Measurement Uncertainty and Errors

Branch note: This page deepens one part of Measurement.

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.

Core Ideas

• Uncertainty describes the likely spread in a measured value.
• Systematic errors affect accuracy; random errors affect precision.
• Repeated independent readings and averaging reduce the random uncertainty in the estimated mean; they do not remove the underlying fluctuations or systematic bias.

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 a stated estimate of the doubt in a measured value. Writing $x\pm \Delta x$ identifies an interval associated with the measurement; at this level it is not a rigorous probability statement or a guarantee that the true value lies inside it.

Key Representations

$$x\pm \Delta x$$ $$\frac{\Delta x}{x}\times 100\%$$

Error vs Uncertainty

Error

In principle, signed error is the difference between a measured value and the true value. Because the true value is generally unknown, experiments often compare with an accepted or reference value; the measured-minus-reference difference is then more precisely described as a deviation from the reference.

$$\text{signed error}=\text{measured value}-\text{true value}$$

Example:

Take the accepted/reference value of $g$ as $9.81{\text{m s}}^{-2}$.

Measured value:

$$9.90{\text{m s}}^{-2}$$

Deviation from the reference:

$$+0.09{\text{m s}}^{-2}$$

Uncertainty

Uncertainty states the estimated interval associated with the measured result.

Example:

$$L=12.4\pm 0.1\text{cm}$$

Associated interval:

$$12.3\text{ cm to }12.5\text{ cm}$$

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.

$$x\pm \Delta x$$

Examples:

• $5.20\pm 0.01\text{s}$
• $25.0\pm 0.5\text{cm}$

Instrument-Based Estimate

For analogue instruments:

• for one directly read value, uncertainty is often estimated as half the smallest scale division when interpolation is reasonable

Example:

Metre rule marked every $1\text{mm}$

Estimated uncertainty:

$$\pm 0.5\text{mm}$$

For digital meters:

• uncertainty often taken as $\pm 1$ least significant digit unless stated otherwise

Always use a stated manufacturer or question uncertainty in preference to these default estimates. If a measured length is the difference of two scale readings, both endpoint readings contribute; do not automatically assign only one half-division to the final difference.

Figure: Conditional analogue-reading rule. For one reading on a suitable analogue scale, half the smallest division is a common estimate. A value obtained from two endpoint readings receives uncertainty contributions from both readings.

Fractional Uncertainty

$$\frac{\Delta x}{x}$$

Example:

$$x=20.0\pm 0.2$$

Then:

$$\frac{\Delta x}{x}=\frac{0.2}{20.0}=0.010$$

Percentage Uncertainty

$$\frac{\Delta x}{x}\times 100\%$$

Using the previous example:

$$\frac{0.2}{20.0}\times 100\%=1.0\%$$

Why Percentage Uncertainty Matters

It allows comparison of quality of different measurements.

Compare:

• $100.0\pm 0.1$ → $0.1\%$
• $2.0\pm 0.1$ → $5\%$

Although both have $\pm 0.1$ absolute uncertainty, the first is much better relatively.

Figure: Random variation produces scatter around a trend. A systematic effect produces a repeatable bias or pattern, so the mean or best-fit relationship can remain displaced from the accepted relationship even after averaging.

Systematic Errors

Systematic errors produce a repeatable bias or pattern under the same conditions. The effect may be a constant offset, a scale-factor error, or another consistent dependence; it need not be the same numerical amount for every reading.

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
• reduced or corrected only when the cause is identified—for example by zero correction, calibration, background subtraction, or a redesigned method

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

• readings fluctuate and, with adequate sampling and no systematic bias, scatter about an underlying mean or trend; a small sample need not look symmetric
• repeated independent measurements reduce the random contribution to the uncertainty of the estimated mean
• averaging does not remove the underlying fluctuations and cannot correct a systematic bias

Precision vs Accuracy

Precision

How close repeated readings are to one another.

High precision means:

• small spread
• low random uncertainty

Accuracy

How close a result, often the mean of repeated readings, is to the accepted/reference value available for comparison.

High accuracy means:

• low systematic error

Typical Cases

High Precision, Low Accuracy

Readings:

$$10.32,10.31,10.33$$

Very close together, but if true value is $9.81$, then inaccurate.

Low Precision, Good Average Accuracy

Readings:

$$9.5,10.1,9.8,9.9$$

Spread is larger, but average may be close to true value.

Repeated Measurements

Repeated readings improve reliability.

Mean Value

$$\bar{x}=\frac{x_1+x_2+x_3+\cdots }{n}$$

Range Estimate

For repeated readings, a practical estimate of uncertainty is sometimes obtained from the spread:

$$\Delta x\approx \frac{\text{max}-\text{min}}{2}$$

This half-range is a simple H2 estimate, not a rigorous statistical confidence interval. It is most informative when the repeats sample the random variation reasonably.

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

$$12.4\pm 0.1\text{cm}$$

Incorrect

$$12.43\pm 0.1\text{cm}$$

(decimal places inconsistent)

Rule for Decimal Places

The measured value should be rounded to the same decimal place as the uncertainty.

Examples:

Correct

$$3.42\pm 0.05$$

Correct

$$15.0\pm 0.2$$

Incorrect

$$15\pm 0.2$$

Practical Examples

Example 1: Measuring Wire Diameter

Using metre rule:

range is suitable, but resolution is too coarse for a thin wire

Using micrometer screw gauge:

finer resolution and smaller read-off uncertainty for this diameter

Choose the instrument whose range, resolution and contact geometry suit the measurement.

Example 2: Measuring Time of Motion

Short motion lasting $0.4\text{s}$ timed manually gives high percentage uncertainty.

Better:

• use light gate
• increase timing interval if possible

Example 3: Cooling Experiment

If thermometer reads consistently $0.5^{\circ }C$ high:

• systematic error

If readings fluctuate by $0.2^{\circ }C$:

• 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
• choose an instrument with suitable range and finer resolution where instrument resolution contributes significantly to measurement uncertainty
• reduce vibrations / drafts

Exam Relevance

In exams, uncertainty and error analysis are often tested through reading instruments, quoting values with uncertainty, distinguishing error types, and suggesting practical improvements.

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 independent readings and averaging reduce the random uncertainty in the estimated mean; they do not remove the underlying fluctuations.
• Quote results as:

$$x\pm \Delta x$$

• Compare quality using percentage uncertainty.

Quick Formula Box

Absolute Form

$$x\pm \Delta x$$

Fractional Uncertainty

$$\frac{\Delta x}{x}$$

Percentage Uncertainty

$$\frac{\Delta x}{x}\times 100\%$$

Mean Value

$$\bar{x}=\frac{\sum x}{n}$$

Links

• Measurement
• Measurement Units and Dimensions
• Uncertainty Propagation Methods
• Measurement Data Presentation
• Measurement Estimation and Experimental Design
• Kinematics
• Forces