Percentages, ratios and numerical data | AQA A-Level Psychology Revision
- Revision Notes
- Aug 4
- 11 min read
Updated: 6 days ago
For 7182 specification, first teach in September 2025
AQA A-Level Psychology | Free Revision Notes
Estimated study time: 50 minutes
This Percentages, ratios and numerical data A-Level Psychology revision page covers the mathematical skills needed to handle psychological data accurately. You will practise using fractions, ratios and percentages, converting between decimal and standard form, and reporting answers using suitable significant figures. These skills form part of AQA Research Methods and may be assessed through unfamiliar data, calculations and research scenarios. They build on measures of central tendency and measures of dispersion, so careful working can earn valuable marks.
Learning Objectives 🎯
By the end of this revision page, you should be able to:
Calculate fractions, ratios and percentages from psychological data.
Convert fractions and percentages into decimal form.
Recognise and use numbers written in decimal and standard form.
Convert numbers between standard form and ordinary decimal notation.
Report numerical results using an appropriate number of significant figures.
Apply numerical skills to unfamiliar psychological research scenarios.
Revision Notes 📚
Why numerical skills matter in A-Level Psychology
AQA requires students to:
recognise and use expressions in decimal and standard form;
use ratios, fractions and percentages;
use an appropriate number of significant figures.
The specification gives examples such as calculating the percentages of cases falling into different observational categories and expressing a correlation coefficient to two or three significant figures.
At least \(10\%\) of the marks in A-Level Psychology assessments require mathematical skills at higher-tier GCSE standard or above. These skills may appear in Research Methods questions or be embedded within questions about other areas of Psychology.
Researchers may use numerical skills to:
summarise the results of an observation;
compare the sizes of different groups;
report the proportion of participants showing a particular response;
convert values before constructing a graph or chart;
report calculated statistics clearly and consistently.
The results of these calculations may later be displayed using tables and graphs.
Fractions
A fraction shows a part of a whole.
A fraction contains:
a numerator, which is the number above the fraction line;
a denominator, which is the number below the fraction line.
For example:
$$\frac{12}{48}$$
In this fraction, \(12\) is the numerator and \(48\) is the denominator.
When calculating the fraction of observations or participants in a particular category, use:
$$\text{Fraction in a category}=\frac{\text{number in the category}}{\text{total number}}$$
Worked example: calculating a fraction
A researcher records \(48\) instances of behaviour. Of these, \(12\) are classified as cooperative.
The fraction classified as cooperative is:
$$\frac{12}{48}$$
Simplify the fraction by dividing both the numerator and denominator by \(12\):
$$\frac{12\div12}{48\div12}=\frac{1}{4}$$
Therefore, the fraction of behaviours classified as cooperative is:
$$\boxed{\frac{1}{4}}$$
A fraction should normally be simplified unless the question requires the original figures to be retained.
Converting fractions into decimals
To convert a fraction into a decimal, divide the numerator by the denominator:
$$\text{Decimal}=\text{numerator}\div\text{denominator}$$
For example:
$$\frac{3}{8}=3\div8=0.375$$
Therefore:
$$\boxed{\frac{3}{8}=0.375}$$
Ratios
A ratio compares the size of one quantity with another quantity.
For example, if a researcher observes \(18\) helpful behaviours and \(12\) unhelpful behaviours, the ratio of helpful to unhelpful behaviours is:
$$18:12$$
Ratios should usually be simplified. Divide both sides by their highest common factor.
In this example, both numbers can be divided by \(6\):
$$18:12=3:2$$
Therefore:
$$\boxed{3:2}$$
This means that for every \(3\) helpful behaviours, there were \(2\) unhelpful behaviours.
The order of a ratio matters
The ratio:
$$3:2$$
is not the same as:
$$2:3$$
Always follow the order stated in the question.
If the question asks for the ratio of helpful to unhelpful behaviour, helpful behaviour must appear first.
