Chemical measurements

What you'll learn

Why every measurement in chemistry has some uncertainty.
How repeated measurements form a distribution of results.
How to calculate a mean and a range from repeat readings.
How to estimate uncertainty using the range about the mean.

Why measurements are never perfectly exact

In chemistry, you often measure things like mass, volume, temperature and time. A measurement is a numerical value found using apparatus, such as a balance, measuring cylinder, thermometer or stopwatch.

Even if you are careful, no measurement is perfectly exact. Apparatus has limits, people read scales slightly differently, and small random changes can happen during an experiment.

Definition

Uncertainty

Uncertainty is the unavoidable doubt in a measured value. It gives an idea of the range in which the true value is likely to lie.

Uncertainty does not mean you made a mistake. It is a normal part of practical chemistry.

For example, if a balance reads to the nearest 0.01 g, it cannot show every possible mass between 2.34 g and 2.35 g. The displayed value is rounded by the balance.

Common Mistake

Uncertainty is not the same as a blunder

A blunder is something like spilling some solution or writing down the wrong number. Uncertainty remains even when the experiment is done carefully.

Repeating measurements

A single measurement gives you one result, but it does not show how much the result might vary. To judge uncertainty better, chemists take repeat measurements.

Definition

Repeat measurements

Repeat measurements are measurements of the same quantity taken more than once, under the same conditions, so the results can be compared.

If the repeat results are close together, the measurement is more precise. If they are spread out, there is more uncertainty.

Definition

Precision

Precision describes how close repeat measurements are to each other. A small spread means high precision.

The mean

When you have repeat measurements, you usually calculate the mean. The mean is the average result.

measurements\text{mean} = \frac{\text{sum of repeat measurements}}{\text{number of measurements}}

The mean is useful because it uses all the repeat readings, so it is usually a better estimate than one reading on its own.

Definition

Anomalous result

An anomalous result is a result that does not fit the pattern of the other results. It may have been caused by an experimental problem or recording error.

If a result is clearly anomalous, you may be expected to leave it out when calculating the mean. The question should make this clear, or the anomaly should be very obvious.

Example

Calculating a mean with an anomalous result

A student measures the volume of gas produced and gets these results:

23.4 cm³, 23.5 cm³, 23.3 cm³, 24.9 cm³, 23.4 cm³

Compare the readings. Four readings are between 23.3 cm³ and 23.5 cm³, but 24.9 cm³ is much higher, so 24.9 cm³ is anomalous.
Calculate the mean using the non-anomalous results:
$cm3\text{mean} = \frac{23.4 + 23.5 + 23.3 + 23.4}{4} = 23.4\ \text{cm}^3$
Use 23.4 cm³ as the best estimate from these repeat measurements, and mention that the anomalous result was not included.

Distribution of results

The distribution of results means how your repeat measurements are spread out.

Definition

Distribution

The distribution of a set of results is the pattern of values: where most results lie, how spread out they are, and whether any result is far from the others.

A dot plot is a simple way to show a distribution. Each dot is one repeated measurement. You can mark the mean and the spread of the results.

Dot plot showing repeated measurements, mean, range and half-range uncertainty

Key Idea

What the spread tells you

A tight cluster of repeat results means low uncertainty and high precision. A wide spread means greater uncertainty.

The range

The range is the difference between the highest and lowest values in a set of results.

Definition

Range

The range is calculated using:

value\text{range} = \text{highest value} - \text{lowest value}

It shows the total spread of the repeat measurements.

The range has the same unit as the measurements. For example, if you measured mass in grams, the range is also in grams.

Estimating uncertainty from repeat measurements

For GCSE Chemistry, you should be able to use the range of repeated measurements about the mean as a measure of uncertainty.

A common estimate is:

uncertainty≈±range2\text{uncertainty} \approx \pm \frac{\text{range}}{2}

This gives the half-range uncertainty. You can then report the result as:

mean±uncertainty\text{mean} \pm \text{uncertainty}

So if the mean mass is 1.82 g and the uncertainty is 0.04 g, you can write:

1.82 \pm 0.04\ \text{g}

This means the result is likely to be somewhere around 1.82 g, with an uncertainty of about 0.04 g either way.

