25.4 - Processing Data - Calculations & Significant Figures

What a calculation question looks like

Practical calculation questions usually award marks for both the calculated value and its presentation.

How the marks split

1. The value (1 mark) - The correct number calculated from the correct data, with the unit.
2. The presentation (1 mark) - The answer given to the required number of significant figures, or consistent with the data used.

A correct answer written without the right unit or rounding won't get full marks, and neither will a well-presented incorrect one.

Error carried forward

• If you calculate an early value incorrectly and then use it correctly in every later part, you still get the later marks (error carried forward or ecf).
• If you're unsure about a reading in part (a), write down your best value and carry on. Leaving later parts blank throws away marks you would have been given.

A typical question

Calculate the resistance $R$ of the wire. Use the equation $R=\frac{V}{I}$. Give your answer to a suitable number of significant figures. [2]

The calculations that keep coming back

Resistance from potential difference and current

$$R=\frac{V}{I}$$

A current of 0.76 A gives a potential difference of 1.00 V across 40.0 cm of wire:

$$R=\frac{1.00}{0.76}=\mathrm{1.3157...}=1.32\mathrm{\Omega }$$

Divide potential difference by current, never current by potential difference.

The period of an oscillation

Timing one swing carries a large reaction-time error, so time multiple swings and divide:

$$T=\frac{t}{n}$$

where $t$ is the total time and $n$ is the number of oscillations. A stopwatch reads 17.76 s for 20 oscillations:

$$T=\frac{17.76}{20}=0.888\text{s}$$

This same method applies to measuring other small quantities, such as the thickness of a sheet of paper or the diameter of a wire.

A mean

Add the values and divide by how many there are. Three readings of 21.3, 21.5, and 21.4 cm:

$$\text{Mean}=\frac{21.3+21.5+21.4}{3}=21.4\text{cm}$$

Keep the mean to the same precision as the readings that went into it.

A ratio

Divide one quantity by another of the same kind. The answer has no unit, because the units cancel. If $d$ = 3.0 cm and $l$ = 7.5 cm:

$$\frac{d}{l}=\frac{3.0}{7.5}=0.40$$

Density

$$\rho =\frac{m}{V}$$

An object of mass 261 g and volume 33.5 cm³:

$$\rho =\frac{261}{33.5}=7.79{\text{g/cm}}^3$$

A rate

A rate is a change divided by the time it took:

$$\text{Rate of cooling}=\frac{\text{temperature drop}}{\text{time taken}}$$

Water that falls from 60 °C to 45 °C in 100 s:

$$\frac{60-45}{100}=0.15{}^{\circ }\text{C/s}$$

Significant figures and units

When the question specifies significant figures

If the question asks for a specific number of significant figures (e.g. "give your answer to two significant figures"), round your calculated value to that number.

When the question does not specify significant figures

Rule one: match the data. Give the answer to the same number of significant figures as the least precise measurement used in the calculation.

• If $V$ = 1.00 V (three significant figures) and $I$ = 0.76 A (two significant figures), give $R$ to two or three significant figures (1.3 Ω or 1.32 Ω).
• When one answer is used in a later step, keep the full calculator value through intermediate steps and round once at the end.

Rule two: be consistent down a table column. All calculated values of the same quantity in a table must be given to a consistent number of significant figures.

• The left-hand column uses three significant figures throughout.
• The right-hand column mixes two and three significant figures, which loses the mark.

Units

• Always include the unit with your final value.
• Build the unit from the calculation:
  • Temperature change in °C ÷ time in s gives °C / s.
  • Resistance in Ω ÷ length in cm gives Ω / cm.
• Quantities calculated as a ratio, magnification, or refractive index have no unit.

Common exam questions

Question 1

A stop-watch reads 17.76 s for 20 oscillations. Calculate the period.

$T=t÷n=17.76÷20=0.888$ s.

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Question 2

Why is the time for 20 oscillations measured rather than the time for one?

Because a single oscillation is short compared with your reaction time when starting and stopping the watch, so the error would be a large fraction of the reading. Timing 20 and dividing spreads that error across all of them and gives a much better value for one.

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Question 3

A student measures $V$ = 3.1 V and $I$ = 0.12 A. Calculate the resistance.

$R=V÷I=3.1÷0.12=25.8$ Ω.

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Question 4

A column of calculated resistances reads 0.658, 1.3, 1.974, 2.6. Every value is correctly rounded. Why does this still lose a mark?

Because the values are not given to a consistent number of significant figures. They should all be to two, or all to three.

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Question 5

A student divides one length by another and writes the answer as 0.40 cm. What is wrong?

The unit. Dividing a length by a length cancels the units, so a ratio has no unit at all and the answer is simply 0.40.

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Question 6

A student gets part (a) wrong and then uses that wrong value correctly in parts (b) and (c). What happens to the marks?

The marks in (b) and (c) are still awarded. This is the error-carried-forward rule, and it is why you should always write down your best value and carry on rather than leaving later parts blank.

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Question 7

Water cools from 58 °C to 43 °C in 300 s. Calculate the rate of cooling and give the unit.

(58 - 43) ÷ 300 = 15 ÷ 300 = 0.05 °C / s.

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Question 8

An object has a mass of 261 g and a volume of 33.5 cm³. Calculate its density.

$\rho =m÷V=261÷33.5=7.79$ g/cm³.

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Question 9

A question asks for an answer to two significant figures and the calculator shows 2.3333. What do you write?

2.3. The rounding instruction carries its own mark, so the full calculator display would lose it.