Converting a ratio into fractions
Suppose the ratio of participants in Condition A to Condition B is:
$$3:2$$
The total number of parts is:
$$3+2=5$$
The fraction in Condition A is:
$$\frac{3}{5}$$
The fraction in Condition B is:
$$\frac{2}{5}$$
These fractions can also be converted into decimals or percentages.
Worked example: using a ratio to calculate group sizes
A researcher divides 40 participants into two conditions using the ratio:
$$3:2$$
First calculate the total number of parts:
$$3+2=5$$
Find the number of participants represented by one part:
$$40\div5=8$$
Condition A contains three parts:
$$3\times8=24$$
Condition B contains two parts:
$$2\times8=16$$
Therefore:
$$\boxed{\text{Condition A}=24}$$
$$\boxed{\text{Condition B}=16}$$
Check the total:
$$24+16=40$$
Percentages
A percentage expresses a quantity as a number out of \(100\).
The formula for calculating a percentage is:
$$\text{Percentage}=\frac{\text{number in the category}}{\text{total number}}\times100$$
The final answer should include the percentage symbol:
$$\%$$
Worked example: calculating a percentage
In a study, \(18\) of \(45\) participants correctly recalled a list of words.
Substitute the values into the formula:
$$\text{Percentage}=\frac{18}{45}\times100$$
Divide the category frequency by the total:
$$\frac{18}{45}=0.4$$
Multiply by \(100\):
$$0.4\times100=40$$
Therefore:
$$\boxed{40\%}$$
Calculating a number from a percentage
You may be told the percentage and asked to calculate the number of participants or observations.
Use:
$$\text{Number}=\frac{\text{percentage}}{100}\times\text{total}$$
Worked example: calculating a frequency from a percentage
A researcher recruits 240 participants. Of these, 35% are allocated to one condition.
Substitute the values:
$$\text{Number}=\frac{35}{100}\times240$$
Convert the percentage into a decimal:
$$\frac{35}{100}=0.35$$
Multiply by the total:
$$0.35\times240=84$$
Therefore:
$$\boxed{84\text{ participants}}$$
A participant count must normally be a whole number because part of a participant cannot be included.
Converting between fractions, decimals and percentages
Fractions, decimals and percentages can represent the same proportion in different forms.
Conversion | Method |
Fraction to decimal | Divide the numerator by the denominator |
Decimal to percentage | Multiply by 100 |
Percentage to decimal | Divide by 100 |
Fraction to percentage | Divide the numerator by the denominator, then multiply by 100 |
Percentage to fraction | Write the percentage over 100, then simplify |
Worked example: fraction, decimal and percentage
Convert the fraction below into a decimal and percentage:
$$\frac{3}{8}$$
To find the decimal:
$$3\div8=0.375$$
To find the percentage:
$$0.375\times100=37.5\%$$
Therefore:
$$\boxed{\frac{3}{8}=0.375=37.5\%}$$
Useful equivalences
Fraction | Decimal | Percentage |
\(\frac{1}{2}\) | 0.5 | 50% |
\(\frac{1}{4}\) | 0.25 | 25% |
\(\frac{3}{4}\) | 0.75 | 75% |
\(\frac{1}{5}\) | 0.2 | 20% |
\(\frac{1}{10}\) | 0.1 | 10% |
These equivalences can help you estimate whether a calculated answer is sensible.
Applying percentages to observational data
The AQA specification gives calculating the percentage of cases falling into different categories in an observation study as an example of using percentages in Psychology.
