1.1.4 - Evaluation, uncertainty and improvements

Conclusions from results

A conclusion is a scientific claim supported by evidence. In practical chemistry, a strong conclusion normally has three parts:

• the pattern or value found in the results
• the chemical meaning of that pattern or value
• a judgement about confidence, based on repeats, uncertainty and limitations

For example, suppose a student finds the concentration of sodium hydroxide by titration. Their calculated concentrations are:

A weak conclusion is:

The concentration is about 0.101 mol dm⁻³.

A stronger conclusion is:

The concentration of sodium hydroxide is about 0.101 mol dm⁻³. The three values are close together, so the result is precise under the conditions used. The conclusion still depends on the accuracy of the volumetric apparatus, endpoint judgement and the concentration of the acid used in the titration.

The second version evaluates the evidence. It does not pretend that repeated values remove every possible error.

When you draw a conclusion, avoid overclaiming. If the data only show a correlation, say so. If the uncertainty is large compared with the difference between two results, say that the evidence is weak. If a method has an obvious systematic limitation, mention that the result may be biased even if the repeats agree.

Worked reasoning:

A student compares two methods for measuring 25.0 cm³ of solution:

Method B gives values that are closer together, so it is more precise. If the pipette is correctly calibrated and used correctly, Method B is also likely to be more accurate because it is designed to deliver a fixed volume with smaller uncertainty. Method A may be acceptable for approximate work, but it is not suitable for a quantitative titration where small volume differences affect the calculated concentration.

A numerical interval helps judge a comparison. Results $24.6\pm0.2$ and $24.9\pm0.2\,\mathrm{cm^3}$ span 24.4–24.8 and 24.7–25.1 cm³. The overlap means these uncertainty estimates do not clearly resolve the small difference. This does not prove the values are equal, and apparatus intervals alone are not a formal statistical significance test.

Anomalies and limitations

An anomaly is a result that does not fit the pattern of the other experimental measurements. It may also be called an outlier. An anomalous result may be caused by a procedural failure or a human error, but you should not remove a result simply because it is inconvenient or different from the expected answer.

Consider these titres:

The trial is not normally used in the mean because it is a rough run. Titrations 1, 2 and 4 are close together. Titration 3 is much higher and should be treated as anomalous if there is a plausible reason, such as overshooting the endpoint or an air bubble in the burette tip. The mean should be calculated from the concordant titres, not from every number collected. Record the suspect value and repeat to investigate it; do not invent a cause or erase it. A well-separated outlier may justify exclusion from a concordant mean even when its exact cause is unknown, but a result must never be removed merely to approach an expected answer.

A limitation is different from an anomaly. A limitation is a weakness in the procedure that can affect the quality of the data even if the student follows the method correctly.

Examples of procedural limitations include:

• heat loss to the surroundings in calorimetry
• a subjective colour-change endpoint in a titration
• gas escaping before a bung is fitted
• a reaction not going to completion
• a thermometer or balance with too large an uncertainty for the size of change measured
• a variable such as temperature, concentration, surface area or timing not being controlled
• a measuring cylinder being used when a volumetric pipette or burette is needed

The key exam skill is to link the limitation to its effect. "Heat loss" alone is weaker than "heat loss makes the measured temperature change too small, so the calculated enthalpy change has too small a magnitude."

Worked reasoning:

A student measures the volume of carbon dioxide produced when a carbonate reacts with acid. The bung is fitted after the acid is added.

Limitation: some carbon dioxide may escape before the bung is fitted.

Effect: the measured gas volume is too low.

Consequence: the calculated amount of gas, and any calculation based on it, is likely to be too low.

Improvement preview: add the acid from a dropping funnel or syringe after the apparatus is sealed, then start timing immediately.

Accuracy, precision and uncertainty

Use measurement language carefully.

A set of readings can be precise but not accurate. For example, a balance with a zero error might repeatedly give masses that are very close to each other but all too high. Repeating the measurement would show good precision, but calibration would be needed to improve accuracy.

Uncertainty describes the range associated with a measurement. If a burette reading is recorded as 12.35 cm³ with an uncertainty of ±0.05 cm³, the margin of error is 0.05 cm³ for that reading. The measured value is best written as:

12.35 ± 0.05 cm³

Percentage uncertainty is different. It compares the uncertainty in a measurement with the size of the measurement.

Practical questions often give the uncertainty in the stem. If not, use the apparatus information:

• use the uncertainty marked on the apparatus where it is given
• for an analogue scale, half the smallest division is a common estimate
• for a digital instrument, use the displayed resolution unless the question states otherwise
• follow any uncertainty convention specified by the question or apparatus manufacturer; when estimating from resolution, state the convention used rather than assuming one rule fits every instrument

Worked comparison: error is not uncertainty

A balance reads 9.92 g for a certified 10.00 g mass, and its stated reading uncertainty is ±0.01 g. The percentage error is $|9.92-10.00|/10.00\times100=0.80\%$. The percentage reading uncertainty is $0.01/9.92\times100\approx0.10\%$. These answer different questions: the first compares the reading with a reference; the second describes its stated measurement uncertainty. The discrepancy exceeds that reading uncertainty, so the stated uncertainty alone does not explain the bias.