Example

Estimating uncertainty from the range

A student repeats a mass measurement five times:

1.84 g, 1.79 g, 1.82 g, 1.86 g, 1.81 g

Calculate the mean:
$g\text{mean} = \frac{1.84 + 1.79 + 1.82 + 1.86 + 1.81}{5} = 1.824\ \text{g}$
Identify the highest and lowest values: the highest is 1.86 g and the lowest is 1.79 g.
Calculate the range:
$g\text{range} = 1.86 - 1.79 = 0.07\ \text{g}$
Estimate the uncertainty using half the range:
$g\text{uncertainty} = \pm \frac{0.07}{2} = \pm 0.035\ \text{g} \approx \pm 0.04\ \text{g}$
Round the mean sensibly and report the result:
$1.82 \pm 0.04\ \text{g}$

Tip

Check what the question asks for

If the question asks for the range, calculate highest minus lowest. If it asks for an uncertainty from repeat measurements, you will often use half the range and write it with a plus-or-minus sign.

Accuracy and precision

Two important words are often used together: accuracy and precision.

Definition

Accuracy

Accuracy describes how close a measurement is to the true value.

Definition

Precision

Precision describes how close repeat measurements are to each other.

A set of measurements can be precise but not accurate. For example, a miscalibrated balance could give very similar repeat readings, but all of them could be too high.

Common Mistake

Small range does not guarantee accuracy

The range of repeat readings mainly shows random uncertainty. It does not prove that the apparatus is correctly calibrated or that the method has no systematic error.

Comparing results with uncertainty

Uncertainty helps you decide whether two results are genuinely different or whether the difference could be due to measurement variation.

If two uncertainty ranges overlap, you should be cautious. The difference might not be significant at GCSE level. If they do not overlap, it is stronger evidence that the results are different.

Example

Comparing two results using uncertainty ranges

Two students measure the mass of product from different methods:

Method A: $2.50 \pm 0.04\ \text{g}$
Method B: $2.62 \pm 0.05\ \text{g}$

Work out the possible range for Method A:
$2.46\ \text{g}$ $2.54\ \text{g}$
Work out the possible range for Method B:
$2.57\ \text{g}$ $2.67\ \text{g}$
Compare the intervals. Method A is 2.46 g to 2.54 g, and Method B is 2.57 g to 2.67 g. They do not overlap, so Method B probably produced a greater mass.

How to reduce uncertainty

You cannot remove uncertainty completely, but you can reduce its effect.

Useful practical habits include:

Repeat the measurement several times.
Calculate a mean.
Identify any obvious anomalous results.
Use apparatus with a smaller scale division when appropriate.
Keep the method the same each time.
Read scales at eye level to avoid reading errors.

Key Idea

The main GCSE skill

You need to be able to look at repeat measurements, calculate the mean, calculate the range, and use the spread of the results to estimate uncertainty.

Exam technique

In the exam

If you are given repeat readings, first check whether any result is clearly anomalous before calculating the mean.
For range, always do highest value minus lowest value, and keep the correct unit.
If asked for uncertainty from repeat measurements, use half the range unless the question specifically asks for the full range.
When comparing two results, convert each result into a lower-to-upper interval and check whether the intervals overlap.

Self review

Check yourself

Five repeat volumes are 18.2 cm³, 18.3 cm³, 18.1 cm³, 18.2 cm³ and 18.4 cm³. What are the mean, range and half-range uncertainty?
Why does a smaller range usually mean a more precise set of measurements?
Two results are $4.50 \pm 0.10\ \text{g}$ and $4.62 \pm 0.08\ \text{g}$ . Do their uncertainty ranges overlap, and what does that suggest?

Chemical measurements

What you'll learn

Why every measurement in chemistry has some uncertainty.
How repeated measurements form a distribution of results.
How to calculate a mean and a range from repeat readings.
How to estimate uncertainty using the range about the mean.

Why measurements are never perfectly exact

Even if you are careful, no measurement is perfectly exact. Apparatus has limits, people read scales slightly differently, and small random changes can happen during an experiment.