Consider the following results:
Behavioural category | Frequency |
Supportive behaviour | 18 |
Neutral behaviour | 12 |
Critical behaviour | 6 |
Total | 36 |
The percentage of supportive behaviours is:
$$\frac{18}{36}\times100=50\%$$
The percentage of neutral behaviours is:
$$\frac{12}{36}\times100=33.333\ldots\%$$
Reported to three significant figures:
$$33.3\%$$
The percentage of critical behaviours is:
$$\frac{6}{36}\times100=16.666\ldots\%$$
Reported to three significant figures:
$$16.7\%$$
The completed table is:
Behavioural category | Frequency | Percentage |
Supportive behaviour | 18 | 50.0% |
Neutral behaviour | 12 | 33.3% |
Critical behaviour | 6 | 16.7% |
Total | 36 | 100.0% |
The ratio of supportive to critical behaviours is:
$$18:6$$
Simplified:
$$\boxed{3:1}$$
Being able to organise calculations like this will support your later work on translating and presenting data.
Checking percentage totals
When categories represent the whole data set, their percentages should total approximately:
$$100\%$$
However, rounded percentages may add to slightly more or less than \(100%.
For example:
$$33.3\%+33.3\%+33.3\%=99.9\%$$
This does not necessarily mean the calculations are incorrect. It may be the result of rounding.
Decimal form
A number is in decimal form when it is written using place value and a decimal point rather than a fraction or power of ten.
Examples include:
$$0.5$$
$$0.037$$
$$1250$$
A decimal may represent:
a proportion, such as 0.4;
a correlation coefficient, such as 0.72;
a measurement, such as 1.35 seconds;
a value converted from standard form.
When reading a results table, check whether the values are ordinary decimals or numbers written in standard form before attempting a calculation or constructing a graph.
Standard form
Standard form is used to represent very large or very small numbers efficiently.
A number in standard form is written as:
$$a\times10^n$$
The value of a must satisfy:
$$1\leq a<10$$
The value of n is a whole-number power. It may be positive, negative or zero.
Examples include:
$$4.82\times10^{-4}$$
and:
$$6.37\times10^5$$
Converting a small decimal into standard form
Convert:
$$0.000482$$
Move the decimal point until the first number is between 1 and 10:
$$4.82$$
The decimal point moved four places to the right, so the power is negative:
$$\boxed{0.000482=4.82\times10^{-4}}$$
A negative power is used because the original value was smaller than 1.
Converting a large number into standard form
Convert:
$$637000$$
Move the decimal point until the first value is between 1 and 10:
$$6.37$$
The decimal point moved five places to the left:
$$\boxed{637000=6.37\times10^5}$$
A positive power is used because the original number was greater than 10.
Converting standard form into decimal form
Convert:
$$3.46\times10^{-3}$$
A power of -3 means move the decimal point three places to the left:
$$\boxed{3.46\times10^{-3}=0.00346}$$
Convert:
$$2.71\times10^4$$
A power of 4 means move the decimal point four places to the right:
$$\boxed{2.71\times10^4=27100}$$
AQA may require you to convert data from standard form into decimal form before using it in another task, such as constructing a data display.
Significant figures
Significant figures show the precision with which a number is reported.
The first significant figure is the first non-zero digit when reading from left to right.
For example, in:
$$0.007846$$
the first significant figure is 7, not any of the zeros before it.
How to round to significant figures
Find the first non-zero digit.
Count the required number of significant figures from that digit.
Look at the next digit.
If the next digit is 5 or more, round up.
If the next digit is less than 5, leave the previous digit unchanged.
Worked example: correlation coefficient
A researcher calculates a correlation coefficient of:
$$0.873624$$
To report this to three significant figures, identify the first three significant digits:
$$8,\ 7,\ 3$$
The next digit is 6, so the third significant digit rounds up:
$$\boxed{0.874}$$
The specification identifies reporting a correlation coefficient to two or three significant figures as an example of appropriate numerical reporting.
This skill becomes particularly important when studying scattergrams and correlation coefficients.
