Representing Data: Charts and Diagrams

What you'll learn

This revision guide covers all the charts and diagrams you need to construct, interpret and compare for AQA GCSE Statistics. You'll learn how to select the most appropriate method for representing different data types, apply accurate construction techniques, and analyse what the visual representation reveals about the data. Understanding these visual methods is essential as they appear in both Paper 1 and Paper 2.

Key terms and definitions

Categorical data — data that can be sorted into non-numerical categories or groups (e.g. eye colour, favourite subject, transport method)

Discrete data — numerical data that can only take specific values, often whole numbers (e.g. shoe size, number of siblings, goals scored)

Continuous data — numerical data that can take any value within a range and is measured rather than counted (e.g. height, temperature, time)

Frequency — the number of times a particular value or category occurs in a dataset

Class interval — a group of continuous values within defined boundaries, used when grouping data (e.g. 10 ≤ h < 20)

Modal class — the class interval with the highest frequency in grouped data

Outlier — a value that lies outside the overall pattern of the data, significantly higher or lower than other values

Skewness — a measure of the asymmetry of a distribution; data can be positively skewed (tail to the right), negatively skewed (tail to the left), or symmetric

Core concepts

Bar charts and composite bar charts

Bar charts display categorical or discrete data using separate bars where the height represents frequency. The bars must be separated by gaps of equal width.

Simple bar charts show one variable:

• Draw bars of equal width with gaps between them
• Height (or length for horizontal bars) represents frequency
• Label both axes clearly with units where appropriate
• Include a title describing what the chart shows

Composite (component) bar charts show how a total is divided into sub-categories:

• Each bar is subdivided into sections representing different categories
• Sections are stacked vertically and use different colours or shading
• A key must be included to identify each component
• Useful for comparing totals and compositions simultaneously

Dual bar charts compare two groups side-by-side:

• Two bars are drawn for each category, placed next to each other
• Different colours distinguish the two groups
• A key identifies which bar represents which group
• Effective for direct comparison between two datasets

Pie charts

Pie charts represent categorical data as sectors of a circle, where the angle of each sector is proportional to its frequency.

Constructing pie charts:

1. Calculate the total frequency
2. For each category, calculate: (frequency ÷ total frequency) × 360°
3. Draw a circle using a pair of compasses
4. Use a protractor to measure and draw each sector angle from the centre
5. Label each sector or provide a key
6. Include a title

Comparative pie charts: When comparing two populations using pie charts, the area of each circle should be proportional to the total it represents. Since area = πr², if population A has twice the frequency of population B, the radius of circle A should be √2 times the radius of circle B.

Pictograms

Pictograms use symbols or pictures to represent data, where each symbol represents a specific quantity.

Construction rules:

• Choose an appropriate value for each symbol (e.g. one symbol = 10 units)
• Use part-symbols for values that aren't exact multiples (half a symbol = half the value)
• Align symbols in neat rows
• Include a clear key stating what one symbol represents
• Ensure all symbols are the same size

Stem-and-leaf diagrams

Stem-and-leaf diagrams preserve actual data values while showing distribution shape. The "stem" represents the leading digit(s) and the "leaf" represents the final digit.

Constructing ordered stem-and-leaf diagrams:

1. Split each value into stem and leaf (e.g. 47 becomes stem 4, leaf 7)
2. List all possible stems in a vertical column
3. Write each leaf in the row next to its stem
4. Order the leaves from smallest to largest in each row
5. Include a key (e.g. 4|7 means 47)
6. Add a title

Back-to-back stem-and-leaf diagrams compare two datasets:

• Use a shared central stem
• Write one dataset's leaves to the left (in descending order)
• Write the other dataset's leaves to the right (in ascending order)
• Excellent for comparing distributions, medians, and spread

Frequency diagrams for continuous data

Histograms display continuous data using bars without gaps, where area (not height) represents frequency.

Equal class widths: When all class intervals have the same width:

• The vertical axis shows frequency
• Height of each bar equals frequency
• Bars touch because data is continuous
• Label axes clearly, including the scale

Unequal class widths: When class intervals have different widths:

• The vertical axis shows frequency density
• Frequency density = frequency ÷ class width
• Area of each bar = frequency
• Essential for accurate representation when intervals vary

Reading histograms:

• To find frequency from a histogram with unequal class widths: frequency = frequency density × class width
• The modal class is the interval with the greatest frequency (largest area)
• To estimate median position, find the value where area is split equally

Frequency polygons: Frequency polygons show the shape of a distribution using connected line segments:

1. Plot points at the midpoint of each class interval at height equal to frequency (or frequency density)
2. Join points with straight lines
3. Close the polygon by joining to the horizontal axis at the midpoint of classes either side
4. Useful for comparing multiple distributions on the same axes

Time series graphs

Time series graphs display how a variable changes over time, with time on the horizontal axis.

Construction:

• Plot time values on the horizontal axis in chronological order
• Plot the variable being measured on the vertical axis
• Join consecutive points with straight lines
• Do not join gaps where data is missing
• Label axes with appropriate time units (hours, months, years)

Interpretation:

• Identify trends (general direction: increasing, decreasing, or stable)
• Spot seasonal variation (regular patterns that repeat at fixed intervals)
• Recognise cyclical patterns (longer-term fluctuations)
• Identify unusual values or sudden changes

Scatter graphs and line graphs

Scatter graphs show the relationship between two numerical variables:

• Each point represents one individual or item
• One variable on each axis
• Points are not joined
• Used to identify correlation (relationship between variables)

Line graphs show how one variable depends on another:

• Used for continuous relationships or mathematical functions
• Points are joined with straight lines or a smooth curve
• Often used in science contexts (e.g. distance-time graphs)

Worked examples

Example 1: Constructing a histogram with unequal class widths

Question: The table shows the time (t minutes) students spent on homework. Draw a histogram to represent this data.