Definition

Uncertainty

Uncertainty is the unavoidable doubt in a measured value. It gives an idea of the range in which the true value is likely to lie.

Uncertainty does not mean you made a mistake. It is a normal part of practical chemistry.

For example, if a balance reads to the nearest 0.01 g, it cannot show every possible mass between 2.34 g and 2.35 g. The displayed value is rounded by the balance.

Common Mistake

Uncertainty is not the same as a blunder

A blunder is something like spilling some solution or writing down the wrong number. Uncertainty remains even when the experiment is done carefully.

Repeating measurements

A single measurement gives you one result, but it does not show how much the result might vary. To judge uncertainty better, chemists take repeat measurements.

Definition

Repeat measurements

Repeat measurements are measurements of the same quantity taken more than once, under the same conditions, so the results can be compared.

If the repeat results are close together, the measurement is more precise. If they are spread out, there is more uncertainty.

Definition

Precision

Precision describes how close repeat measurements are to each other. A small spread means high precision.

The mean

When you have repeat measurements, you usually calculate the mean. The mean is the average result.

measurements\text{mean} = \frac{\text{sum of repeat measurements}}{\text{number of measurements}}

The mean is useful because it uses all the repeat readings, so it is usually a better estimate than one reading on its own.

Definition

Anomalous result

An anomalous result is a result that does not fit the pattern of the other results. It may have been caused by an experimental problem or recording error.

If a result is clearly anomalous, you may be expected to leave it out when calculating the mean. The question should make this clear, or the anomaly should be very obvious.

Example

Calculating a mean with an anomalous result

A student measures the volume of gas produced and gets these results:

23.4 cm³, 23.5 cm³, 23.3 cm³, 24.9 cm³, 23.4 cm³

Compare the readings. Four readings are between 23.3 cm³ and 23.5 cm³, but 24.9 cm³ is much higher, so 24.9 cm³ is anomalous.
Calculate the mean using the non-anomalous results:
$cm3\text{mean} = \frac{23.4 + 23.5 + 23.3 + 23.4}{4} = 23.4\ \text{cm}^3$
Use 23.4 cm³ as the best estimate from these repeat measurements, and mention that the anomalous result was not included.

Distribution of results

The distribution of results means how your repeat measurements are spread out.

Definition

Distribution

The distribution of a set of results is the pattern of values: where most results lie, how spread out they are, and whether any result is far from the others.

A dot plot is a simple way to show a distribution. Each dot is one repeated measurement. You can mark the mean and the spread of the results.

Dot plot showing repeated measurements, mean, range and half-range uncertainty

Key Idea

What the spread tells you

A tight cluster of repeat results means low uncertainty and high precision. A wide spread means greater uncertainty.

The range

The range is the difference between the highest and lowest values in a set of results.

Definition

Range

The range is calculated using:

value\text{range} = \text{highest value} - \text{lowest value}

It shows the total spread of the repeat measurements.

The range has the same unit as the measurements. For example, if you measured mass in grams, the range is also in grams.

Estimating uncertainty from repeat measurements

For GCSE Chemistry, you should be able to use the range of repeated measurements about the mean as a measure of uncertainty.

A common estimate is:

uncertainty≈±range2\text{uncertainty} \approx \pm \frac{\text{range}}{2}

This gives the half-range uncertainty. You can then report the result as:

mean±uncertainty\text{mean} \pm \text{uncertainty}

So if the mean mass is 1.82 g and the uncertainty is 0.04 g, you can write:

1.82 \pm 0.04\ \text{g}

This means the result is likely to be somewhere around 1.82 g, with an uncertainty of about 0.04 g either way.

Example

Estimating uncertainty from the range

A student repeats a mass measurement five times:

1.84 g, 1.79 g, 1.82 g, 1.86 g, 1.81 g

Calculate the mean:
$g\text{mean} = \frac{1.84 + 1.79 + 1.82 + 1.86 + 1.81}{5} = 1.824\ \text{g}$
Identify the highest and lowest values: the highest is 1.86 g and the lowest is 1.79 g.
Calculate the range:
$g\text{range} = 1.86 - 1.79 = 0.07\ \text{g}$
Estimate the uncertainty using half the range:
$g\text{uncertainty} = \pm \frac{0.07}{2} = \pm 0.035\ \text{g} \approx \pm 0.04\ \text{g}$
Round the mean sensibly and report the result:
$1.82 \pm 0.04\ \text{g}$

Tip

Check what the question asks for

Accuracy and precision

Two important words are often used together: accuracy and precision.