Worked example: a small measured value
A participant’s response time is:
$$0.007846\text{ seconds}$$
To three significant figures, the significant digits are:
$$7,\ 8,\ 4$$
The next digit is 6, so the 4 rounds up to 5:
$$\boxed{0.00785\text{ seconds}}$$
Significant figures are not decimal places
Consider:
$$0.004728$$
To three decimal places:
$$0.005$$
To three significant figures:
$$0.00473$$
These answers are different because decimal places are counted from the decimal point, whereas significant figures are counted from the first non-zero digit.
Worked example: rounding a percentage
A percentage calculation produces:
$$\frac{17}{36}\times100=47.2222\ldots\%$$
To three significant figures:
$$\boxed{47.2\%}$$
To one decimal place:
$$\boxed{47.2\%}$$
In this example, the answers happen to be the same, but this will not always be the case.
Choosing an appropriate level of precision
The question may tell you how many significant figures or decimal places to use. Follow that instruction exactly.
When no specific instruction is given:
retain enough figures to represent the data accurately;
use a consistent level of precision when comparing similar results;
avoid reporting an unnecessarily long string of digits;
keep unrounded values during your working;
round the final answer rather than rounding at every stage.
For exact counts, such as the number of participants, a whole-number answer is normally required.
For calculated results, such as a mean or correlation coefficient, decimal places or significant figures may be appropriate.
Broader estimation, algebra and numerical problem-solving skills are developed further in mathematical skills in Psychology.
Key Words 🔑
Key word | Student-friendly definition | How it may be used in an exam |
Fraction | A way of showing a part of a whole using a numerator and denominator. | You may calculate the fraction of participants or observations in a category. |
Numerator | The number above the fraction line, showing how many parts are being considered. | It is usually the number in the category when calculating a proportion. |
Denominator | The number below the fraction line, showing the total number of equal parts. | It is usually the total number of participants, scores or observations. |
Ratio | A comparison between two or more quantities. | You may be asked to calculate or simplify the ratio between two conditions or categories. |
Percentage | A proportion expressed as a number out of 100. | You may calculate the percentage of cases falling into an observational category. |
Decimal form | A number written using place value and a decimal point. | Data may need to be converted into decimal form before being used in a calculation or graph. |
Standard form | A way of writing a number as \(a\times10^n\), where \(1\leq a<10\). | You may need to convert a value between standard form and ordinary decimal form. |
Power of ten | The exponent showing how many places a decimal point has effectively moved. | Positive and negative powers are used when converting very large or very small values. |
Significant figure | A digit that contributes to the precision of a reported number, beginning with the first non-zero digit. | You may be instructed to report a calculated result to two or three significant figures. |
Decimal place | A digit positioned after the decimal point. | A question may instruct you to round a result to a specified number of decimal places. |
Proportion | A part of a total, which can be expressed as a fraction, decimal or percentage. | You may compare the proportions of participants showing different responses. |
Frequency | The number of times a score, response or behaviour occurs. | Frequencies may be converted into fractions or percentages for comparison. |
Hints from the Examiner Reports 💡
No lesson-specific examiner guidance was identified in the provided reports.
Common Mistakes ⚠️
Mistake: Using the number in the category as the denominator.
Why this is incorrect:
The denominator should normally be the total number of cases, not the number within the category being calculated.
How to improve:
Write the calculation in words before inserting the values:
$$\frac{\text{number in the category}}{\text{total number}}$$
Mistake: Forgetting to multiply by 100 when calculating a percentage.
Why this is incorrect:
Dividing the category frequency by the total produces a decimal proportion, not a percentage.
How to improve:
Use the full formula every time:
$$\text{Percentage}=\frac{\text{category frequency}}{\text{total frequency}}\times100$$
Mistake: Giving a percentage without the percentage symbol.
Why this is incorrect:
A value such as 35 is not the same as 35%.
How to improve:
Check the required form of the answer and include:
$$\%$$
Mistake: Simplifying only one side of a ratio.
Why this is incorrect:
Both parts of a ratio must be divided by the same value. Otherwise, the comparison changes.