Time (t minutes)	Frequency
0 ≤ t < 10	8
10 ≤ t < 30	40
30 ≤ t < 40	15
40 ≤ t < 70	30

Solution:

Step 1: Calculate class widths

• 0 ≤ t < 10: width = 10
• 10 ≤ t < 30: width = 20
• 30 ≤ t < 40: width = 10
• 40 ≤ t < 70: width = 30

Step 2: Calculate frequency density for each class

Time (t minutes)	Frequency	Class width	Frequency density
0 ≤ t < 10	8	10	0.8
10 ≤ t < 30	40	20	2.0
30 ≤ t < 40	15	10	1.5
40 ≤ t < 70	30	30	1.0

Step 3: Draw the histogram with touching bars, vertical axis labelled "Frequency density" and horizontal axis labelled "Time (t minutes)". Plot bars with heights matching frequency density values.

Mark scheme notes: 1 mark for calculating frequency density correctly; 1 mark for appropriate scale and axis labels; 1 mark for accurate bars with correct heights.

Example 2: Interpreting a back-to-back stem-and-leaf diagram

Question: The back-to-back stem-and-leaf diagram shows test scores for Class A and Class B.

Class A          |  Class B
                 |
        9 8 5 3  | 4 | 5 7
    9 7 6 4 2 1  | 5 | 1 2 4 8 9
        8 5 3 0  | 6 | 0 3 6 7
            7 2  | 7 | 1 5 8
                 | 8 | 2

Key: 4|5|7 means 45 for Class A and 57 for Class B

Compare the distributions of test scores for the two classes.

Solution:

Class A:

• Range: 87 - 43 = 44 marks
• Median position: (12 + 1) ÷ 2 = 6.5th value = (56 + 57) ÷ 2 = 56.5 marks
• Data is fairly symmetric with most scores in the 50s and 60s

Class B:

• Range: 82 - 45 = 37 marks
• Median position: (13 + 1) ÷ 2 = 7th value = 63 marks
• Data is positively skewed with most scores in the upper range

Comparison: Class B generally performed better with a higher median (63 vs 56.5 marks). Class B has a smaller range (37 vs 44) suggesting more consistent performance. Class A shows more symmetric distribution while Class B is skewed with more high scores.

Mark scheme notes: 1 mark for correct statistical comparison using median or range; 1 mark for comment on shape/distribution; 1 mark for contextualised conclusion.

Example 3: Choosing appropriate diagrams

Question: A researcher collects data on: a) The favourite sport of 200 students b) The heights of 50 plants measured to the nearest centimetre c) Daily temperature in London throughout January

For each dataset, suggest an appropriate diagram and justify your choice. (3 marks)

Solution:

a) Favourite sport: Bar chart or pie chart. The data is categorical, so a bar chart would clearly show frequencies for each sport category. A pie chart would show proportions of the total, useful for comparing relative popularity. (1 mark)

b) Plant heights: Histogram (with grouped class intervals). The data is continuous and measured, requiring a histogram to accurately represent the distribution. With 50 values, grouping into class intervals would show the shape effectively. (1 mark)

c) Daily temperature: Time series graph. The data varies over time, so plotting temperature against date would show trends, patterns, and any unusual days clearly. This allows identification of warming or cooling trends throughout the month. (1 mark)

Common mistakes and how to avoid them

• Leaving gaps in histograms — Continuous data requires touching bars because values can fall anywhere in the range. Only bar charts (for categorical/discrete data) have gaps.

• Confusing frequency with frequency density — When class widths are unequal, always use frequency density on the vertical axis. Remember: area represents frequency, not height.

• Incorrect pie chart calculations — Always use (frequency ÷ total) × 360° for each sector angle. A common error is using the frequency directly or forgetting to multiply by 360.

• Unordered stem-and-leaf diagrams — Leaves must be written in ascending order for the diagram to be useful for finding median and quartiles. Always reorder after initial plotting.

• Wrong scale on comparative pie charts — When comparing populations of different sizes, the radius (not area or diameter) should be proportional to the square root of the frequency ratio.

• Joining points on scatter graphs — Points on scatter graphs should never be joined as each represents a separate individual. Only join points on line graphs and time series where there's a continuous relationship.

Exam technique for "Representing Data: Charts and Diagrams"

• Understand command words precisely: "Draw" requires accurate construction with ruler, compasses, and protractor; "Sketch" needs correct shape but not precise measurements; "Compare" requires statements about both datasets using statistics or distribution features; "Describe" means identify key features like trend, shape, or unusual values.

• Show your working for histogram questions: Always create a table showing frequency, class width, and frequency density calculations. This earns method marks even if you make an arithmetic error.

• Use the data form to select diagrams: Categorical data → bar chart or pie chart; Discrete numerical → bar chart or stem-and-leaf; Continuous data → histogram or frequency polygon; Change over time → time series graph.

• Read scales carefully: Exam questions often use non-standard scales (e.g. 1 square = 5 units). Check before reading values or plotting points, and state what scale you're using when constructing diagrams.

Quick revision summary

Charts and diagrams convert numerical data into visual representations. Choose bar charts for categorical data, histograms for continuous data (using frequency density when class widths vary), stem-and-leaf diagrams when preserving actual values, and time series for temporal data. Pie charts show proportions using sector angles of (frequency ÷ total) × 360°. Always label axes, include keys where needed, and use appropriate scales. For comparison questions, calculate statistical measures and comment on shape, spread, and central tendency. Accurate construction with proper equipment earns marks in exams.