Measurement, Uncertainty and Presenting Results

AS · 14 min

About half the marks in Paper 3 do not depend on the chemistry at all: they are for recording data correctly, drawing tables and graphs to the conventions Cambridge sets out, giving answers to a sensible number of significant figures, stating uncertainties, and evaluating the experiment. These marks are the easiest to secure, because the rules are written down in the syllabus and do not change from paper to paper. This note sets out every one of those rules with examples, so that you can apply them automatically in the exam.

Recording data in tables

The syllabus requires numerical data and observations to be presented in a single table with headings and units, drawn before you take the readings.

Key result

Rules for tables

Each column heading has the quantity and its unit, separated by a solidus or brackets: volume / cm³, volume (cm³) or volume in cm³. Units do not appear next to each number in the body of the table.
Record raw readings to the same precision, matching the instrument: all burette readings to 0.05 cm³ (two decimal places), all masses from a 2 d.p. balance to 0.01 g, temperatures from a 1 °C thermometer to the nearest 0.5 °C.
A measuring cylinder calibrated at 1 cm³ is read to the nearest 0.5 cm³.
Record observations to the same level of detail, using simple colour words and "pale" or "dark", with comparisons where helpful ("darker brown than at 3 minutes").
Calculated quantities (such as $1/t$ or mean titre) can go in extra columns, with their own units.

A correctly drawn table for a rate experiment:

volume of $\ce{Na2S2O3}$ / cm³	volume of water / cm³	time / s	$\dfrac{1}{t}$ / s⁻¹
50.0	0.0	20	0.050
25.0	25.0	41	0.024

Significant figures in calculated values

Key result

A calculated quantity should be given to the same number of significant figures as, or one more than, the smallest number of significant figures in the data used.

Examples from the syllabus: a titre of 23.45 cm³ (4 s.f.) gives a concentration to 4 s.f., such as $1.305\ \text{mol dm}^{-3}$ . If the concentration of the other reagent is given as $0.100\ \text{mol dm}^{-3}$ (3 s.f.), the answer may be given to 3 or 4 s.f.: $0.131$ or $0.1305\ \text{mol dm}^{-3}$ .

Keep full values in your calculator through intermediate steps and round only at the end. Rounding at each step can change the final answer enough to lose the accuracy mark.

Uncertainty in measurements

Definition

For this syllabus, the maximum uncertainty in a quantitative measurement is half the difference between the closest calibrations of the instrument (that is, half the smallest division), unless an uncertainty is given.

instrument	smallest division	uncertainty per reading
thermometer	1 °C	±0.5 °C
thermometer	0.2 °C	±0.1 °C
burette	0.1 cm³	±0.05 cm³
measuring cylinder (100 cm³)	1 cm³	±0.5 cm³
balance (2 d.p.)	0.01 g	±0.005 g
pipette, volumetric flask	single mark	tolerance stated on the glassware (e.g. ±0.06 cm³ for a 25 cm³ pipette)

Measurements that need two readings

A titre is the difference between two burette readings, a temperature change between two thermometer readings, and a mass by difference between two balance readings. Each reading carries its own uncertainty, so the uncertainty of the difference is double:

titre: $2 \times 0.05 = \pm 0.10\ \text{cm}^3$ ;
temperature change with a 1 °C thermometer: $2 \times 0.5 = \pm 1.0\ ^\circ\text{C}$ ;
mass by difference on a 2 d.p. balance: $2 \times 0.005 = \pm 0.01\ \text{g}$ .

Percentage uncertainty

Key result

\text{percentage uncertainty} = \frac{\text{uncertainty}}{\text{measured value}} \times 100

The syllabus example: a temperature change of 14.0 °C measured with a thermometer calibrated at 1 °C has a maximum percentage error of $\dfrac{2 \times 0.5}{14.0} \times 100 = 7.14\%$ .