The same absolute uncertainty matters more when the measured value is small. That is why a 0.10 cm³ uncertainty is a tiny issue for a 25.00 cm³ titre, but a much larger issue for a 2.50 cm³ volume.

Worked example:

A student measures 2.56 g on a balance with uncertainty ±0.01 g.

To a sensible number of significant figures, this is 0.39%.

If the same balance is used to measure only 0.12 g:

The absolute uncertainty is the same, but the percentage uncertainty is much larger because the measured mass is small.

Combined uncertainties

M1.3 requires simple uncertainty handling when data are combined. The most important case in this lesson is a value found by difference.

If a measurement is found by subtracting one reading from another, both readings have uncertainty. Add the absolute uncertainties before converting to a percentage uncertainty.

This applies to:

• titre = final burette reading - initial burette reading
• temperature change = final temperature - initial temperature
• mass lost = initial mass - final mass
• mass transferred by difference = mass before transfer - mass after transfer

Worked example 1: burette titre

A titre is calculated from two burette readings:

• initial reading = 1.20 cm³
• final reading = 22.80 cm³
• uncertainty in each reading = ±0.05 cm³

First calculate the titre:

The titre uses two readings, so:

Now convert to a percentage:

So the percentage uncertainty in the titre is about 0.46%.

Common error: using only 0.05 cm³, which gives half the correct percentage uncertainty.

Worked example 2: comparing two uncertainty sources

A student measures a mass of solid by difference:

• mass of bottle + solid = 16.58 g
• mass of empty bottle = 15.34 g
• uncertainty in each mass reading = ±0.005 g

Mass of solid:

Uncertainty in the mass by difference:

Percentage uncertainty:

The same experiment has a temperature change:

• initial temperature = 20.5 °C
• final temperature = 55.5 °C
• uncertainty in each temperature reading = ±0.1 °C

Temperature change:

Uncertainty in the temperature change:

Percentage uncertainty:

The mass has the greater percentage uncertainty because 0.806% is greater than 0.571%.

These sums give a conservative maximum uncertainty; they are not a statistical confidence interval. For multiplication or division, add the percentage uncertainties in independent measured inputs using the simple maximum-uncertainty convention. For a power, multiply the percentage uncertainty by the absolute value of the power. Use any different procedure explicitly supplied by a question.

For example, concentration $c=n/V$ uses an amount with 0.60% uncertainty and a volume with 0.20% uncertainty. The estimated maximum percentage uncertainty in $c$ is 0.80%. If $c=0.1000\,\mathrm{mol\,dm^{-3}}$, its absolute uncertainty is $0.1000\times0.80/100=0.0008\,\mathrm{mol\,dm^{-3}}$. This propagates measurement uncertainty; it does not include an unrecognised loss of solute or incorrect chemical assumptions.

Improving experimental design

An improvement must match the weakness. Generic answers such as "repeat it", "use better apparatus" or "be more careful" often miss the mark unless they explain what problem is being solved.

Use this three-step structure:

1. Identify the limitation or uncertainty.
2. Suggest a specific change to apparatus or procedure.
3. Explain how the change improves the evidence.

Examples:

Be precise about whether an improvement affects accuracy or precision.

• More repeats usually improve confidence in the mean and help identify anomalies. They do not remove a systematic error such as an uncalibrated balance.
• More precise apparatus reduces apparatus uncertainty. It does not automatically remove procedural bias.
• Increasing the size of the measured change often reduces percentage uncertainty, but it must not create a new limitation, such as unsafe heating or a reaction becoming too vigorous.

Worked example: same apparatus improvement

A student determines an enthalpy change using the same balance, thermometer and polystyrene cup. The temperature change is only 4.0 °C. The uncertainty in each temperature reading is ±0.1 °C.

Current percentage uncertainty in temperature change:

If the student safely increases the amount reacting while keeping the total solution volume and heat capacity approximately unchanged, for example by using more concentrated solutions where suitable, and the temperature change becomes 12.0 °C, the absolute uncertainty in the temperature change is still 0.2 °C:

Increasing every reactant amount and the solution volume in the same proportion need not increase the temperature rise: both released heat and heat capacity increase. The improvement above is valid only when it makes the measured temperature change larger, so the same absolute uncertainty is a smaller percentage of the result. The student should still consider other limitations, such as heat loss and whether the reaction remains safe and complete.

Quick Recap

• Repeats reveal precision; agreement alone does not establish accuracy or remove bias.
• Investigate anomalies and justify exclusions while retaining the raw record.
• Percentage error compares with an accepted value; percentage uncertainty compares uncertainty with the measured value.
• For the stated maximum-uncertainty convention, add absolute uncertainties for differences and percentage uncertainties for products/quotients.
• Match each improvement to a specific limitation and explain its effect. A larger signal helps only if the method remains safe and comparable.