Statistical diagrams: bar charts, pie charts, pictograms, stem-and-leaf diagrams, frequency diagrams, histograms with equal class widths

What you'll learn

Statistical diagrams form a major component of the CIE IGCSE Mathematics data handling syllabus. You must be able to construct, interpret and compare six core diagram types: bar charts, pie charts, pictograms, stem-and-leaf diagrams, frequency diagrams and histograms with equal class widths. Exam questions typically allocate 4-6 marks per diagram construction and require you to extract information, calculate statistics or make comparisons between datasets.

Key terms and definitions

Bar chart — a diagram using separate rectangular bars of equal width to represent discrete data, with height proportional to frequency.

Pie chart — a circular diagram divided into sectors, where each sector angle is proportional to the frequency it represents (angle = frequency/total × 360°).

Pictogram — a diagram using identical symbols or icons to represent data, where each symbol represents a fixed number of items and partial symbols show fractional values.

Stem-and-leaf diagram — an ordered display showing individual data values split into a 'stem' (leading digit(s)) and 'leaf' (trailing digit), preserving the original data while showing distribution shape.

Frequency diagram — a graph plotting frequency against discrete values, using vertical lines or crosses to show how often each value occurs.

Histogram — a diagram using adjacent rectangular bars to represent continuous data grouped into class intervals, where bar area (not just height) is proportional to frequency when class widths are equal.

Class interval — a range of values used to group continuous data (e.g., 10 ≤ x < 20).

Frequency density — the height of a histogram bar, calculated as frequency ÷ class width, ensuring bar area represents frequency correctly.

Core concepts

Bar charts for discrete and categorical data

Bar charts display discrete data (countable values) or categorical data (named groups). Each bar represents one category or value, with gaps between bars emphasising the discrete nature.

Construction requirements:

• Equal-width bars with consistent spacing
• Bars drawn to accurate height using a scale
• Clearly labelled axes with units where appropriate
• Title describing what the chart represents

Key features for CIE IGCSE Mathematics:

• Vertical bar charts are standard (horizontal acceptable for some contexts)
• Dual/compound bar charts show two datasets side-by-side for comparison
• The frequency (or other measure) is read from the bar height
• Never join the tops of bars — this confuses bar charts with frequency polygons

Exam questions often ask you to:

• Complete partially drawn bar charts from a frequency table
• Read values or calculate totals from given bar charts
• Compare two datasets shown on compound bar charts

Pie charts and sector angle calculations

Pie charts represent data as proportions of a whole circle. The sector angle for each category must be calculated precisely.

Calculation method:

1. Find the total frequency
2. Calculate the angle for each category: (frequency ÷ total) × 360°
3. Draw sectors accurately using a protractor
4. Label each sector clearly

Example calculation: If 45 students chose football from a total of 120 students: Angle = (45 ÷ 120) × 360° = 135°

CIE IGCSE exam requirements:

• Show all angle calculations when constructing pie charts
• Check angles sum to 360° (allow 1° tolerance for rounding)
• Use a ruler for each radius line
• Include a key or labels identifying each sector
• When interpreting pie charts, work backwards: angle ÷ 360° × total = frequency

Common contexts:

• Survey results (favourite subjects, transport methods)
• Market share data
• Budget allocations

Pictograms and fractional symbols

Pictograms use repeated symbols where each icon represents a fixed value stated in a key.

Construction rules:

• Choose an appropriate symbol value (e.g., 1 symbol = 10 items)
• Draw whole symbols for complete multiples
• Use accurate fractions of symbols: ½ symbol, ¼ symbol, ¾ symbol
• Align symbols neatly in rows
• Always include a clear key

Example: If 1 car symbol = 8 cars and the data shows 26 cars: 26 ÷ 8 = 3.25, so draw 3 complete symbols + ¼ of a symbol

Exam focus:

• Questions test your ability to interpret fractional symbols correctly
• You may need to complete a pictogram given some data
• Calculate totals by counting whole and partial symbols
• Choose sensible symbol values that avoid too many fractional symbols

Stem-and-leaf diagrams for ordered data

A stem-and-leaf diagram (or stemplot) displays individual data values while showing the distribution shape.