The percentage uncertainty is smaller when the measured quantity is larger. This is the basis of most "suggest an improvement" answers: use a larger volume of titrant, a larger mass of solid, or more concentrated reagents to give a bigger temperature change.

When a result is calculated by multiplying or dividing several measurements, the percentage uncertainties are added to estimate the total percentage uncertainty in the result.

Errors: systematic and random

Definition

A systematic error shifts every reading in the same direction by a similar amount; it is not reduced by repeating. A random error makes readings scatter unpredictably above and below the true value; its effect is reduced by repeating and averaging.

type	examples from Paper 3	how to deal with it
systematic	a balance with a zero error (use the same balance for all weighings so the error cancels in differences); a thermometer reading 1 °C too high; heat loss in every enthalpy experiment; $\ce{CO2}$ dissolving in water when collected over water	change the method or apparatus, calibrate, use the same instrument throughout
random	variation in room temperature during a rate experiment; judging when a cross disappears; judging the end-point colour	repeat readings and take a mean; use an instrument (light sensor) instead of judgement

"Human error" is not an acceptable answer. Name the specific source, for example "difficulty in judging the exact moment the cross disappears", which the syllabus classes as a random error.

Accuracy and precision

An accurate result is close to the true value.
Precise results are close to each other (small spread). Concordant titres are precise.
Results can be precise but inaccurate if there is a systematic error: four titres agreeing within 0.05 cm³ but all made with a burette rinsed with water will all be too large.

Drawing graphs

The syllabus gives the rules explicitly.

Key result

Rules for graphs

Plot the independent variable on the $x$ -axis and the dependent variable on the $y$ -axis, unless told otherwise.
Label both axes with quantity and unit, using the same convention as the table (time / s).
Choose scales that are easy to read: 1, 2 or 5 units to a 20 mm square (or multiples of 10 of these). Never use scales of 3, 6 or 7 per square.
The plotted points must occupy at least half the grid in both directions. The axes do not have to start at zero unless you need the origin or an intercept.
Plot points with a cross (×) or circled dot (⊙), accurately, with a sharp pencil.
Draw a straight line or smooth curve of best fit with an even balance of points on either side along its whole length. Do not join the dots.
Identify anomalous points (circle and label them) and ignore them when drawing the line.

Reading a graph

Gradient of a straight line: draw a large triangle, using two points on the line that are more than half the length of the line apart. Gradient $= \dfrac{\Delta y}{\Delta x}$ , with units.
Intercept or intersection: read where the line crosses an axis or another line.
Extrapolation: extend the line beyond the data (for example back to the time of mixing in an enthalpy experiment).

y = 0.99 x (10, 10.0) (20, 19.5) (30, 30.0) (35, 31.0) (40, 39.5) (50, 49.5) (5, 4.95) -- (50, 4.95) (50, 4.95) -- (50, 49.5)

This graph shows the volume of hydrogen (cm³, $y$ -axis) produced when different masses of magnesium (mg, $x$ -axis) react with excess acid. The point at 35 mg is anomalous (well below the line) and is ignored. The triangle uses points far apart on the line.

Evaluation

Identifying the most significant source of error

Compare the percentage uncertainties of the measurements, or think about which systematic error matters most. In an enthalpy experiment with a 1 °C thermometer, the temperature change often has a much larger percentage uncertainty than the volumes, and heat loss is the largest systematic error.

Suggesting improvements

Improvements must be realistic and specific, and must address the error you identified:

problem	realistic improvement
large percentage uncertainty in a temperature change	use a thermometer with 0.2 °C (or 0.1 °C) divisions; use more concentrated reagents for a larger $\Delta T$
heat loss	add a lid; insulate the cup; use the extrapolation method
small titre (large percentage uncertainty)	use a more dilute titrant or a larger pipetted volume, so the titre is larger
judging the end of a rate experiment	use a light sensor / colorimeter; repeat and take a mean
gas lost before collecting	use a divided flask or a small tube inside the flask, so reagents mix only after the bung is in
incomplete decomposition	heat to constant mass

"Use more accurate apparatus" or "be more careful" scores nothing on its own.

