Errors and Uncertainties — AQA A-level Physics

SPEC 3.1.2 - MEASUREMENTS AND ERRORS

Limitations of Physical Measurements

Errors, precision, accuracy, uncertainty, and combining uncertainties

Learning Objectives

By the end of this lesson, you will be able to:

Objectives

Distinguish between random and systematic errors

Define precision, accuracy, repeatability, reproducibility, resolution

Calculate absolute, fractional, and percentage uncertainties

Combine uncertainties correctly for different operations

Determine uncertainty in gradients from graphs

Types of Error

All measurements have some degree of uncertainty. Errors fall into two main categories:

Random Errors

Affect precision. Cause differences in measurements - a spread about the mean.

You cannot get rid of ALL random errors.

Systematic Errors

Affect accuracy. Cause all results to be too high or too low by the same amount.

Due to apparatus or faults in method.

Key Difference

Random errors can be reduced by repeating and averaging. Systematic errors cannot be reduced by repeating - they require identifying and eliminating the source.

Random Errors

Example: Electronic noise in circuits

To Reduce Random Errors

Take at least 3 repeats and calculate the mean (also identifies anomalies)

Use computers/data loggers/cameras to reduce human error

Use appropriate equipment with higher resolution

Systematic Errors

Examples: Zero error (balance not zeroed), parallax error (reading at wrong angle)

To Reduce Systematic Errors

Calibrate apparatus by measuring a known value

Correct for background radiation in radiation experiments

Read the meniscus at eye level to reduce parallax

Use controls in experiments

Worked Example 1

1

Identifying Error Types

A student measures the period of a pendulum 10 times. The results in seconds are: 2.03, 1.98, 2.05, 1.97, 2.02, 2.01, 1.99, 2.04, 2.00, 2.01. The true period is 2.01 s.

(a) Describe the precision and accuracy of these results.

(b) What type of error is present?

(a)

Precision: Range = 2.05 - 1.97 = 0.08 s. Small spread → precise.

•

Accuracy: Mean = 2.01 s = true value → accurate.

(b)

Scatter around mean indicates random error (likely human reaction time). No systematic error since mean = true value.

Key Definitions

TermDefinition

PrecisionMeasurements are consistent - they fluctuate slightly about a mean value. Does NOT indicate accuracy.

RepeatabilitySame experimenter, same equipment, same method → same results

ReproducibilityDifferent person, different techniques/equipment → same results

ResolutionThe smallest change in quantity that gives a recognisable change in reading

AccuracyA measurement close to the true value

Precision vs Accuracy

Think of measurements like arrows hitting a target:

Precise but NOT Accurate (Systematic Error)

Accurate but NOT Precise (Random Error)

Precise AND Accurate (The Goal!)

Remember

Precision = clustering (low random error). Accuracy = closeness to true value (low systematic error).

Worked Example 2

2

Precision vs Accuracy Analysis

Two students measure g (true value = 9.81 m s⁻²):

A: 9.2, 9.3, 9.1, 9.2, 9.3 m s⁻²

B: 9.5, 10.1, 9.8, 10.0, 9.6 m s⁻²

Compare the precision and accuracy of each student's results.

A:

Range = 0.2 m s⁻² → High precision. Mean = 9.22, error = 0.59 → Low accuracy.

B:

Range = 0.6 m s⁻² → Lower precision. Mean = 9.80, error = 0.01 → High accuracy.

•

Conclusion: A has systematic error (precise but inaccurate). B has random error only (less precise but accurate).

Worked Example 3

3

Resolution Selection

A student needs to measure the thickness of a human hair (approximately 0.08 mm). Three instruments are available:

A: 30 cm ruler (1 mm divisions)

B: Vernier caliper (0.1 mm resolution)

C: Micrometer (0.01 mm resolution)

Which instrument is most appropriate? Justify your answer.

1

Ruler (1 mm): Resolution > measurement. Unsuitable - cannot detect the measurement.

2

Vernier caliper (0.1 mm): 0.08 mm ≈ resolution. Marginal - would give ±63% uncertainty.

3

Micrometer (0.01 mm): 0.08 mm = 8× resolution. Most appropriate - gives ±6% uncertainty.

