Distributions | AQA A-Level Psychology Revision
- Revision Notes
- Aug 5
- 15 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 Distributions A-Level Psychology revision page explains how psychological scores may be arranged across a data set. You will learn to recognise a normal distribution, distinguish positively and negatively skewed distributions, and identify the likely positions of the mean, median and mode. These skills form part of AQA Research Methods and help psychologists interpret patterns within quantitative data. A secure understanding of measures of central tendency will make it much easier to explain how unusual scores affect the shape of a distribution.
Learning Objectives 🎯
By the end of this revision page, you should be able to:
Define a distribution.
Describe the characteristics of a normal distribution.
Distinguish positively and negatively skewed distributions.
Identify the tail of a skewed distribution.
Explain how extreme scores affect the mean.
Identify the likely positions of the mean, median and mode in normal and skewed distributions.
Apply knowledge of distributions to unfamiliar psychological data.
Revision Notes 📚
Distributions in A-Level Psychology
A distribution shows how scores are spread across the possible values in a data set.
It can show:
which scores occur most frequently;
where most of the scores are concentrated;
how widely the scores are spread;
whether the scores are arranged symmetrically;
whether there is a longer tail at one end;
the likely positions of the mean, median and mode.
Distributions are commonly represented using a frequency table, frequency diagram or histogram. The construction and interpretation of these displays are covered in tables and graphs.
The AQA specification requires students to know:
the characteristics of normal distributions;
the characteristics of skewed distributions;
the likely position of the mean, median and mode within a distribution.
Frequency and the shape of a distribution
A frequency is the number of times a score occurs.
Consider the following memory scores:
Memory score | Frequency |
\(2\) | \(1\) |
\(3\) | \(1\) |
\(4\) | \(2\) |
\(5\) | \(3\) |
\(6\) | \(2\) |
\(7\) | \(1\) |
\(8\) | \(1\) |
The score of \(5\) has the highest frequency. Scores become progressively less frequent as they move away from \(5\) in either direction.
When these frequencies are plotted, they form a roughly symmetrical distribution.
The total frequency is:
$$1+1+2+3+2+1+1=11$$
This means that the data set contains \(11\) scores.
Measures of central tendency in distributions
The position and shape of a distribution can be understood using three measures of central tendency:
the mean;
the median;
the mode.
The mean
The mean is calculated by adding all the scores and dividing by the number of scores:
$$\text{Mean}=\frac{\sum x}{N}$$
where:
\(\sum x\) represents the total of all scores;
\(N\) represents the number of scores.
The mean uses every score in the data set. This means that unusually high or low scores can move the mean towards the tail of a skewed distribution.
The median
The median is the middle score when the scores are arranged in numerical order.
If there is an even number of scores, the median is found from the two middle scores.
The median depends on the position of scores rather than their exact values. It is therefore less affected by extreme scores than the mean.
The mode
The mode is the most frequently occurring score.
On a frequency graph, the mode is located at the highest point or tallest bar.
There may be:
one mode;
more than one mode;
no mode, if every score occurs equally often.
Normal distribution
A normal distribution is a symmetrical distribution in which most scores are concentrated around the centre and progressively fewer scores occur towards either extreme.
When represented graphically, a normal distribution forms a symmetrical bell-shaped curve.
Characteristics of a normal distribution
A normal distribution has the following characteristics:
It is symmetrical around its centre.
The left and right sides are mirror images.
Most scores are concentrated around the middle.
Scores become less frequent towards each extreme.
The two tails extend in opposite directions.
The mean, median and mode have the same value.
The highest point of the distribution is at the centre.
The relationship between the measures of central tendency is:
$$\text{Mean}=\text{Median}=\text{Mode}$$
Worked example: a symmetrical distribution
Consider the scores:
$$2,\ 3,\ 4,\ 4,\ 5,\ 5,\ 5,\ 6,\ 6,\ 7,\ 8$$
There are \(11\) scores.
Calculating the mean
Add the scores:
$$2+3+4+4+5+5+5+6+6+7+8=55$$
Divide by the number of scores:
$$\text{Mean}=\frac{55}{11}=5$$
Finding the median
The middle score is the sixth score:
$$2,\ 3,\ 4,\ 4,\ 5,\ \boxed{5},\ 5,\ 6,\ 6,\ 7,\ 8$$
Therefore:
$$\text{Median}=5$$
Finding the mode
The most frequent score is \(5\):
$$\text{Mode}=5$$
Therefore:
$$\boxed{\text{Mean}=\text{Median}=\text{Mode}=5}$$
The equal measures of central tendency are consistent with a symmetrical distribution.
