Errors and uncertainties

1. Overview

In experimental physics, the fundamental principle is that no measurement is perfectly exact. Every physical quantity determined in a laboratory is subject to some degree of doubt, which is scientifically quantified as uncertainty. The objective of an experiment is not merely to find a number, but to determine the true value of a quantity within a specific range of confidence.

A result is only scientifically valid if it is accompanied by an estimate of its reliability. This requires a rigorous understanding of the limitations of measuring instruments and the experimental environment. By distinguishing between random and systematic errors, and by mastering the mathematical propagation of uncertainties, physicists can evaluate the validity of their conclusions. In the context of the Cambridge 9702 syllabus, this topic is foundational for Paper 2 (Theory), Paper 3 (Advanced Practical Skills), and Paper 5 (Planning, Analysis and Evaluation).

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Key Definitions

• Accuracy: The degree of closeness of a measured value (or the mean of a set of values) to the true value of the quantity. A measurement is accurate if the systematic error is small.
• Precision: The degree of agreement or closeness between repeated measurements. It indicates the reproducibility or "consistency" of the data. A measurement is precise if the random error is small.
• Random Error: Unpredictable fluctuations in readings that cause measurements to be scattered around a mean value. These are caused by factors such as environmental changes or human limitations in judgment. They affect the precision of the result.
• Systematic Error: Errors that cause all readings to be consistently shifted from the true value by a fixed amount in the same direction. These are caused by faulty equipment or flawed experimental design. They affect the accuracy of the result.
• Zero Error: A specific type of systematic error where an instrument provides a non-zero reading when the actual quantity being measured is zero.
• Absolute Uncertainty: The actual range ($\pm \Delta x$) within which the true value is estimated to lie. It is always expressed in the same units as the measurement itself.
• Fractional Uncertainty: The ratio of the absolute uncertainty to the measured value ($\frac{\Delta x}{x}$). It is a dimensionless ratio.
• Percentage Uncertainty: The fractional uncertainty expressed as a percentage ($\frac{\Delta x}{x} \times 100%$). This allows for the comparison of the relative "size" of uncertainties across different physical quantities.

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Content

3.1 Random and Systematic Errors

Random Errors Random errors are inherent in the measurement process and result from unpredictable variations in experimental conditions or the observer's technique.

• Characteristics: They cause readings to be spread on both sides of the mean. If a large number of readings are taken, they typically follow a Gaussian (normal) distribution.
• Examples:
  • Parallax error: Reading a scale from slightly different angles each time.
  • Environmental fluctuations: Small changes in air temperature or pressure during a gas experiment.
  • Human reaction time: Variations in starting and stopping a manual stopwatch.
• Reduction: The effect of random errors is reduced by taking multiple repeat readings and calculating a mean. This allows positive and negative fluctuations to partially cancel out. It also aids in identifying anomalies (outliers) which must be excluded from the mean calculation.

Systematic Errors Systematic errors are constant and predictable. They do not average out, regardless of how many times the experiment is repeated.

• Characteristics: They shift the entire data set away from the true value by a constant magnitude and direction.
• Examples:
  • Zero error: A micrometer screw gauge that reads $0.01\text{ mm}$ when fully closed.
  • Calibration error: A wooden meter rule that has expanded due to humidity.
  • Theoretical assumptions: Ignoring air resistance in a free-fall experiment or ignoring the internal resistance of a battery.
• Identification and Reduction: Systematic errors are often identified by plotting a graph. If the theory predicts a line through the origin ($y = mx$) but the graph shows a non-zero intercept, a systematic error is likely present. They can only be eliminated by recalibrating instruments, applying a correction factor, or improving the experimental design.

3.2 Precision vs. Accuracy: The Target Analogy

To distinguish between these two concepts, imagine a target:

1. High Accuracy, High Precision: All arrows are clustered tightly in the bullseye. (Small random error, small systematic error).
2. Low Accuracy, High Precision: All arrows are clustered tightly together but are shifted away from the bullseye. (Small random error, large systematic error).
3. High Accuracy, Low Precision: The arrows are scattered widely, but their average position is the bullseye. (Large random error, small systematic error).
4. Low Accuracy, Low Precision: The arrows are scattered widely and are far from the bullseye. (Large random error, large systematic error).

Exam Note: In 9702, you must use the term "true value" when discussing accuracy and "repeated measurements" or "consistency" when discussing precision.

3.3 Assessment of Uncertainties in Derived Quantities

When base measurements (like mass and length) are used to calculate a derived quantity (like density), the uncertainties in the base measurements "propagate" through the calculation.

1. Absolute, Fractional, and Percentage Uncertainty For a measurement $x \pm \Delta x$:

• Absolute Uncertainty = $\Delta x$
• Fractional Uncertainty = $\frac{\Delta x}{x}$
• Percentage Uncertainty = $\left( \frac{\Delta x}{x} \right) \times 100%$

2. Rules for Combining Uncertainties These rules are essential for Paper 2 and Paper 3.

• Addition and Subtraction: If $y = a + b$ or $y = a - b$, the absolute uncertainties are added. $$\Delta y = \Delta a + \Delta b$$ Note: Even if you are subtracting the values, you still add the uncertainties because the "doubt" in the result increases.

