Graph Scale

Graph Scale Selection & The Danger Zone

1. Definitions

• Large square: $2\text{ cm}$ i.e. 10 small squares ($2\text{ mm}$)
• $N$: Total number of large squares on the axis:
  • $8$ on x-axis
  • $12$ on y-axis
• $R$: Range of data to be plotted on that axis ($R=\text{max}-\text{min}$)
• $s$: Value represented by 1 large square (e.g. 1 large square represents $50\Omega$)

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2. Core Graph Rules

Rule 1: Points must fit on the grid:

$\text{Total axis range}\ge \text{Data range}$ $N\times s\ge R⟹s\ge \frac{R}{N}$

Rule 2: Points must occupy at least half the grid (50% rule)

$\text{Squares used}\ge \frac{N}{2}$ $\frac{R}{s}\ge \frac{N}{2}⟹s\le \frac{2R}{N}$

TIP

Combining both

$\frac{R}{N}\le s\le \frac{2R}{N}$

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3. Standard Scales Rule

The mark scheme requires sensible scales: $s\text{ must be }1\times 10^n,2\times 10^n,\text{ or }5\times 10^n$

WARNING

Awkward scales like 3, 4, 6, 7, 8 or 2.5 lose marks

Derivation of the “Danger Zone”

We want to find when no standard scale can satisfy $\frac{R}{N}\le s\le \frac{2R}{N}$

Notice the ratio between consecutive standard scales:

• From $1$ to $2$: factor of $\frac{2}{1}=2.0$
• From $5$ to $10$: factor of $\frac{10}{5}=2.0$
• From $2$ to $5$: factor of $\frac{5}{2}=2.5$

Because our acceptable range only allows a factor of $2$ ($\text{upper limit}=2\times \text{lower limit}$), a problem occurs only between $2\times 10^n$ and $5\times 10^n$.

For the entire interval to fall in the gap between $2\times 10^n$ and $5\times 10^n$:

1. The lower limit must be greater than $2\times 10^n$: $\frac{R}{N}>2\times 10^n$

2. The upper limit must be less than $5\times 10^n$: $\frac{2R}{N}<5\times 10^n⟹\frac{R}{N}<2.5\times 10^n$

Combining these two inequalities: $2\times 10^n<\frac{R}{N}<2.5\times 10^n$

Multiply everything by $N$: $2N\times 10^{\mathbf{n}}<\mathbf{R}<2.5N\times 10^{\mathbf{n}}$

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4. Summary for Exam Planning

If your range $R$ falls in this interval, you cannot choose a standard scale that obeys both marking points.

• For $N=8$ squares: (x-axis) $16\times 10^n<R<20\times 10^n$ (Avoid ranges whose leading digits are between $1.6$ and $2.0$)

• For $N=12$ squares: (y-axis) $24\times 10^n<R<30\times 10^n$ (Avoid ranges whose leading digits are between $2.4$ and $3.0$)

TIP

How to Check Leading Digits (Excluding Leading Zeros)

Because the standard scales just repeat every power of 10, the $\times 10^n$ only shifts decimal places.

1. Ignore units, prefixes (milli, micro, kilo), and $\times 10^{\text{power}}$.
2. Skip past all leading zeros until you reach the first non-zero number.
3. Read the first two digits to see if they land in the danger zone:
   • $0.00018\text{ A}\to 18$ (falls between $16$ and $20$) ❌
   • $1.9\times 10^{-6}\text{ V}\to 19$ (falls between $16$ and $20$) ❌
   • $0.027\text{ mA}\to 27$ (falls between $24$ and $30$) ❌
   • $0.085\text{ V}\to 85$ (Safe)

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5. Pre-Experiment Workflow & $R_y$ Calibration

Since modern exam graphs are linear, you can calibrate your range after testing just your two extreme points.

Step 1: Test Extremes First

1. Measure your lowest and highest independent variable ($x$) points first.
2. Ensure $R_x$ is wide and its leading digits are not between $1.6$ and $2.0$.
3. Calculate $R_{y,\text{old}}=y_{\max }-y_{\min }$ and check if its leading digits fall between $2.4$ and $3.0$.

Step 2: Proportional Scaling Formula

If $R_y$ is in the Danger Zone ($2.4\to 3.0$), adjust $R_x$. Since gradient $m\approx \frac{R_y}{R_x}$ is constant (the intercept $c$ cancels out):

$\frac{R_{y,\text{new}}}{R_{x,\text{new}}}=\frac{R_{y,\text{old}}}{R_{x,\text{old}}}⟹{\mathbf{R}}_{\mathbf{x},\text{new}}={\mathbf{R}}_{\mathbf{x},\text{old}}\times \left(\frac{{\mathbf{R}}_{\mathbf{y},\text{new}}}{{\mathbf{R}}_{\mathbf{y},\text{old}}}\right)$

• Normal adjustment (Expand): Choose a safe $R_{y,\text{new}}$ with leading digits $\ge 3.1$ and expand $R_x$.
• If $R_x$ is already at the physical or specified max (Shrink): Choose a safe $R_{y,\text{new}}$ with leading digits below $2.4$ (aim for $\approx 2.1\to 2.3$) and shrink $R_x$.

Step 3: Complete the Table

Once your extreme points guarantee safe ranges for both axes, fill in your 4 to 5 intermediate readings evenly.

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6. Critical Practical Warnings

WARNING

1. Transformed Graph Axes (e.g. $1/V$, $L^2$, $\sqrt{h}$)

Always calculate $R$ using the actual plotted values, not the raw apparatus readings:

• If the axis is $1/V$, then $R=\left|\frac{1}{V_{\min }}-\frac{1}{V_{\max }}\right|$.
• If the axis is $L^2$, then $R=|L_{\max }^2-L_{\min }^2|$. Run the danger zone check and calibration formula on these transformed numbers.

WARNING

2. Don’t Lose the “Range” Mark

When shrinking $R_x$ to drop $R_y$ below $2.4$, be careful not to make $R_x$ too small, or you will fail the mark scheme’s minimum range requirement (e.g. $\Delta L\ge 70\text{ cm}$). Only shrink it by the small amount needed to hit leading digits $\approx 2.2$.

WARNING

3. Re-Check the New $R_x$

After calculating $R_{x,\text{new}}$, quickly verify that its own leading digits haven’t landed between $1.6$ and $2.0$.

Danger

Do not start or end your x-axis with your min and max values e.g. if you need to plat values from 6.5, …, 45.6 do not start your graph with 6.5. start it with a multiple of $s$ like 5 in that case