Worked examples

Percentage uncertainty in a titre

A titre is 24.40 cm³ and the burette readings are each uncertain by ±0.05 cm³. The pipette used for the 25.0 cm³ portion has an uncertainty of ±0.06 cm³. Calculate the percentage uncertainty in the titre and in the pipetted volume.

Solution

Titre uncertainty $= 2 \times 0.05 = 0.10\ \text{cm}^3$ (two readings).

\frac{0.10}{24.40} \times 100 = 0.41\%

Pipette:

\frac{0.06}{25.0} \times 100 = 0.24\%

Both are small, which is why titration results can be quoted to 3 or 4 significant figures.

Uncertainty in a temperature change

In an enthalpy experiment, the temperature rose from 19.5 °C to 33.5 °C, measured with a thermometer calibrated at 1 °C intervals. Calculate the percentage uncertainty in the temperature change and suggest two ways of reducing it.

Solution

$\Delta T = 33.5 - 19.5 = 14.0\ ^\circ\text{C}$ . Each reading is ±0.5 °C, so $\Delta T$ is ±1.0 °C.

\frac{1.0}{14.0} \times 100 = 7.14\%

Improvements: use a thermometer calibrated at 0.2 °C (or 0.1 °C) intervals; use larger amounts (higher concentrations) of reactants to produce a larger temperature change in the same volume.

Classifying errors

For each, state whether the error is systematic or random: (a) every mass is read from a balance that shows 0.02 g with nothing on it; (b) the room warms up during a series of rate experiments; (c) heat is lost from a polystyrene cup in every enthalpy experiment.

Solution

(a) Systematic: every reading is 0.02 g too high. It cancels when masses are found by difference on the same balance.

(b) Random: the temperature varies unpredictably between runs, changing the rates by different amounts.

(c) Systematic: heat is always lost, so the measured temperature rise is always too small (and $|\Delta H|$ always underestimated).

Exam-style: gradient and a result from a graph

Using the graph of volume of hydrogen against mass of magnesium above, find the gradient and use it to determine the relative atomic mass of magnesium. Assume $\ce{Mg + 2HCl -> MgCl2 + H2}$ and a molar volume of $24.0\ \text{dm}^3\ \text{mol}^{-1}$ .

Solution

Two points on the line far apart: $(5, 4.9)$ and $(50, 49.4)$ .

\text{gradient} = \frac{49.4 - 4.9}{50 - 5} = \frac{44.5}{45} = 0.989\ \text{cm}^3\ \text{mg}^{-1}

For a mass $m$ (in mg) of magnesium, $n(\ce{Mg}) = \dfrac{m}{1000\,A_r}$ mol, and $V(\ce{H2})$ in $\text{cm}^3 = n \times 24\,000 = \dfrac{24\,m}{A_r}$ .

So the gradient $= \dfrac{24}{A_r}$ , and

A_r = \frac{24}{0.989} = 24.3

Exam-hard: evaluating an enthalpy experiment

A student measures the enthalpy change of reaction between zinc powder and $25.0\ \text{cm}^3$ of copper(II) sulfate solution (measured with a $50\ \text{cm}^3$ measuring cylinder with 1 cm³ graduations) in an uncovered glass beaker, using a thermometer calibrated at 1 °C. The temperature rises by 5.0 °C. The student weighs the zinc on a 2 d.p. balance.

(a) Calculate the percentage uncertainty in the volume and in the temperature change.

(b) Identify the most significant source of error and suggest three specific improvements.

Solution

(a) Volume: $\dfrac{0.5}{25.0} \times 100 = 2.0\%$ . Temperature change: $\dfrac{1.0}{5.0} \times 100 = 20\%$ .

