
Short motion lasting timed manually gives high percentage uncertainty.
Better:
- use light gate
- increase timing interval if possible
Example 3: Cooling Experiment
If thermometer reads consistently high:
- systematic error
If readings fluctuate by :
- random error
How to Reduce Uncertainty
Reduce Systematic Error
- calibrate instrument
- correct zero error
- improve insulation
- reduce friction
- better alignment
Reduce Random Error
- repeat measurements
- average readings
- use larger measured interval
- use more precise instrument
- reduce vibrations / drafts
Common Exam Mistakes
- saying repeated readings remove systematic error
- confusing precision with accuracy
- quoting too many decimal places
- omitting units in uncertainty
- giving percentage uncertainty without calculation
- using unsuitable instrument
Fast Revision Summary
- All measurements have uncertainty.
- Systematic errors affect accuracy.
- Random errors affect precision.
- Repeated readings reduce random effects.
- Quote results as:
- Compare quality using percentage uncertainty.