•

Conclusion: Use the micrometer. Resolution should be significantly smaller than the quantity being measured.

What is Uncertainty?

Definition

The uncertainty of a measurement is the bounds in which the accurate value can be expected to lie.

Example: A temperature reading of 20°C ± 2°C means the true value could be within 18-22°C.

Absolute Uncertainty

A fixed quantity

7 ± 0.6 V

Fractional Uncertainty

As a fraction

7 ±

\frac{3}{35}

V

Percentage Uncertainty

As a percentage

7 ± 8.6% V

Key Formula

percentage uncertainty = \frac{uncertainty}{value} \times 100%

Readings vs Measurements

Reading

When one value is found.

Example: Reading a thermometer

Uncertainty = ± half the smallest division

Measurement

When difference between 2 readings is found.

Example: Using a ruler (start and end point)

Uncertainty = at least ± 1 smallest division

Examples

Thermometer (1°C divisions): uncertainty = ±0.5°C

Ruler (1 mm divisions): uncertainty = ±0.5 mm + ±0.5 mm = ±1 mm

Digital Readings

Uncertainty is ± the last significant digit. E.g., 3.2 V has uncertainty ±0.1 V

Uncertainty from Repeated Data

Rule

For repeated data, the uncertainty is half the range:

uncertainty = \frac{largest - smallest}{2}

Express as: mean ± (range/2)

Example

Readings: 5.2, 5.4, 5.1, 5.3, 5.5 m

Mean = 5.3 m

Range = 5.5 - 5.1 = 0.4 m

Result: 5.3 ± 0.2 m

Reducing Uncertainty

Fix one end of a ruler → only one reading uncertainty

Measure multiple instances (e.g., time 10 swings, divide by 10 → uncertainty also divides by 10)

Measure larger quantities → smaller percentage uncertainty

Worked Example 4

4

Zero Error Correction

A micrometer reads +0.03 mm when fully closed (zero error). The student records readings of: 0.58 mm, 0.61 mm, 0.59 mm, 0.60 mm, 0.58 mm.

(a) What is the corrected mean diameter?

(b) What type of error is the zero reading?

(a)

Mean = (0.58+0.61+0.59+0.60+0.58)/5 = 0.592 mm

•

Zero error = +0.03 mm (positive → subtract)

•

Corrected diameter = 0.592 - 0.03 = 0.56 mm

(b)

This is a systematic error - all readings are too high by the same amount. Cannot be reduced by repeating.

Combining Uncertainties: Add/Subtract

Adding or Subtracting Data → ADD ABSOLUTE UNCERTAINTIES

E1

Temperature Difference

A thermometer with uncertainty ±0.5 K shows temperature falling from 298 ± 0.5 K to 273 ± 0.5 K. What is the difference in temperature?

Difference: 298 - 273 = 25 K

Uncertainty: 0.5 + 0.5 = 1 K

Answer: 25 ± 1 K

Important

When subtracting, you still ADD the absolute uncertainties. Uncertainties always add up - they never cancel!

Combining Uncertainties: Multiply/Divide

Multiplying or Dividing Data → ADD PERCENTAGE UNCERTAINTIES

E2

Calculating Acceleration

A force of 91 ± 3 N is applied to a mass of 7 ± 0.2 kg. Find the acceleration.

a = F/m = 91/7 = 13 m s⁻²

% uncertainty in F: (3/91) × 100 = 3.3%

% uncertainty in m: (0.2/7) × 100 = 2.9%

Total % uncertainty: 3.3% + 2.9% = 6.2%

Absolute uncertainty: 6.2% of 13 = 0.8 m s⁻²

Answer: a = 13 ± 0.8 m s⁻² (or 13 ± 6.2%)

Combining Uncertainties: Powers

Raising to a Power → MULTIPLY PERCENTAGE UNCERTAINTY BY THE POWER

E3

Area of a Circle

The radius of a circle is 5 ± 0.3 cm. Find the percentage uncertainty in the area.

Area = πr² = π × 25 = 78.5 cm²

% uncertainty in r: (0.3/5) × 100 = 6%

% uncertainty in r²: 6% × 2 = 12%

Answer: 78.5 ± 12% cm²

Why ×2?