• Multiplication and Division: If $y = ab$ or $y = \frac{a}{b}$, the percentage uncertainties are added. $$\frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$$ (Or: $% \Delta y = % \Delta a + % \Delta b$)

• Powers: If $y = a^n$, the percentage uncertainty is multiplied by the power. $$\frac{\Delta y}{y} = |n| \left( \frac{\Delta a}{a} \right)$$ (Or: $% \Delta y = |n| \times % \Delta a$) Note: This applies to square roots ($n = 0.5$) and reciprocals ($n = -1$). Always use the absolute value of the power $n$.

3.4 Practical Considerations: Resolution and Human Error

• Instrument Resolution: The smallest change in a quantity that an instrument can detect.
  • Digital instruments: The uncertainty is usually taken as the smallest significant digit (e.g., $\pm 0.01\text{ g}$ for a digital balance).
  • Analogue instruments: The uncertainty is often taken as half the smallest scale division. However, for a ruler, there is an uncertainty at both ends of the object, so the total uncertainty is usually the smallest division (e.g., $\pm 1\text{ mm}$).
• Human Reaction Time: When using a stopwatch, the uncertainty is rarely the resolution of the watch ($0.01\text{ s}$). It is dominated by human reaction time, typically estimated as $\pm 0.1\text{ s}$ to $\pm 0.2\text{ s}$ in 9702 practicals.
• Uncertainty from Repeated Readings: If you have a set of repeated measurements, the absolute uncertainty is estimated as half the range: $$\Delta x = \frac{x_{max} - x_{min}}{2}$$
• Reducing Percentage Uncertainty: To make a measurement more reliable, you should aim to reduce the percentage uncertainty. This can be done by:
  1. Using an instrument with a higher resolution (smaller $\Delta x$).
  2. Measuring a larger magnitude of the quantity (larger $x$). For example, measuring the time for 20 oscillations of a pendulum instead of just one, or measuring the thickness of 50 pages of a book instead of one.

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3.5 Worked Examples

Worked Example 1 — Density of a Metal Cylinder

A student determines the density $\rho$ of a metal cylinder. The following measurements are taken: Mass $m = (50.0 \pm 0.1) \text{ g}$ Length $L = (4.50 \pm 0.02) \text{ cm}$ Diameter $d = (1.20 \pm 0.01) \text{ cm}$

Calculate the density and its absolute uncertainty.

Step 1: Calculate the value of density. Volume $V = \frac{\pi d^2 L}{4} = \frac{\pi \times (1.20)^2 \times 4.50}{4} = 5.08938 \text{ cm}^3$ $\rho = \frac{m}{V} = \frac{50.0}{5.08938} = 9.82438 \text{ g cm}^{-3}$

Step 2: Calculate percentage uncertainties for each measurement. $% \Delta m = \frac{0.1}{50.0} \times 100 = 0.2%$ $% \Delta L = \frac{0.02}{4.50} \times 100 = 0.444%$ $% \Delta d = \frac{0.01}{1.20} \times 100 = 0.833%$

Step 3: Combine percentage uncertainties. Using the formula $\rho = \frac{4m}{\pi d^2 L}$: $% \Delta \rho = % \Delta m + (2 \times % \Delta d) + % \Delta L$ $% \Delta \rho = 0.2% + (2 \times 0.833%) + 0.444% = 2.31%$

Step 4: Convert back to absolute uncertainty. $\Delta \rho = \frac{2.31}{100} \times 9.82438 = 0.2269 \text{ g cm}^{-3}$

Step 5: Final Answer. The uncertainty is quoted to 1 sig fig: $\Delta \rho = \pm 0.2 \text{ g cm}^{-3}$. The value must match the decimal places of the uncertainty: $\rho = (9.8 \pm 0.2) \text{ g cm}^{-3}$

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Worked Example 2 — Resistivity of a Wire

A student measures the resistivity $\rho$ of a wire using the formula $\rho = \frac{RA}{L}$, where $A = \frac{\pi d^2}{4}$. The measurements are: Resistance $R = (25.4 \pm 0.2) \text{ }\Omega$ Length $L = (1.500 \pm 0.002) \text{ m}$ Diameter $d = (0.45 \pm 0.01) \text{ mm}$

Calculate the percentage uncertainty in the resistivity.

