# Non-linear data analysis

> CIE A-Level Physics · Practical skills (A2)
> Source: https://www.owlsprep.com/study/cie-9702-u30-non-linear-data-analysis/

This module covers linearization of non-linear relationships for CIE A-Level Physics Paper 5 analysis. You will learn how to transform equations, plot correct axes, extract unknown constants, and calculate their uncertainties.

**Prerequisites:** [Linear graph analysis and gradient calculation](https://www.owlsprep.com/study/cie-9702-u20-linear-data-analysis/); Logarithm rules for base 10 and natural logs

## Learning objectives

- Transform common non-linear relationships into linear form for graphical analysis
- Identify transformed axes, gradient and intercept to find unknown physical constants
- Calculate uncertainties in unknown constants derived from transformed graphs
- Distinguish between log-log plots for power laws and semi-log plots for exponentials

## Principles of Linearization

When experimental data follows a non-linear relationship, you cannot directly find unknown constants by drawing a straight line through plotted points. Instead, you transform the variables to produce a straight line that matches the standard form $Y = mX + c$, where $Y$ and $X$ are transformed variables, $m$ is gradient, and $c$ is the y-intercept.

**Linearization** — The process of rearranging a non-linear mathematical relationship into standard straight line form to enable graphical calculation of unknown constants.

*Example:* Rearranging $y = kx^n$ to get $\lg y = n \lg x + \lg k$

**Worked example:** A quantity $P$ varies with $Q$ as $P = \frac{a}{Q + b}$. Transform this into linear form and state $Y$, $X$, $m$ and $c$ in terms of $a$ and $b$.

1. Start with the original non-linear relationship:
2. $$P = \frac{a}{Q + b}$$
3. Take reciprocals of both sides and rearrange:
4. $$\frac{1}{P} = \frac{Q + b}{a} = \frac{1}{a} Q + \frac{b}{a}$$
5. Compare to $Y = mX + c$ to identify all terms: $Y = \frac{1}{P}$, $X = Q$, gradient $m = \frac{1}{a}$, intercept $c = \frac{b}{a}$

> **Exam tip:** Always show your full rearrangement working in Paper 5. You will get marks for correctly identifying $Y$ and $X$ even if you make a small mistake later.

## Power Law Relationships ($y = kx^n$)

Power laws are the most common non-linear relationship tested in CIE Paper 5. Any power law can be linearized using logarithms, regardless of the value of $n$. This requires a log-log plot, with both axes transformed to logarithmic scales.

$$\lg y = \lg (k x^n) = n \lg x + \lg k$$

For this relationship, the gradient of the straight line equals $n$ (the power), and the intercept equals $\lg k$. You can reverse the logarithm to find $k = 10^{\text{intercept}}$.

**Worked example:** The period $T$ of a pendulum varies with length $l$ as $T = k l^n$. A best fit line of $\lg T$ against $\lg l$ has gradient 0.498 and y-intercept 0.116. Find $n$ and $k$.

1. Confirm linearized form: $\lg T = n \lg l + \lg k$, so gradient = $n$, intercept = $\lg k$
2. Read gradient directly to get $n = 0.498 \approx 0.50$ (2 significant figures)
3. Reverse the logarithm to find $k$:
4. $$k = 10^{0.116} \approx 1.3$$

> **tip**
>
> If your line of best fit does not cross the y-axis within your plot area, calculate the intercept using $c = Y - mX$ with any point on your line, do not guess.

## Exponential Relationships

Exponential relationships are common in decay, cooling, and growth experiments. They have the form $y = y_0 e^{kx}$ or $y = y_0 a^x$, and require a semi-log plot (one axis logarithmic, one linear) to linearize.

**Semi-log plot** — A linear plot for exponential relationships where only the dependent variable is plotted on a logarithmic scale. This contrasts with log-log plots for power laws, where both variables are logarithmic.

**Worked example:** Radioactive activity $A$ varies with time $t$ as $A = A_0 e^{-\lambda t}$. Transform this to linear form and explain how to find $\lambda$.

1. Take natural logarithm of both sides of the original equation:
2. $$\ln A = \ln (A_0 e^{-\lambda t})$$
3. Simplify using logarithm rules to get linear form:
4. $$\ln A = -\lambda t + \ln A_0$$
5. Compare to $Y = mX + c$: $Y = \ln A$, $X = t$, gradient $m = -\lambda$. Calculate the gradient of the best fit line, then take the negative value to get $\lambda$.