Definition

Accuracy

Accuracy describes how close a measurement is to the true value.

Definition

Precision

Precision describes how close repeat measurements are to each other.

A set of measurements can be precise but not accurate. For example, a miscalibrated balance could give very similar repeat readings, but all of them could be too high.

Common Mistake

Small range does not guarantee accuracy

The range of repeat readings mainly shows random uncertainty. It does not prove that the apparatus is correctly calibrated or that the method has no systematic error.

Comparing results with uncertainty

Uncertainty helps you decide whether two results are genuinely different or whether the difference could be due to measurement variation.

If two uncertainty ranges overlap, you should be cautious. The difference might not be significant at GCSE level. If they do not overlap, it is stronger evidence that the results are different.

Example

Comparing two results using uncertainty ranges

Two students measure the mass of product from different methods:

Method A: $2.50 \pm 0.04\ \text{g}$
Method B: $2.62 \pm 0.05\ \text{g}$

Work out the possible range for Method A:
$2.46\ \text{g}$ $2.54\ \text{g}$
Work out the possible range for Method B:
$2.57\ \text{g}$ $2.67\ \text{g}$
Compare the intervals. Method A is 2.46 g to 2.54 g, and Method B is 2.57 g to 2.67 g. They do not overlap, so Method B probably produced a greater mass.

How to reduce uncertainty

You cannot remove uncertainty completely, but you can reduce its effect.

Useful practical habits include:

Repeat the measurement several times.
Calculate a mean.
Identify any obvious anomalous results.
Use apparatus with a smaller scale division when appropriate.
Keep the method the same each time.
Read scales at eye level to avoid reading errors.

Key Idea

The main GCSE skill

You need to be able to look at repeat measurements, calculate the mean, calculate the range, and use the spread of the results to estimate uncertainty.

Exam technique

In the exam

If you are given repeat readings, first check whether any result is clearly anomalous before calculating the mean.
For range, always do highest value minus lowest value, and keep the correct unit.
If asked for uncertainty from repeat measurements, use half the range unless the question specifically asks for the full range.
When comparing two results, convert each result into a lower-to-upper interval and check whether the intervals overlap.

Self review

Check yourself

Five repeat volumes are 18.2 cm³, 18.3 cm³, 18.1 cm³, 18.2 cm³ and 18.4 cm³. What are the mean, range and half-range uncertainty?
Why does a smaller range usually mean a more precise set of measurements?
Two results are $4.50 \pm 0.10\ \text{g}$ and $4.62 \pm 0.08\ \text{g}$ . Do their uncertainty ranges overlap, and what does that suggest?

Chemical measurements

What you'll learn

Why measurements are never perfectly exact

Repeating measurements

The mean

Distribution of results

The range

Estimating uncertainty from repeat measurements

Accuracy and precision

Comparing results with uncertainty

How to reduce uncertainty

You've reached the end

Chemical measurements

What you'll learn

Why measurements are never perfectly exact

Repeating measurements

The mean

Distribution of results

The range

Estimating uncertainty from repeat measurements

Accuracy and precision

Comparing results with uncertainty

How to reduce uncertainty

You've reached the end

Chemical measurements

What you'll learn

Why measurements are never perfectly exact

Repeating measurements

The mean

Distribution of results

The range

Estimating uncertainty from repeat measurements

Accuracy and precision

Comparing results with uncertainty

How to reduce uncertainty

You've reached the end

Chemical measurements Revision Guide

Chemical measurements

What you'll learn

Why measurements are never perfectly exact

Repeating measurements

The mean

Distribution of results

The range

Estimating uncertainty from repeat measurements

Accuracy and precision

Comparing results with uncertainty

How to reduce uncertainty

You've reached the end

Chemical measurements Revision Guide