Step 1: Identify the relationship. $\rho = \frac{R \pi d^2}{4L}$

Step 2: Calculate individual percentage uncertainties. $% \Delta R = \frac{0.2}{25.4} \times 100 = 0.787%$ $% \Delta L = \frac{0.002}{1.500} \times 100 = 0.133%$ $% \Delta d = \frac{0.01}{0.45} \times 100 = 2.222%$

Step 3: Combine using the power rule and product rule. $% \Delta \rho = % \Delta R + (2 \times % \Delta d) + % \Delta L$ $% \Delta \rho = 0.787% + (2 \times 2.222%) + 0.133%$ $% \Delta \rho = 0.787% + 4.444% + 0.133% = 5.364%$

Step 4: Final Answer. Quoting to 2 significant figures: Percentage uncertainty in $\rho = 5.4%$

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Worked Example 3 — Acceleration of Free Fall

In an experiment to determine $g$, a ball is dropped from rest. The distance fallen $s$ is $(2.000 \pm 0.001) \text{ m}$ and the time taken $t$ is $(0.64 \pm 0.02) \text{ s}$. Using $s = \frac{1}{2}gt^2$, calculate $g$ and its absolute uncertainty.

Step 1: Calculate $g$. $g = \frac{2s}{t^2} = \frac{2 \times 2.000}{(0.64)^2} = 9.7656 \text{ m s}^{-2}$

Step 2: Calculate percentage uncertainties. $% \Delta s = \frac{0.001}{2.000} \times 100 = 0.05%$ $% \Delta t = \frac{0.02}{0.64} \times 100 = 3.125%$

Step 3: Combine percentage uncertainties. $% \Delta g = % \Delta s + (2 \times % \Delta t)$ $% \Delta g = 0.05% + (2 \times 3.125%) = 6.3%$

Step 4: Calculate absolute uncertainty. $\Delta g = \frac{6.3}{100} \times 9.7656 = 0.615 \text{ m s}^{-2}$

Step 5: Final Answer. $\Delta g = \pm 0.6 \text{ m s}^{-2}$ (1 sig fig). $g = (9.8 \pm 0.6) \text{ m s}^{-2}$

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Key Equations

Concept	Equation	Notes
Sum / Difference	$\Delta y = \Delta a + \Delta b$	Add absolute uncertainties. (Memorise)
Product / Quotient	$\frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$	Add fractional/percentage uncertainties. (Memorise)
Power Rule	**$\frac{\Delta y}{y} =	n
Percentage Uncertainty	$% \text{Unc} = \frac{\Delta x}{x} \times 100$	Standard definition. (Memorise)
Repeated Readings	$\Delta x = \frac{\text{Max} - \text{Min}}{2}$	Used for multiple trials. (Memorise)

Note: These equations are not provided on the 9702 Data and Formulae sheet. You must be able to apply them from memory.

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Common Mistakes to Avoid

• ❌ Wrong: Subtracting uncertainties when two quantities are subtracted (e.g., calculating a change in temperature $\Delta \theta = \theta_2 - \theta_1$).
• ✅ Right: Always add absolute uncertainties for both addition and subtraction. Uncertainty represents the "total doubt," which always increases when combining measurements.
• ❌ Wrong: Adding absolute uncertainties when multiplying or dividing (e.g., calculating $V = IR$).
• ✅ Right: You must convert to percentage uncertainties first, add them, and then convert back to an absolute uncertainty if required.
• ❌ Wrong: Forgetting to multiply the percentage uncertainty by the power (e.g., in $A = \pi r^2$, using $% \Delta A = % \Delta r$).
• ✅ Right: The power rule states $% \Delta A = 2 \times % \Delta r$.
• ❌ Wrong: Including constants like $\pi$, $2$, or $1/2$ in the uncertainty propagation.
• ✅ Right: Pure numbers have zero uncertainty. They affect the calculated value but do not contribute to the percentage uncertainty of the result.
• ❌ Wrong: Quoting an uncertainty to many significant figures (e.g., $\pm 0.12345$).
• ✅ Right: Limit absolute uncertainty to 1 or 2 significant figures, then round your main value to the same number of decimal places.

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Exam Tips

1. The "Line of Best Fit" Test: In Paper 3, if your graph of $y$ vs $x$ is a straight line that does not pass through the origin when the theory ($y=kx$) says it should, state that this indicates a systematic error.
2. Significant Figures Consistency: This is a major marking point. If your absolute uncertainty is $\pm 0.1$ (1 decimal place), your value must be written to 1 decimal place (e.g., $5.0 \pm 0.1$, not $5 \pm 0.1$).
3. Read the Question Carefully: Does it ask for the absolute uncertainty or the percentage uncertainty? Providing the wrong type will lose a mark.
4. Stopwatch Uncertainty: If a question involves timing an event manually and the uncertainty isn't given, use $\pm 0.1\text{ s}$ or $\pm 0.2\text{ s}$ and justify it by mentioning human reaction time.
5. Multiple Readings: If you are given a table of repeated readings, the absolute uncertainty is half the range. If the range is zero (all readings are the same), use the resolution of the instrument as the uncertainty.
6. Combining Rules: For complex formulas like $y = \frac{a^2}{\sqrt{b}}$, the total percentage uncertainty is $% \Delta y = (2 \times % \Delta a) + (0.5 \times % \Delta b)$. Break it down step-by-step to avoid errors.
7. Units: Always check that your absolute uncertainty has the same units as the measured value. Percentage uncertainty has no units.