**Check your understanding**

Test your understanding of axis selection:

1. What axes would you use to linearize $y = 2 \times 5^x$?

   - Plot $\ln y$ against $x$
   - Plot $y$ against $\ln x$
   - Plot $\lg y$ against $\ln x$
   - Plot $1/y$ against $x$

   *Answer:* Plot $\ln y$ against $x$

   *Why:* Exponential relationships always require plotting the log of the dependent variable against the untransformed independent variable.

## Uncertainties in Transformed Constants

After finding the uncertainty in the gradient and intercept of your transformed line, you need to propagate this uncertainty to find the uncertainty in your final unknown constant. The simplest and most reliable method for CIE exams is the maximum/minimum method.

**Worked example:** You have found $k = 10^c$, where $c = 0.116 \pm 0.012$. Calculate the absolute uncertainty in $k$.

1. Calculate the maximum possible value of $k$ when $c$ is at its maximum:
2. $$k_{max} = 10^{0.116 + 0.012} = 10^{0.128} \approx 1.34$$
3. Calculate the minimum possible value of $k$ when $c$ is at its minimum:
4. $$k_{min} = 10^{0.116 - 0.012} = 10^{0.104} \approx 1.27$$
5. Find the absolute uncertainty as half the difference between max and min:
6. $$\Delta k = \frac{k_{max} - k_{min}}{2} = \frac{1.34 - 1.27}{2} \approx 0.04$$
7. Final result: $k = 1.3 \pm 0.04$

> **tip**
>
> Always use the maximum/minimum method for transformed constants. It avoids mistakes from incorrect application of percentage error approximation rules.

> **Exam tip:** You will lose marks if you only give the uncertainty in gradient/intercept and do not propagate it to the final unknown constant.

## Common pitfalls

- **Wrong:** Mixing up which variable goes on the x-axis and y-axis
  - Why it fails: You did not write out the full rearranged equation before plotting
  - Correct: Always write the full rearranged linear equation, then match $Y$ to the y-axis and $X$ to the x-axis before plotting
- **Wrong:** Reporting the intercept as your final value of the constant, forgetting to reverse the logarithm
  - Why it fails: You stopped after reading the intercept off the graph
  - Correct: If $c = \lg k$, always calculate $k = 10^c$ (or $k = e^c$ for natural logs) before reporting your final answer
- **Wrong:** Mixing log bases (base 10 for one variable, natural log for the other) in a log-log plot
  - Why it fails: You thought log base did not matter as long as both are logs
  - Correct: Always use the same log base for all transformed logarithmic variables. Mixing bases changes the gradient and gives an incorrect result
- **Wrong:** Using the small percentage uncertainty approximation for log-transformed constants
  - Why it fails: You remembered the approximation for linear relationships and applied it incorrectly here
  - Correct: Always use the maximum/minimum method to calculate uncertainty for transformed constants, it is simpler and more reliable for exams
- **Wrong:** Picking two close points to calculate gradient, instead of using the full line length
  - Why it fails: You used data points from the table instead of your best fit line
  - Correct: Always calculate gradient using two points on your best fit line at opposite ends of the line, to reduce the absolute uncertainty in your gradient value

## Cheatsheet

| Original Relationship | Y (y-axis) | X (x-axis) | Gradient | Intercept |
| --- | --- | --- | --- | --- |
| $y = kx^n$ | $\lg y$ / $\ln y$ | $\lg x$ / $\ln x$ | $n$ | $\lg k$ / $\ln k$ |
| $y = y_0 e^{kx}$ | $\ln y$ | $x$ | $k$ | $\ln y_0$ |
| $y = y_0 a^x$ | $\lg y$ / $\ln y$ | $x$ | $\lg a$ / $\ln a$ | $\lg y_0$ / $\ln y_0$ |
| $y = \frac{a}{x + b}$ | $1/y$ | $x$ | $1/a$ | $b/a$ |
| $y^2 = ax + b$ | $y^2$ | $x$ | $a$ | $b$ |

## What's next

Non-linear data analysis is a core skill for CIE A-Level Physics Paper 5, which accounts for 20% of your final A-Level grade. Mastering linearization allows you to test theoretical physical models against collected experimental data and extract unknown constants from your results, which is the foundation of all practical physics work. The skills you learned here are applied across all experimental topics, from radioactive decay to pendulum motion and inverse square law experiments. Next, you can build on this knowledge by studying uncertainty propagation for calculated values and experimental planning, which are the other core skills tested in Paper 5.

- [Advanced Uncertainty Analysis](https://www.owlsprep.com/study/cie-9702-u30-advanced-uncertainty-analysis/)
- [Procedure evaluation](https://www.owlsprep.com/study/cie-9702-u30-procedure-evaluation/)

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