B.1 Handling data

Learning objectives

—B.1.1—Significant figuresUse an appropriate number of significant figures and report calculated results consistently with the precision of the raw data and the least accurate measurement.—B.1.2—Arithmetic and weighted meansCalculate arithmetic and weighted means, including relative atomic mass from isotopic abundances, and select concordant titration data after identifying outliers.—B.1.3—Measurement uncertaintyIdentify measurement uncertainties and use simple techniques to determine uncertainty when data are combined, including uncertainty from two burette readings used to calculate a titre.

Significant figures communicate how precisely a value is supported, not how many digits a calculator can display. Keep extra digits through intermediate working, then round the final result once, using the raw measurements and any explicit instruction to decide the justified precision.

Situation	Reporting guide
multiplication or division	match the measured input with the fewest significant figures
addition or subtraction	match the least precise decimal place
exact count or stoichiometric coefficient	does not limit significant figures
stated answer precision	follow the instruction after completing the calculation

For $80.0-15.846=64.154\,\mathrm{kg}$ , the subtraction is limited by $80.0\,\mathrm{kg}$ to the tenths place, so $64.2\,\mathrm{kg}$ is appropriate. Writing $64.154\,\mathrm{kg}$ claims precision that the first measurement did not provide.

Precision must also fit the quantity. A calculated number of protons, neutrons, atoms or molecules represents a count, so a physically interpreted answer may need to be a whole number even if earlier data contain several significant figures.

Do not round every intermediate value to the final precision: accumulated rounding can change the answer. Decimal places and significant figures are different, and trailing zeros after a decimal may be essential evidence of precision.

\bar{x}=\frac{\sum x}{n}\qquad \text{weighted mean}=\frac{\sum(x_iw_i)}{\sum w_i}

Data situation	Mean to use
repeated comparable measurements	arithmetic mean of the accepted values
isotopes with different abundances	mass weighted by abundance
mixture components with stated fractions	property weighted by component fraction
titration containing an outlier	mean of concordant accurate titres only

For isotopes of masses 35 and 37 with abundances 75% and 25%, $A_r=(35 imes75+37 imes25)/100=35.5$ . The result lies between the isotope masses and closer to the more abundant isotope, providing a useful check.

Calculate each titre as final minus initial burette reading. Exclude the rough value, then identify concordant accurate titres: in this Edexcel chemistry context, accepted titres agree within $0.20\,\mathrm{cm^3}$ . For $24.30$ , $23.80$ and $24.20\,\mathrm{cm^3}$ , use $24.30$ and $24.20$ only, giving a mean of $24.25\,\mathrm{cm^3}$ .

Do not average every recorded value automatically. Excluding a value requires a stated concordance or outlier rule, while a weighted mean must divide by the total weight rather than merely by the number of entries.

\text{percentage uncertainty}=\frac{\text{absolute uncertainty}}{|\text{measured change or value}|}\times100%

Derived value	Simple uncertainty treatment
one direct reading	use the stated uncertainty for that reading
difference of two readings	add their absolute uncertainties
mass by difference	include both balance readings
titre or temperature change	include both initial and final readings

A titre is final burette reading minus initial reading. If each reading is $\pm0.05\,\mathrm{cm^3}$ , the titre uncertainty is $\pm0.10\,\mathrm{cm^3}$ . For an $18.95\,\mathrm{cm^3}$ titre, the percentage uncertainty is $(0.10/18.95) imes100=0.53\%$ .

If a reported mass of $9.53\,\mathrm{g}$ came from two balance readings, each with $\pm0.01\,\mathrm{g}$ uncertainty, the mass by difference is $9.53\pm0.02\,\mathrm{g}$ , giving the possible range $9.51$ to $9.55\,\mathrm{g}$ .

With the same apparatus, reduce percentage uncertainty by measuring a larger change while keeping the chemistry valid—for example, a larger temperature rise makes a fixed thermometer uncertainty a smaller fraction of the result.

Subtract readings to obtain the measured change, but add their absolute uncertainties. Repeating can reveal scatter, yet it does not halve the stated uncertainty of each instrument reading.

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—B.2 AlgebraRead

B.1 Handling data

Learning objectives

Report only the precision supported by the data

Average only values that answer the same question

Combine reading uncertainties before judging the result

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