Identifying a normal distribution
When interpreting a graph, look for:
A single central peak.
Similar frequencies on corresponding sides of the centre.
Frequencies decreasing as scores move away from the centre.
Tails extending on both sides.
The mean, median and mode occupying the same central position.
A graph does not need to be perfectly smooth to show an approximately normal distribution. Data collected from a finite sample may show small variations while still forming an approximately symmetrical pattern.
Symmetry in a normal distribution
A distribution is symmetrical when the pattern on one side of the centre reflects the pattern on the other side.
For example, scores equally far from the centre may have similar frequencies.
If the centre is \(10\), the frequencies of scores at:
$$9\text{ and }11$$
may be similar, as may the frequencies at:
$$8\text{ and }12$$
The mean is not pulled more strongly towards either side because unusually high and unusually low scores are balanced.
Skewed distributions
A skewed distribution is an asymmetrical distribution in which scores extend further towards one end of the scale than the other.
The extended end is called the tail.
There are two types of skewed distribution:
positive skew;
negative skew.
The name of the skew is determined by the direction of the tail, not by the location of most of the scores.
This is one of the most important rules to remember.
Positive skew
A positively skewed distribution has a long tail extending towards the higher, more positive values on the right-hand side of the graph.
Most scores are concentrated towards the lower end of the scale.
A small number of unusually high scores stretch the distribution towards the right.
A positive skew may be described as a right skew because its tail extends to the right.
Characteristics of a positive skew
In a positively skewed distribution:
most scores are relatively low;
the highest point is towards the lower end;
a smaller number of high scores form the tail;
the tail extends towards the right;
the mean is pulled towards the high scores;
the mode has the lowest value;
the mean has the highest value.
The likely order is:
$$\text{Mode}<\text{Median}<\text{Mean}$$
Equivalently:
$$\text{Mean}>\text{Median}>\text{Mode}$$
Why the mean moves in a positive skew
The mean uses every score in the data set. A few unusually high scores increase the total of the scores and pull the mean towards the positive tail.
The median depends on the position of the middle score, so it is affected less.
The mode remains at the most frequently occurring score, usually near the main concentration of lower scores.
Worked example: positive skew
Consider the scores:
$$2,\ 2,\ 2,\ 3,\ 3,\ 4,\ 4,\ 5,\ 8,\ 12,\ 20$$
Most scores are concentrated at the lower end, but there are several high scores, including \(20\).
Calculating the mean
Add the scores:
$$2+2+2+3+3+4+4+5+8+12+20=65$$
Divide by the number of scores:
$$\text{Mean}=\frac{65}{11}=5.909\ldots$$
To three significant figures:
$$\text{Mean}=5.91$$
Finding the median
The sixth score is:
$$2,\ 2,\ 2,\ 3,\ 3,\ \boxed{4},\ 4,\ 5,\ 8,\ 12,\ 20$$
Therefore:
$$\text{Median}=4$$
Finding the mode
The most frequent score is \(2\):
$$\text{Mode}=2$$
The order is:
$$2<4<5.91$$
Therefore:
$$\boxed{\text{Mode}<\text{Median}<\text{Mean}}$$
This is the expected pattern for a positively skewed distribution.
Psychological example of positive skew
Suppose a group completes a very difficult memory test.
Most participants may receive low scores, while a small number receive much higher scores.
The distribution would have:
most scores towards the lower end;
a tail extending towards the higher scores;
a positive skew.
The long tail would point to the right.
Negative skew
A negatively skewed distribution has a long tail extending towards the lower, more negative values on the left-hand side of the graph.
Most scores are concentrated towards the higher end of the scale.
A small number of unusually low scores stretch the distribution towards the left.
A negative skew may be described as a left skew because its tail extends to the left.
Characteristics of a negative skew
In a negatively skewed distribution:
most scores are relatively high;
the highest point is towards the higher end;
a smaller number of low scores form the tail;
the tail extends towards the left;
the mean is pulled towards the low scores;
the mean has the lowest value;
the mode has the highest value.
The likely order is:
$$\text{Mean}<\text{Median}<\text{Mode}$$
Why the mean moves in a negative skew
A few unusually low scores reduce the total of the scores. Because the mean uses every score, it is pulled towards the negative tail.
The median is less affected because it is based on the position of the middle score.
The mode remains near the concentration of the most frequent, higher scores.