How to improve:
Identify a common factor and divide every part of the ratio by it:
$$18:12=(18\div6):(12\div6)=3:2$$
Mistake: Reversing the order of a ratio.
Why this is incorrect:
A ratio of Condition A to Condition B is not interchangeable with a ratio of Condition B to Condition A.
How to improve:
Underline the order used in the question and write the category labels above your figures before simplifying.
Mistake: Writing standard form with a first number greater than or equal to 10.
Why this is incorrect:
In standard form, the first value must satisfy:
$$1\leq a<10$$
For example:
$$48.2\times10^3$$
is not correctly written in standard form.
How to improve:
Rewrite it as:
$$4.82\times10^4$$
Mistake: Using the wrong sign for a standard-form power.
Why this is incorrect:
Very small values usually require a negative power, while large values usually require a positive power.
How to improve:
Check whether the original value is smaller than 1 or larger than 10.
For example:
$$0.0034=3.4\times10^{-3}$$
but:
$$3400=3.4\times10^3$$
Mistake: Confusing significant figures with decimal places.
Why this is incorrect:
Decimal places are counted from the decimal point. Significant figures are counted from the first non-zero digit.
How to improve:
Circle the first non-zero digit before beginning to count significant figures.
Mistake: Rounding values too early in a multi-stage calculation.
Why this is incorrect:
Early rounding can produce a less accurate final answer.
How to improve:
Keep the full calculator value during your working and round only the final answer to the required precision.
Mistake: Giving part of a participant as a final answer.
Why this is incorrect:
A participant count normally needs to be a whole number.
How to improve:
Check whether the answer represents a measurement or a count. Counts of people or observations should usually be whole numbers.
Exam-Style Questions ✍️
Question 1
Define the term percentage.[1 mark]
Question 2
In an observation study, a researcher records 15 examples of cooperative behaviour from a total of 60 recorded behaviours.
Calculate the percentage of behaviours classified as cooperative. Show your working.[2 marks]
Question 3
A researcher records the following behaviour in a controlled observation.
Behavioural category | Frequency |
Helping another person | 18 |
Ignoring another person | 12 |
Criticising another person | 6 |
a) Express the number of helping behaviours as a fraction of the total. Give your answer in its simplest form.[2 marks]
b) Calculate the ratio of helping behaviours to criticising behaviours. Give your answer in its simplest form.[2 marks]
c) Calculate the percentage of behaviours classified as ignoring another person. Give your answer to three significant figures.[3 marks]
Question 4
Convert the following number into standard form:
$$0.000482$$
[1 mark]
Question 5
Convert the following number from standard form into ordinary decimal form:
$$6.37\times10^5$$
[1 mark]
Question 6
A researcher calculates a correlation coefficient of:
$$0.873624$$
Report this value to three significant figures.[1 mark]
Question 7
A researcher recruits 240 participants. Of these participants, 35% are placed in Condition A.
Calculate the number of participants placed in Condition A. Show your working.[2 marks]
Question 8
Two researchers investigate whether participants correctly identify an emotional facial expression.
Study | Correct identifications | Total participants |
Study A | 27 | 45 |
Study B | 32 | 50 |
Calculate the percentage of correct identifications in each study.
Use your answers to state which study produced the higher proportion of correct identifications. Show your working.[5 marks]
Question 9
A researcher measures the response time of a participant as:
$$0.007846\text{ seconds}$$
Report the response time to three significant figures.[1 mark]
Question 10
A psychologist records 70 responses to a questionnaire.
Response category | Frequency |
Agree | 14 |
Neither agree nor disagree | 21 |
Disagree | 35 |
a) Calculate the percentage of responses in each category.[3 marks]
b) Express the ratio of agree responses to disagree responses in its simplest form.[2 marks]
c) Express the proportion of participants who disagreed as a fraction in its simplest form.[1 mark]
d) Convert the proportion of participants who neither agreed nor disagreed into decimal form.[1 mark]
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