Because r² means r × r. Each r contributes 6% uncertainty, so total = 6% + 6% = 12%. The power tells you how many times to count the uncertainty.

Combining Uncertainties: Summary

OperationRuleExample

a + b or a - bAdd absolute uncertaintiesΔ(a±b) = Δa + Δb

a × b or a ÷ bAdd percentage uncertainties%total = %a + %b

aⁿMultiply % uncertainty by n%total = n × %a

Significant Figures

Uncertainties should be given to the same number of significant figures as the data.

Worked Example 5

5

Combining Uncertainties

Calculate the kinetic energy of a ball: mass = 2.0 ± 0.1 kg, velocity = 5.0 ± 0.2 m s⁻¹. Use KE = ½mv².

1

KE = ½ × 2.0 × 5.0² = 25 J

2

% in m: (0.1/2.0) × 100 = 5%

3

% in v: (0.2/5.0) × 100 = 4%

4

% in v²: 4% × 2 = 8% (power rule)

5

Total %: 5% + 8% = 13% (multiply/divide rule)

6

Absolute: 13% × 25 = 3.25 ≈ 3 J

•

Answer: KE = 25 ± 3 J

Uncertainties on Graphs

Uncertainties are shown as error bars on graphs.

Rules

If uncertainty is 5 mm, draw error bars 5 squares either side of the point

Line of best fit should go through all error bars (excluding anomalous points)

Uncertainty in Gradient

The uncertainty in a gradient is found using lines of best and worst fit:

Method

Draw steepest and shallowest lines that must go through all error bars

Calculate gradient of best fit and worst fit lines

% uncertainty = \frac{∣ best gradient - worst gradient ∣}{best gradient} \times 100%

Uncertainty in Y-Intercept

When best and worst fit lines have different y-intercepts:

% uncertainty = \frac{∣ best y-intercept - worst y-intercept ∣}{best y-intercept} \times 100%

Alternative Method

Using maximum and minimum gradient lines:

% uncertainty = \frac{max gradient - min gradient}{2} \times 100%

Modulus Lines

The |...| symbols mean "absolute value" - always take the positive difference.

Worked Example 6

6

Gradient Uncertainty

From a graph of velocity vs time: best fit gradient = 4.5 m s⁻², steepest line = 5.1 m s⁻², shallowest line = 4.0 m s⁻². Find the percentage uncertainty in the gradient.

1

Worst gradient = steepest OR shallowest (whichever differs more from best)

•

|5.1 - 4.5| = 0.6 vs |4.0 - 4.5| = 0.5

•

Use worst = 5.1 m s⁻²

2

% uncertainty = |4.5 - 5.1|/4.5 × 100 = 0.6/4.5 × 100 = 13%

•

Answer: gradient = 4.5 ± 13% m s⁻² (or 4.5 ± 0.6 m s⁻²)

Practice Question

Question: A student times 10 swings of a pendulum and gets 6.2 ± 0.1 s. Calculate the period and its uncertainty.

1

Period = 6.2 ÷ 10 = 0.62 s

2

Uncertainty also divides by 10: 0.1 ÷ 10 = 0.01 s

3

Period = 0.62 ± 0.01 s

•

This is why timing multiple swings reduces uncertainty!

Summary

Key Rules

Random errorsAffect precision; reduce by repeating

Systematic errorsAffect accuracy; identify and eliminate source

Reading uncertainty± half smallest division

Measurement uncertainty± 1 smallest division (two readings)

Repeated datamean ± (range/2)

Add/subtractAdd absolute uncertainties

Multiply/divideAdd percentage uncertainties

Powers (xⁿ)Multiply % uncertainty by n

Next Steps

What's Next

In the next topic, we'll apply these measurement principles to mechanics - starting with scalars, vectors, and motion.

Before You Go

Test your understanding with the practice questions and end-of-lesson test. Focus on:

Identifying random vs systematic errors

Calculating uncertainties from repeated data

Combining uncertainties correctly (know which rule to use!)

Finding gradient uncertainty from graphs