Worked example: negative skew
Consider the scores:
$$1,\ 5,\ 8,\ 9,\ 10,\ 10,\ 11,\ 11,\ 12,\ 12,\ 12$$
Most scores are concentrated at the higher end, but there are several lower scores, including \(1\).
Calculating the mean
Add the scores:
$$1+5+8+9+10+10+11+11+12+12+12=101$$
Divide by the number of scores:
$$\text{Mean}=\frac{101}{11}=9.1818\ldots$$
To three significant figures:
$$\text{Mean}=9.18$$
Finding the median
The sixth score is:
$$1,\ 5,\ 8,\ 9,\ 10,\ \boxed{10},\ 11,\ 11,\ 12,\ 12,\ 12$$
Therefore:
$$\text{Median}=10$$
Finding the mode
The most frequent score is \(12\):
$$\text{Mode}=12$$
The order is:
$$9.18<10<12$$
Therefore:
$$\boxed{\text{Mean}<\text{Median}<\text{Mode}}$$
This is the expected pattern for a negatively skewed distribution.
Psychological example of negative skew
Suppose participants complete a very easy recognition task.
Most participants may achieve high scores, while a small number achieve much lower scores.
The distribution would have:
most scores towards the higher end;
a tail extending towards the lower scores;
a negative skew.
The long tail would point to the left.
Comparing normal and skewed distributions
Feature | Normal distribution | Positive skew | Negative skew |
Shape | Symmetrical | Asymmetrical | Asymmetrical |
Position of most scores | Around the centre | Towards the lower end | Towards the higher end |
Direction of tail | Tails extend on both sides | Tail extends to the right | Tail extends to the left |
Effect on mean | Balanced at the centre | Pulled towards high scores | Pulled towards low scores |
Mean, median and mode | \(\text{Mean}=\text{Median}=\text{Mode}\) | \(\text{Mode}<\text{Median}<\text{Mean}\) | \(\text{Mean}<\text{Median}<\text{Mode}\) |
A quick method for identifying skew
Use the following three-step method.
Step 1: Find the tail
Identify the end where the distribution extends furthest.
Step 2: Name the skew from the tail
Tail to the right means positive skew.
Tail to the left means negative skew.
Step 3: Position the mean
The mean is pulled towards the tail.
Therefore:
positive tail means the mean is furthest to the right;
negative tail means the mean is furthest to the left.
A useful rule is:
The mean follows the tail.
The positions of the mean, median and mode
The positions of the three measures can help you identify the shape of a distribution.
Normal distribution
$$\text{Mean}=\text{Median}=\text{Mode}$$
All three measures are at the centre.
Positive skew
$$\text{Mode}<\text{Median}<\text{Mean}$$
From left to right, the order is:
$$\text{Mode},\ \text{Median},\ \text{Mean}$$
Negative skew
$$\text{Mean}<\text{Median}<\text{Mode}$$
From left to right, the order is:
$$\text{Mean},\ \text{Median},\ \text{Mode}$$
Why the median remains between the mean and mode
In a skewed distribution:
the mode stays close to the most common group of scores;
the mean is pulled furthest towards the extreme tail;
the median usually lies between them because it is based on the central position.
This produces the typical arrangements:
$$\text{Mode}<\text{Median}<\text{Mean}$$
for positive skew, and:
$$\text{Mean}<\text{Median}<\text{Mode}$$
for negative skew.
Selecting a measure of central tendency for skewed data
The specification requires students to understand the mean, median and mode and select an appropriate measure for a given set of data.
In a skewed distribution, the mean may not represent the typical score well because it is affected by extreme values.
The median may sometimes be more representative because:
it identifies the middle score;
it is less affected by unusually high or low values.
The mode may be useful when the most frequently occurring score is important, although it does not use all the values in the data set.
Your choice should depend on:
the distribution of the data;
the presence of unusual scores;
the level of measurement;
what the researcher wants to describe.
The relationship between data type and statistical decisions is developed further in levels of measurement.
Distribution and dispersion
The shape of a distribution is connected to how widely the scores are spread.
Two data sets could have the same mean but different levels of dispersion.
For example:
Data set A
$$4,\ 5,\ 5,\ 5,\ 6$$
Data set B
$$1,\ 3,\ 5,\ 7,\ 9$$
Both have a mean of:
$$5$$
However, Data set B is more widely spread.
Measures such as the range and standard deviation help describe this spread. These are covered in measures of dispersion.