(b) The temperature change has by far the largest uncertainty, and heat loss from an uncovered glass beaker is a large systematic error. Improvements:

use a polystyrene cup with a lid instead of a glass beaker, to reduce heat loss;
use a thermometer with 0.2 °C (or 0.1 °C) divisions;
use a more concentrated copper(II) sulfate solution (zinc in excess) to give a larger temperature rise, and/or record temperatures over time and extrapolate the cooling curve back to the time of mixing. (The mass of zinc is not important because zinc is in excess; using a pipette or burette for the volume would also help but the volume is not the main problem.)

(c) Less negative (smaller magnitude). Heat lost to the surroundings and absorbed by the glass beaker means the measured temperature rise is smaller than it should be, so the calculated energy released per mole is smaller.

Watch out

Do not quote uncertainties to more precision than the measurement: "±0.05 cm³", not "±0.050 cm³ per reading" mixed with "24.4 cm³".
Do not forget to double the uncertainty for differences (titre, $\Delta T$ , mass by difference).
A graph with points crowded into one corner, or scales of 3 units per square, loses marks even if the plotting is accurate.
Do not force a line through the origin unless the science requires it, and do not include anomalous points in the line.
Never write "human error". Name the actual source.

Exam tip

The table and graph rules are mark-scheme items: headings with units, consistent decimal places, sensible scales, half the grid used, best-fit line, anomaly identified.
For gradients, show the triangle on the graph, the coordinates used, and the calculation with units.
When asked for the "most significant source of error", compare percentage uncertainties or justify with the size of the effect. A one-word answer rarely scores.
Improvements must link to the problem: "use a lid to reduce heat loss" scores; "use a lid" alone may not.
When asked to "suggest how the investigation could be extended", propose a new independent variable to investigate, keeping the same method (for example, the effect of temperature after investigating concentration).

Practical skills

A quick checklist to run through at the end of every Paper 3 quantitative question:

Is every reading in the table, with headings and units, and to the right precision?
Are the concordant titres identified, or the anomalous points marked?
Is every calculation step written down, with the final answer to 3 or 4 s.f. and a unit?
Have I given a sign for $\Delta H$ ?
If asked for errors: have I named a specific source, said whether it is systematic or random where relevant, and given a realistic improvement linked to it?

Summary

Single table, headings "quantity / unit", raw readings to the instrument's precision (burette 0.05 cm³, 1 °C thermometer 0.5 °C).
Calculated answers: same number of significant figures as the least precise data, or one more.
Uncertainty = half the smallest division; double it for a difference of two readings.
Percentage uncertainty = uncertainty / value × 100; larger measurements give smaller percentage uncertainties.
Systematic errors shift all readings one way; random errors scatter them and are reduced by repeats.
Graphs: labelled axes, scales of 1, 2 or 5 per 20 mm, points over at least half the grid, line of best fit, anomalies identified, large gradient triangle.
Improvements must be realistic, specific and linked to the identified error.

Practice

Question

Write a suitable column heading for a column of temperatures measured in degrees Celsius, and state how many decimal places readings from a 1 °C thermometer should have.
Calculate the percentage uncertainty in a titre of 12.50 cm³ read from a burette with 0.1 cm³ graduations. Suggest how the titration could be changed to reduce it.
A mass of 1.26 g is found by difference on a balance reading to 0.01 g. Calculate the percentage uncertainty.
A student gives a concentration calculated from a titre of 21.35 cm³ and a concentration of 0.0500 mol dm⁻³ as 0.0427125 mol dm⁻³. Rewrite it to an appropriate number of significant figures.
Explain the difference between accuracy and precision using titration results as an example.
In a gas-collection experiment, the volume of carbon dioxide is 44 cm³, read from a measuring cylinder with 1 cm³ graduations. Calculate the percentage uncertainty, and identify a systematic error in collecting this gas over water.
State three features of a well-drawn graph for Paper 3, other than labelled axes.
A student writes "the results were inaccurate because of human error" in an evaluation of a disappearing-cross experiment. Rewrite this as a creditworthy answer and suggest an improvement.
In an experiment to find the enthalpy change of solution of a salt, 2.00 g of solid is dissolved in 50.0 cm³ of water and the temperature falls by 2.5 °C (thermometer ±0.5 °C). Calculate the percentage uncertainty in $\Delta T$ , compare it with that in the mass, and suggest the single most effective improvement.
The concentration of a sodium hydroxide solution is found by titration: $25.0\ \text{cm}^3$ (pipette, $\pm 0.06\ \text{cm}^3$ ) is titrated with $0.100\ \text{mol dm}^{-3}$ HCl, whose concentration is stated to be accurate to $\pm 0.5\%$ . The mean titre is $24.40\ \text{cm}^3$ (burette readings $\pm 0.05\ \text{cm}^3$ ). The calculated concentration is $0.0976\ \text{mol dm}^{-3}$ . Estimate the total percentage uncertainty and the absolute uncertainty in this concentration, and identify which measurement contributes most.

