Significant Figures

Significant figures communicate the meaningful precision of a value. OCR requires you to provide answers to an appropriate number of significant figures in quantitative chemistry. This applies whether you are calculating amounts of substance, concentrations, yields or atom economy.

Rounding changes how precisely a value is reported; it does not make the original measurement more accurate. Accuracy concerns closeness to the true value, whereas precision concerns the closeness of repeated measurements and the level of detail supported by the measurement.

Counting significant figures

Start at the first non-zero digit and count from left to right.

• Every non-zero digit is significant: 347 has three significant figures.
• Zeros between significant digits count: 305 has three significant figures.
• Zeros before the first non-zero digit do not count: 0.00457 has three significant figures, because the significant digits are 4, 5 and 7.
• Trailing zeros after a decimal point count when they are written to show precision: 2.50 has three significant figures, while 2.5 has two.

For a whole number such as 12,000, trailing zeros can be ambiguous unless its precision is stated. Standard form removes that ambiguity: $1.2 \times 10^4$ has two significant figures, whereas $1.200 \times 10^4$ has four.

Rounding to a stated number of significant figures

Suppose 0.0045678 must be given to three significant figures.

1. Ignore the leading zeros and locate the first three significant digits: 4, 5 and 6.
2. Inspect the next digit, 7.
3. Because 7 is 5 or more, increase the final retained digit from 6 to 7.
4. Remove the remaining digits.

The answer is 0.00457.

The decision rule is always the same:

• if the next digit is 0–4, leave the final retained digit unchanged;
• if the next digit is 5–9, increase the final retained digit by one.

For example, 10.376 to three significant figures is 10.4. The significant digits are 1, 0 and 3; the next digit is 7, so the 3 rounds up. The zero between 1 and 3 is significant.

For large whole numbers, digits removed by rounding are replaced with zeros to preserve place value. Thus 12,345 to two significant figures becomes 12,000, or more clearly $1.2 \times 10^4$.

Significant figures are not decimal places

Decimal places are counted from the decimal point, but significant figures are counted from the first non-zero digit. This distinction matters especially for very small numbers.

For 0.006783:

• to three significant figures, the answer is 0.00678;
• to three decimal places, the answer is 0.007.

The two instructions ask for different kinds of rounding, so read the wording carefully.

Choosing an appropriate precision

If a question states the required number of significant figures, use that number. If it does not, the answer should reflect the precision of the supplied measurements rather than the full calculator display. A common approach for calculations involving measured quantities is to report the result to the smallest number of significant figures used in the relevant data. In many school science contexts, three significant figures is a sensible reporting convention when no different precision is indicated, but the context and the precision of the data take priority.

Exact counts and defined conversion factors do not normally restrict precision in the same way as measured values. For example, the 60 in “60 seconds in one minute” is an exact conversion, not a measurement made to one significant figure.

A final zero may be necessary to communicate the chosen precision. If a calculated rate is 1 bubble per second and the answer is required to two significant figures, write 1.0 bubbles per second. Writing simply 1 would show only one significant figure.

Round only at the end

Carry the full calculator value through intermediate stages, then round the final answer once. Repeated early rounding can accumulate differences and produce an incorrect final value.

For instance, if 61 bubbles are counted in 60 s, the calculator gives a rate of $61 \div 60 = 1.0166\ldots$ bubbles per second. To two significant figures, the reported rate is 1.0 bubbles per second. The zero is important: it shows that the result has been reported to two significant figures.

Always retain the unit when rounding. Rounding changes the numerical precision, not the quantity being calculated or its unit.