Distribution and histograms
A histogram can show the shape of a distribution because:
the horizontal axis represents numerical values or intervals;
the height of each adjoining bar shows frequency;
the overall pattern reveals symmetry or skew.
To identify the shape:
Locate the highest frequencies.
Examine how the frequencies change on either side.
Identify whether the graph is symmetrical.
Look for a longer tail.
Use the direction of the tail to name the skew.
Do not identify the skew from the side containing the tallest bars. Identify it from the direction of the tail.
Interpreting distributions in examination questions
You may be given:
a histogram;
a frequency diagram;
a table of scores;
values for the mean, median and mode;
a description of where scores are concentrated;
an unfamiliar psychological scenario.
You may then be asked to:
identify the type of distribution;
justify your answer;
describe the position of the mean, median and mode;
explain the effect of extreme scores;
select an appropriate measure of central tendency;
compare two distributions.
Writing a complete interpretation
A complete answer should use evidence from the information provided.
For example:
The distribution is positively skewed because most scores are concentrated towards the lower end and the tail extends towards the higher values. The mean is likely to be greater than the median and mode because the unusually high scores pull the mean towards the positive tail.
For a negative skew:
The distribution is negatively skewed because most scores are concentrated towards the higher end and the tail extends towards the lower values. The mean is likely to be lower than the median and mode because the unusually low scores pull it towards the negative tail.
Comparing two distributions
When comparing distributions, consider:
whether each is normal or skewed;
the direction of any skew;
the position of the peak;
where most scores are concentrated;
the positions of the mean, median and mode;
the spread of the scores;
whether one distribution contains more extreme scores.
Use comparative language such as:
more symmetrical;
more positively skewed;
more widely spread;
concentrated around a higher score;
has a longer negative tail.
Avoid simply describing each distribution separately when the question asks for a comparison.
Distributions and inferential statistics
Understanding the shape and level of measurement of data helps researchers make decisions about analysis.
The AQA specification requires students to select inferential tests using factors including:
level of measurement;
experimental design;
whether the research investigates a difference, association or correlation.
These decisions are introduced in choosing an inferential test.
Key Words 🔑
Key word | Student-friendly definition | How it may be used in an exam |
Distribution | The way scores are spread across the possible values in a data set. | You may identify or describe the pattern of scores in a graph or table. |
Normal distribution | A symmetrical distribution in which most scores occur around the centre and fewer occur towards the extremes. | You may identify its bell shape and state the positions of the mean, median and mode. |
Symmetrical | Having matching patterns on either side of the centre. | A normal distribution is symmetrical. |
Skewed distribution | An asymmetrical distribution with a longer tail at one end. | You may identify whether the skew is positive or negative. |
Positive skew | A distribution with a tail extending towards the higher values on the right. | You may explain that the mean is pulled towards the positive tail. |
Negative skew | A distribution with a tail extending towards the lower values on the left. | You may explain that the mean is pulled towards the negative tail. |
Tail | The extended part of a distribution containing relatively few scores. | The direction of the tail determines the name of the skew. |
Mean | The total of all scores divided by the number of scores. | You may identify how extreme values pull the mean towards a tail. |
Median | The middle score after scores have been arranged in order. | It usually lies between the mean and mode in a skewed distribution. |
Mode | The most frequently occurring score. | It is located at the highest point of a distribution. |
Frequency | The number of times a particular score occurs. | Frequencies determine the height of bars or points in a distribution. |
Extreme score | A value positioned a considerable distance from most other scores. | An extreme score may pull the mean towards one end of a distribution. |
Bell-shaped curve | The symmetrical shape associated with a normal distribution. | You may use this phrase when describing a normal distribution. |
Hints from the Examiner Reports 💡
No lesson-specific examiner guidance was identified in the provided reports.
Common Mistakes ⚠️
Mistake: Naming the skew from the position of most scores.
Why this is incorrect:
The skew is named after the direction of the tail, not the side containing most of the scores.
How to improve:
Find the long tail first:
right tail means positive skew;
left tail means negative skew.
Mistake: Saying that a positive skew has most scores at the positive end.
Why this is incorrect:
In a positive skew, most scores are usually concentrated towards the lower end. The smaller number of high scores form the positive tail.
How to improve:
Remember that the tail, not the main group of scores, gives the skew its name.
Mistake: Saying that a negative skew has most scores at the negative end.
Why this is incorrect:
In a negative skew, most scores are usually concentrated towards the higher end. A smaller number of low scores form the negative tail.
How to improve:
Locate the tail before describing the concentration of scores.