Answers

temperature / °C (or temperature (°C)); readings to the nearest 0.5 °C, so one decimal place (e.g. 21.0, 21.5).
Uncertainty $= 2 \times 0.05 = 0.10\ \text{cm}^3$ . $0.10 / 12.50 \times 100 = 0.80\%$ . Use a more dilute titrant (or pipette a larger volume of the other solution) so the titre is larger, around 20–25 cm³.
Uncertainty $= 2 \times 0.005 = 0.01\ \text{g}$ (two weighings). $0.01 / 1.26 \times 100 = 0.79\%$ .
The data have 4 s.f. (21.35) and 3 s.f. (0.0500), so 3 or 4 s.f.: $0.0427$ or $0.04271\ \text{mol dm}^{-3}$ .
Precision is how close repeated results are to each other (e.g. titres of 24.35 and 24.40 cm³ are precise, concordant). Accuracy is how close a result is to the true value. Titres can be precise but inaccurate if, for example, the burette was rinsed with water so every titre is too large.
$0.5 / 44 \times 100 = 1.1\%$ . Systematic error: carbon dioxide is slightly soluble in water, so some dissolves and the volume collected is always too small (also gas lost before the bung is inserted).
Any three: scales of 1, 2 or 5 units per 20 mm square; points covering at least half the grid in each direction; points plotted accurately as × or ⊙; a single straight line or smooth curve of best fit with points evenly distributed on either side; anomalous points identified and ignored.
"It was difficult to judge exactly when the cross disappeared, which is a random error that affects each time by a different amount." Improvement: use a light sensor and data logger (or colorimeter) to detect a fixed decrease in light transmission, and repeat each concentration and take a mean.
$\Delta T$ : $1.0 / 2.5 \times 100 = 40\%$ . Mass: $0.01 / 2.00 \times 100 = 0.5\%$ . The temperature change is by far the largest uncertainty. The most effective improvement is to increase $\Delta T$ : use a larger mass of solid in the same volume of water (or the same mass in less water), and/or use a thermometer with 0.1 °C divisions.
Titre: $0.10 / 24.40 \times 100 = 0.41\%$ . Pipette: $0.06 / 25.0 \times 100 = 0.24\%$ . Acid concentration: $0.5\%$ . The concentration is calculated by multiplying and dividing these quantities, so the percentage uncertainties add: $0.41 + 0.24 + 0.5 = 1.15\% \approx 1.2\%$ . Absolute uncertainty $= 0.0976 \times 0.0115 = \pm 0.0011\ \text{mol dm}^{-3}$ , so $c = 0.0976 \pm 0.0011\ \text{mol dm}^{-3}$ . The largest contribution is the uncertainty in the concentration of the standard acid, not the titration readings.

Recording data in tables

Significant figures in calculated values

Uncertainty in measurements

Measurements that need two readings

Percentage uncertainty

Errors: systematic and random

Accuracy and precision

Drawing graphs

Reading a graph

Evaluation

Identifying the most significant source of error

Suggesting improvements

Worked examples

Practice

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