Construction process:

1. Split each data value into stem (leading digit(s)) and leaf (final digit)
2. List stems vertically in numerical order
3. Write leaves horizontally in numerical order for each stem
4. Include a key explaining the notation (e.g., "2|3 means 23")
5. Add a title

Example for data: 23, 27, 31, 35, 38, 42:

Stem | Leaf
  2  | 3 7
  3  | 1 5 8
  4  | 2
Key: 2|3 means 23

Key advantages:

• Preserves actual data values (unlike grouped frequency tables)
• Shows distribution shape, gaps and clusters
• Easy to find median, mode and range
• Can create back-to-back stemplots to compare two datasets

CIE examination requirements:

• Leaves must be in ascending order (unordered leaves lose marks)
• Always include the key
• Use consistent stem intervals
• For back-to-back diagrams, leaves for one dataset extend left, the other extends right

Frequency diagrams for discrete data distributions

Frequency diagrams plot frequency on the vertical axis against discrete data values on the horizontal axis.

Two common formats:

1. Vertical line diagram — vertical lines drawn from each x-value to the appropriate frequency
2. Cross/point diagram — a cross or point plotted at (value, frequency)

Construction steps:

• Draw and label both axes with equal scales
• Plot a line or cross for each data value at its frequency
• Do not join points or fill areas (this would incorrectly suggest continuous data)
• Include a title

Reading information:

• The mode is the value(s) with the highest frequency
• Calculate mean using: Σ(value × frequency) ÷ Σfrequency
• Find median by counting to the middle position
• Total frequency = sum of all line heights

Histograms with equal class widths

Histograms display continuous data grouped into class intervals. For CIE IGCSE, you work with equal class widths (unequal widths are extended level content).

Critical difference from bar charts:

• Bars are adjacent (no gaps) because data is continuous
• Horizontal axis shows a continuous scale, not discrete categories
• Bar area represents frequency (when widths are equal, height also represents frequency)

Construction requirements:

1. Draw a continuous horizontal axis covering the full data range
2. Mark class boundaries precisely (not just class limits)
3. Draw bars with heights equal to frequency
4. Label axes: typically "frequency" and the variable name with units
5. Bars must be adjacent with no gaps or overlaps

Understanding class intervals:

• Notation like "10 ≤ h < 20" means h can equal 10 but not 20
• The lower boundary is 10, upper boundary is 20
• Width = 20 - 10 = 10

Equal class widths: When all intervals have the same width, frequency density = frequency ÷ class width gives the same relative value for each bar, so you can simply use frequency as the height.

Interpreting histograms:

• Frequency = bar height × class width (for equal widths, this simplifies)
• Modal class is the interval with the highest bar
• Estimate mean by using class midpoints
• Total frequency = sum of all bar frequencies

Exam contexts:

• Heights of students
• Test scores grouped into ranges
• Waiting times
• Distances travelled

Worked examples

Example 1: Pie chart construction

The table shows how 80 students travel to school:

Construct a pie chart to represent this data.

Solution:

Total = 32 + 28 + 16 + 4 = 80 students

Calculate angles:

• Walk: (32 ÷ 80) × 360° = 144°
• Bus: (28 ÷ 80) × 360° = 126°
• Car: (16 ÷ 80) × 360° = 72°
• Cycle: (4 ÷ 80) × 360° = 18°

Check: 144° + 126° + 72° + 18° = 360° ✓

[Draw a circle, use protractor to measure and draw each sector accurately, label each sector or provide a key]

Example 2: Stem-and-leaf diagram interpretation

The stem-and-leaf diagram shows test scores for 15 students:

Stem | Leaf
  4  | 2 5 8
  5  | 1 3 6 7 9
  6  | 2 4 7 8 9
  7  | 3 5
Key: 4|2 means 42 marks

Find (a) the median score, (b) the range, (c) the number of students scoring above 60.