Mistake: Placing the mean at the highest point of every distribution.
Why this is incorrect:
The highest point represents the mode because it is the most frequent score.
In a skewed distribution, the mean is pulled towards the tail.
How to improve:
Use these patterns:
$$\text{Positive skew: Mode}<\text{Median}<\text{Mean}$$
$$\text{Negative skew: Mean}<\text{Median}<\text{Mode}$$
Mistake: Assuming that the mean, median and mode are always equal.
Why this is incorrect:
They occupy the same position in a normal distribution, but they are separated in a skewed distribution.
How to improve:
First identify whether the distribution is symmetrical or skewed.
Mistake: Reversing the positions of the measures in a positive skew.
Why this is incorrect:
High extreme scores pull the mean to the right, so the mean has the greatest value.
How to improve:
Remember:
$$\text{Mode}<\text{Median}<\text{Mean}$$
Mistake: Reversing the positions of the measures in a negative skew.
Why this is incorrect:
Low extreme scores pull the mean to the left, so the mean has the smallest value.
How to improve:
Remember:
$$\text{Mean}<\text{Median}<\text{Mode}$$
Mistake: Describing a graph as normal only because it has one peak.
Why this is incorrect:
A normal distribution must also be symmetrical, with progressively fewer scores towards both extremes.
How to improve:
Check the whole shape rather than focusing only on the highest point.
Mistake: Giving no evidence for the identified distribution.
Why this is incorrect:
A question asking you to explain or justify an answer requires reference to the shape, tail or positions of the measures.
How to improve:
State both the type and the evidence:
The distribution is positively skewed because its tail extends towards the higher scores.
Mistake: Assuming that the median is pulled as far as the mean by extreme values.
Why this is incorrect:
The median is based on the middle position and is less affected by the exact size of extreme scores.
How to improve:
Remember that the mean uses every score, so it is pulled furthest towards the tail.
Exam-Style Questions ✍️
Question 1
Define the term normal distribution.[2 marks]
Question 2
State the relationship between the mean, median and mode in a normal distribution.[1 mark]
Question 3
A distribution has most scores concentrated towards the lower end of the scale and a long tail extending towards the higher values.
a) Identify the type of distribution.[1 mark]
b) Explain your answer.[2 marks]
c) State the likely order of the mean, median and mode, from lowest to highest.[1 mark]
Question 4
A researcher calculates the following measures for a set of questionnaire scores:
Measure | Value |
Mean | \(14.2\) |
Median | \(16\) |
Mode | \(18\) |
Identify the likely shape of the distribution. Explain your answer.[3 marks]
Question 5
A psychologist records the following scores:
$$2,\ 3,\ 4,\ 4,\ 5,\ 5,\ 5,\ 6,\ 6,\ 7,\ 8$$
a) Calculate the mean.[2 marks]
b) Identify the median.[1 mark]
c) Identify the mode.[1 mark]
d) Explain what these values suggest about the shape of the distribution.[2 marks]
Question 6
The scores below were obtained from a difficult memory task:
$$1,\ 1,\ 2,\ 2,\ 2,\ 3,\ 4,\ 5,\ 9,\ 13,\ 19$$
a) Identify the mode.[1 mark]
b) Identify the median.[1 mark]
c) Calculate the mean. Give your answer to three significant figures.[2 marks]
d) Use the positions of the mean, median and mode to identify the likely distribution.[2 marks]
Question 7
Explain why a small number of unusually high scores may cause a distribution to become positively skewed. Refer to the mean in your answer.[3 marks]
Question 8
A researcher gives a very easy recognition test to a group of participants. Most participants achieve high scores, but a small number achieve much lower scores.
a) Identify the likely shape of the distribution.[1 mark]
b) Describe the likely position of the mean in relation to the median and mode.[2 marks]
c) Explain why the mean occupies this position.[2 marks]
Question 9
A student makes the following statements:
Statement A:“A positive skew has most scores at the positive end of the scale.”
Statement B:“In a negative skew, the mean is likely to have a lower value than the median and mode.”
Statement C:“In a normal distribution, the mean, median and mode occupy the same position.”
Identify which statements are correct. Explain why any incorrect statement is wrong.[5 marks]
Question 10
Two groups complete the same psychological test.
Group A produces a symmetrical distribution with a central peak.
Group B produces a distribution in which most scores are high and the tail extends towards the lower scores.
Compare the two distributions. In your answer, refer to:
the shape of each distribution;
the direction of any skew;
the likely positions of the mean, median and mode.
[6 marks]
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