# Evaluation of Results

> CIE A-Level Physics · 9702 AS Practical Skills
> Source: https://www.owlsprep.com/study/cie-9702-u15-evaluation-of-results/

This sub-topic covers how to quantify uncertainty in raw and calculated experimental results, distinguish between error types, and evaluate the accuracy of practical work, a core skill for CIE AS Physics practical assessment.

**Prerequisites:** [Basic experimental measurement techniques](https://www.owlsprep.com/study/cie-9702-u15-measurement-techniques/); [Graph plotting for practical data](https://www.owlsprep.com/study/cie-9702-u15-graphical-analysis/)

## Learning objectives

- Distinguish between random and systematic errors in experimental measurements
- Calculate absolute, fractional and percentage uncertainty for raw and calculated data
- Find uncertainty in gradients and intercepts from experimental graphs
- Evaluate the accuracy of results and justify conclusions with reference to uncertainty

## Types of Error and Uncertainty

**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.

> **tip**
>
> To reduce random error: take multiple readings and calculate a mean. Systematic error cannot be reduced by repeating measurements; it requires correcting your experimental procedure.

**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

## Combining Uncertainties

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

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

1. Addition/subtraction: add **absolute** uncertainties
2. Multiplication/division: add **fractional/percentage** uncertainties
3. 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%

## Uncertainty in Graph Gradients

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.

**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 $\frac{2}{24} \times 100\% \approx 8\%$

## Evaluating Experimental Conclusions

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

**Exam command terms**

Common command terms for evaluation have specific exam expectations:

- **Evaluate** — Comment on accuracy/reliability and suggest relevant improvements *(State if the result supports the hypothesis, referencing uncertainty)*

- **Justify** — You must explicitly reference uncertainty to support your conclusion *(Justify if your result is accurate by checking if true value is within your uncertainty range)*

**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.

## Common pitfalls

- **Wrong:** Adding percentage uncertainties when adding or subtracting measured quantities
  - Why it fails: Uncertainties add as absolute values for addition/subtraction, not percentage
  - Correct: Convert all uncertainties to absolute values, sum the absolute uncertainties, then calculate percentage uncertainty from the total
- **Wrong:** Forgetting to multiply uncertainty by the exponent when raising to a power
  - Why it fails: If $y = x^n$, the fractional uncertainty scales by $n$, so it cannot stay the same as for $x$
  - Correct: Always multiply the fractional uncertainty of $x$ by the power of $x$ in the expression for $y$
- **Wrong:** Drawing worst-fit lines outside error bars to get a larger uncertainty
  - Why it fails: Worst-fit lines must still pass through all error bars, just at the extreme of the possible range
  - Correct: Draw the steepest and shallowest lines that pass through every error bar, then calculate uncertainty from these
- **Wrong:** Claiming a result is inaccurate just because it is not exactly equal to the true value
  - Why it fails: All experiments have uncertainty, so small deviations are expected within the error range
  - Correct: Check if the true value falls within your $\text{mean} \pm \text{uncertainty}$ range, and conclude accuracy from that
- **Wrong:** Quoting uncertainty to more than 1 or 2 significant figures
  - Why it fails: Uncertainty is an estimate, so extra significant figures are meaningless
  - Correct: Round uncertainty to one significant figure, then round the best estimate to the same decimal place as the uncertainty

## Cheatsheet

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

## 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.

- [Motion in a circle](https://www.owlsprep.com/study/cie-9702-u16-overview/)
- [Angular displacement and speed](https://www.owlsprep.com/study/cie-9702-u16-angular-displacement-and-speed/)
- [Centripetal acceleration](https://www.owlsprep.com/study/cie-9702-u16-centripetal-acceleration/)

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