Evaluation of Results

CIE A-Level Physics· Unit 15: Practical Skills (AS)· 15 min read

1. Types of Error and Uncertainty★★☆☆☆⏱ 4 min

📘 Definition

Random and Systematic Errors

Random errors cause scatter of readings around the true value, arising from uncontrollable fluctuations in measurement conditions. Systematic errors shift all measurements consistently away from the true value, from faulty equipment or incorrect procedure.

Example:

Variation in reaction time when timing a pendulum is random; consistent parallax error reading a scale is systematic.

Uncertainty is the margin of doubt in a measurement, quantifying the effect of errors. Reducing random error improves precision, while correcting systematic error improves accuracy.

📐 Worked Example

A student measures the time for 10 pendulum oscillations three times: 12.2 s, 12.5 s, 12.1 s. Classify the variation between readings and find the uncertainty in the mean time.

1
Variation between repeated measurements of the same quantity is caused by random error.
2
Calculate the mean time:
3
$t_{\text{mean}} = \frac{12.2 + 12.5 + 12.1}{3} = 12.27 \approx 12.3 \ \text{s}$
4
Uncertainty is half the range of the readings:
5
$\Delta t = \frac{t_{\text{max}} - t_{\text{min}}}{2} = \frac{12.5 - 12.1}{2} = 0.2 \ \text{s}$
6
Final result: $t = 12.3 \pm 0.2$ s

2. Combining Uncertainties★★☆☆☆⏱ 5 min

📘 Definition

Uncertainty Classifications

For measurement $x$ , absolute uncertainty: $Δ x$ , fractional: $\frac{Δ x}{x}$ , percentage: $\frac{Δ x}{x} \times 100%$

Absolute uncertainty has the same units as the measured quantity. Fractional and percentage uncertainty are dimensionless, describing uncertainty relative to the measured value.

Addition/subtraction: add absolute uncertainties
Multiplication/division: add fractional/percentage uncertainties
Powers ( $x^{n}$ ): multiply fractional uncertainty by $n$

📐 Worked Example

A sphere has diameter $d = 2.00 \pm 0.02$ cm. Calculate the percentage uncertainty in its volume.

1
Volume of a sphere is:
2
$V = \frac{\pi d^3}{6}$
3
Calculate percentage uncertainty in diameter:
4
$\% \Delta d = \frac{0.02}{2.00} \times 100\% = 1\%$
5
Since $V \propto d^{3}$ , multiply percentage uncertainty by the exponent 3:
6
$\% \Delta V = 3 \times 1\% = 3\%$
7
Final result: percentage uncertainty in volume is 3%

3. Uncertainty in Graph Gradients★★★☆☆⏱ 5 min

For linear graphs, the uncertainty in gradient and intercept is found by drawing worst-fit lines alongside your best-fit line. Worst-fit lines are the extreme possible lines that still pass through all error bars on your plotted points.

📘 Definition

Uncertainty in Gradient

The uncertainty is half the difference between the gradient of the steepest worst-fit and shallowest worst-fit line.

📐 Worked Example

A student plots force against extension for a spring. Best-fit gradient is $24$ Nm⁻¹, maximum gradient is $26$ Nm⁻¹, minimum gradient is $22$ Nm⁻¹. Find the uncertainty in the spring constant.

1
Uncertainty is half the range of the gradients:
2
$\Delta k = \frac{k_{\text{max}} - k_{\text{min}}}{2} = \frac{26 - 22}{2} = 2 \ \text{Nm}^{-1}$
3
Record the final gradient as:
4
$k = 24 \pm 2 \ \text{Nm}^{-1}$
5
Percentage uncertainty is $\frac{2}{24} \times 100% \approx 8%$

4. Evaluating Experimental Conclusions★★★☆☆⏱ 4 min

To evaluate the accuracy of a result, you check if the accepted true value lies within the range of your result: $x_{true} \in [x_{mean} - Δ x, x_{mean} + Δ x]$ . If it does, the result is accurate within experimental uncertainty.

📐 Worked Example

A student calculates $g = 9.3 \pm 0.6$ ms⁻². Accepted value is $9.81$ ms⁻². Comment on the accuracy of the result.

1
Find the range of the student's result: $9.3 - 0.6 = 8.7$ to $9.3 + 0.6 = 9.9$ ms⁻²
2
Check if the accepted value falls within this range: $8.7 < 9.81 < 9.9$ , so the accepted value is inside the uncertainty range.
3
Conclusion: Within the uncertainty of the experiment, the result is accurate.

5. Common Pitfalls

Wrong move:

Adding percentage uncertainties when adding or subtracting measured quantities

Why:

Uncertainties add as absolute values for addition/subtraction, not percentage

Correct move:

Convert all uncertainties to absolute values, sum the absolute uncertainties, then calculate percentage uncertainty from the total

Wrong move:

Forgetting to multiply uncertainty by the exponent when raising to a power

Why:

If $y = x^{n}$ , the fractional uncertainty scales by $n$ , so it cannot stay the same as for $x$

Correct move:

Always multiply the fractional uncertainty of $x$ by the power of $x$ in the expression for $y$

Wrong move:

Drawing worst-fit lines outside error bars to get a larger uncertainty

Why:

Worst-fit lines must still pass through all error bars, just at the extreme of the possible range

Correct move:

Draw the steepest and shallowest lines that pass through every error bar, then calculate uncertainty from these

Wrong move:

Claiming a result is inaccurate just because it is not exactly equal to the true value

Why:

All experiments have uncertainty, so small deviations are expected within the error range

Correct move:

Check if the true value falls within your $mean \pm uncertainty$ range, and conclude accuracy from that

Wrong move:

Quoting uncertainty to more than 1 or 2 significant figures

Why:

Uncertainty is an estimate, so extra significant figures are meaningless

Correct move:

Round uncertainty to one significant figure, then round the best estimate to the same decimal place as the uncertainty

6. Quick Reference Cheatsheet

Rule	Operation	Uncertainty Calculation
Add/Subtract	$y = a \pm b$	$Δ y = Δ a + Δ b$
Multiply/Divide	$y = \frac{ab}{c}$	$\frac{Δ y}{y} = \frac{Δ a}{a} + \frac{Δ b}{b} + \frac{Δ c}{c}$
Power	$y = a x^{n}$	$\frac{Δ y}{y} = n \frac{Δ x}{x}$
Graph Gradient	$m_{b}$ (best), $m_{ma x}$ , $m_{min}$	$Δ m = \frac{m _{ma x} - m _{min}}{2}$
Repeated Readings	Mean of $n$ measurements	$Δ x = \frac{x _{ma x} - x _{min}}{2}$

7. Frequently Asked

What is the difference between accuracy and precision?

Accuracy refers to how close a measured value is to the true accepted value, while precision describes how close repeated measurements are to each other. High precision does not guarantee high accuracy: a zeroed incorrectly balance will give precise but inaccurate readings.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 3
Uncertainty in resistance calculation
2023 · 3
Comment on accuracy of result
2021 · 3
Percentage uncertainty in gradient

Going deeper

syllabusOfficial CIE 9702 SyllabusFull practical assessment requirements

What's Next

Mastery of evaluation of results is required for all practical and data analysis questions in both AS and A2 CIE Physics, and is a core skill for any scientific investigation. Understanding uncertainty helps you design better experiments, critically evaluate results, and justify your conclusions clearly in exam answers. This sub-topic builds directly on measurement skills and graphical analysis, and prepares you for more advanced experimental planning and evaluation at A2.