Solution:

(a) There are 15 values, so median is the 8th value when ordered. Counting from the stem-and-leaf: 42, 45, 48, 51, 53, 56, 57, 59 Median = 59 marks

(b) Range = highest value - lowest value = 75 - 42 = 33 marks

(c) Scores above 60: 62, 64, 67, 68, 69, 73, 75 Number of students = 7

Example 3: Histogram construction

Draw a histogram to represent this data on waiting times:

Solution:

All class widths equal 5 minutes, so use frequency as bar height directly.

Draw horizontal axis from 0 to 20 labelled "Waiting time (minutes)" Draw vertical axis from 0 to 16 labelled "Frequency"

Plot bars:

• 0 to 5: height 8
• 5 to 10: height 14
• 10 to 15: height 11
• 15 to 20: height 7

Bars must be adjacent (no gaps). Label both axes clearly.

Common mistakes and how to avoid them

Mistake: Leaving gaps between bars on a histogram. Correction: Histograms represent continuous data — bars must be adjacent with no spaces. Only bar charts (for discrete data) have gaps.

Mistake: Not ordering leaves in a stem-and-leaf diagram. Correction: Leaves must be in ascending numerical order along each row. Write an unordered draft first, then rewrite with ordered leaves for your final answer.

Mistake: Calculating pie chart angles incorrectly or forgetting to show calculations. Correction: Always use the formula (frequency ÷ total) × 360° and show all working. Check your angles sum to 360° before drawing.

Mistake: Misreading fractional symbols in pictograms. Correction: Use the key to find the value of one whole symbol, then calculate: ½ symbol = half that value, ¼ symbol = a quarter, etc. Count carefully.

Mistake: Drawing bar charts when the question asks for a histogram. Correction: Check whether data is discrete (bar chart) or continuous/grouped (histogram). Look for class intervals (like 10-20) as a histogram indicator.

Mistake: Forgetting the key on a stem-and-leaf diagram or pictogram. Correction: Always include a key showing what your notation means (e.g., "3|4 means 34" or "🚗 = 10 cars"). The key is a mandatory element worth marks.

Exam technique for statistical diagrams

Command words and their meanings:

• "Construct" or "Draw" — accurately create the diagram using given data, showing all necessary calculations and labels
• "Complete" — finish a partially drawn diagram (check scales and patterns carefully)
• "Find" or "Use the diagram to..." — extract specific values or perform calculations from a given diagram
• "Compare" — make specific numerical comparisons between two datasets, stating differences with values

Answer structure tips:

• For construction questions: show angle calculations for pie charts, organise data before drawing stem-and-leaf diagrams, use a ruler and protractor for accuracy
• For interpretation: quote specific values from the diagram, show calculation steps clearly
• Allocate time based on marks: a 4-mark diagram needs accuracy and complete labelling; don't rush

Mark allocation patterns:

• Pie chart construction: typically 1 mark for correct angles, 1 mark for accurate drawing, 1 mark for labels
• Stem-and-leaf: 1 mark for correct stems, 1 mark for ordered leaves, 1 mark for key
• Reading from diagrams: usually 1 mark per correctly extracted value

Final check before moving on:

• Are both axes labelled with units?
• Does your diagram have a title or key where required?
• Have you shown working for calculations?
• For histograms, are bars adjacent? For bar charts, are there gaps?

Quick revision summary

Six diagram types appear in CIE IGCSE Mathematics: bar charts (discrete data, bars have gaps), pie charts (angles = frequency/total × 360°), pictograms (with fractional symbols and a key), stem-and-leaf diagrams (ordered leaves, must include key), frequency diagrams (vertical lines for discrete values), and histograms (continuous data, adjacent bars). Know when to use each type: discrete versus continuous data determines chart versus histogram. Always label axes, show calculations for pie charts, order leaves in stemplots, and include keys where required. Practice extracting information and comparing datasets across all six formats.