Significant Figures

Rules for Rounding

When calculations produce a long string of decimals on your calculator, you must use rounding to shorten the number to the correct precision.

• Look at the digit immediately to the right of your last required significant figure.
• If this digit is 5 or greater, round up (increase the last significant figure by 1).
• If this digit is 4 or less, round down (keep the last significant figure the same).
• For example, rounding $4.8271$ to 3 s.f. gives $4.83$, while $0.07142$ to 3 s.f. gives $0.0714$.

Multi-Step Calculations and Precision

In chemistry, you will frequently perform multi-step calculations. The most critical rule is the "least precise" rule: for multiplication and division, your final answer must be rounded to match the least significant measurement in the provided data.

However, you must never round intermediate values during the steps of your calculation. You should carry the full calculator display (or at least two extra "guard digits") through every step. Rounding too early causes a rounding error, which will shift your final answer away from the true value. Note that exact values, such as the large numbers used for balancing equations or unit conversions (like dividing by $1000$ to turn $\text{cm}^3$ into $\text{dm}^3$), have infinite precision and do not affect your final significant figures.

Worked Example

A student performs a titration. They find that $25.0\,\text{cm}^3$ of $0.125\,\text{mol/dm}^3$ sodium hydroxide (NaOH) neutralises $18.55\,\text{cm}^3$ of sulfuric acid ($\text{H}_2\text{SO}_4$). Calculate the concentration of the sulfuric acid. Give your answer to an appropriate number of significant figures.

Step 1: Identify the significant figures in the provided data.

• $25.0$ has 3 s.f.

• $0.125$ has 3 s.f.

• $18.55$ has 4 s.f.

• The lowest number of significant figures is 3. The final answer must be given to 3 s.f.

Step 2: Calculate the moles of NaOH.

• $\text{Moles} = \text{concentration} \times \text{volume in dm}^3$

• $\text{Moles} = 0.125 \times (25.0 / 1000)$

• $\text{Moles} = 0.003125\,\text{mol}$

Step 3: Use the molar ratio to find the moles of $\text{H}_2\text{SO}_4$.

• The balanced equation is $2\text{NaOH} + \text{H}_2\text{SO}_4 \rightarrow \text{Na}_2\text{SO}_4 + 2\text{H}_2\text{O}$.

• The ratio is $2:1$, so divide the moles of NaOH by 2.

• $\text{Moles of } \text{H}_2\text{SO}_4 = 0.003125 / 2 = 0.0015625\,\text{mol}$

• (Note: We do not round this intermediate value).

Step 4: Calculate the concentration of the sulfuric acid.

• $\text{Concentration} = \text{moles} / \text{volume in dm}^3$

• $\text{Concentration} = 0.0015625 / (18.55 / 1000)$

• $\text{Concentration} = 0.084231805...\,\text{mol/dm}^3$

Step 5: Round the final answer to the appropriate number of significant figures (3 s.f.).

• Final Answer: $0.0842\,\text{mol/dm}^3$

Precision of Apparatus

The number of significant figures you can record in an experiment depends directly on the precision of the equipment you are using.

• Burettes: Readings must always be recorded to two decimal places, ending in either $.00$ or $.05$ (e.g., $24.30\,\text{cm}^3$). This typically gives 4 significant figures.

• Pipettes: A standard laboratory pipette measures exactly $25.0\,\text{cm}^3$, providing 3 significant figures.

• Measuring cylinders: A standard reading of $25\,\text{cm}^3$ from a basic cylinder only provides 2 significant figures.

When calculating an average titre in OCR practicals, you must only use concordant results (results within $0.10\,\text{cm}^3$ of each other). If you make a mathematical error early in a multi-step question, examiners apply the Error Carried Forward principle, but you will still lose marks if your final answer has the wrong number of significant figures.