Statistics: Histograms with unequal class widths and frequency density

What you'll learn

This topic addresses how to construct and interpret histograms when data is grouped into classes of different widths. Understanding frequency density is essential for representing data accurately on histograms, a skill regularly tested in CIE IGCSE Mathematics Paper 2 and Paper 4. You'll learn to calculate frequency density, draw histograms correctly, and extract information from completed diagrams.

Key terms and definitions

Histogram — a statistical diagram where the area of each bar represents the frequency of that class, with no gaps between bars for continuous data.

Class width — the range of values covered by a single group or interval, calculated as upper boundary minus lower boundary.

Frequency density — the frequency per unit of class width, calculated using the formula: frequency density = frequency ÷ class width.

Class boundaries — the precise limits of each class interval; for continuous data, these connect adjacent classes without gaps (e.g., 0-10 has boundaries 0 and 10).

Frequency — the number of data values that fall within a particular class interval.

Unequal class widths — class intervals of varying sizes, requiring frequency density calculations to ensure the histogram represents data proportionally.

Modal class — the class interval with the highest frequency density, represented by the tallest bar on a histogram.

Core concepts

Why frequency density matters

Standard bar charts plot frequency directly on the vertical axis, which works perfectly when all bars represent equal ranges. Histograms differ because they represent continuous data and must account for varying class widths.

Consider two classes:

• Class A: 0-10 (width 10) with frequency 20
• Class B: 10-30 (width 20) with frequency 30

If you plotted frequency directly, Class B would appear taller despite having a lower concentration of data. Class A has 2 values per unit width, while Class B has only 1.5. Frequency density corrects this distortion by converting all frequencies to a common scale.

The fundamental principle: area represents frequency in a histogram. Since area = height × width, we need height = frequency ÷ width.

Calculating frequency density

The formula appears simple but requires careful application:

Frequency density = Frequency ÷ Class width

Follow these steps systematically:

1. Identify the class boundaries clearly
2. Calculate the class width (upper boundary − lower boundary)
3. Divide the frequency by the class width
4. Record the frequency density to at least 1 decimal place

Example calculation:

For the second class:

• Class boundaries: 10 and 25
• Class width = 25 − 10 = 15
• Frequency = 45
• Frequency density = 45 ÷ 15 = 3

Constructing a histogram from grouped data

CIE IGCSE Mathematics papers frequently ask candidates to complete or construct histograms. Follow this methodical approach:

Step 1: Create a frequency density column

Add a column to your frequency table and calculate frequency density for each class using the formula above.

Step 2: Set up axes correctly

• Horizontal axis: continuous scale showing the data variable (age, height, time, etc.)
• Vertical axis: frequency density (label it correctly — marks are lost for "frequency")
• Choose an appropriate scale that accommodates your maximum frequency density value
• Ensure equal spacing on both axes

Step 3: Draw the bars

• Each bar extends from the lower boundary to the upper boundary of its class
• Bar height equals the frequency density (not the frequency)
• No gaps between bars for continuous data
• Use a ruler for accuracy

Step 4: Label fully

• Both axes must be clearly labelled with units where applicable
• Title the graph if required

Reading information from a histogram

Exam questions regularly test whether you can extract data from a completed histogram. Remember the core principle: area = frequency.

To find frequency from a histogram:

Frequency = Frequency density × Class width

Example: A bar spans from 20 to 35 on the horizontal axis with height 2.4 on the frequency density axis.

• Class width = 35 − 20 = 15
• Frequency density = 2.4
• Frequency = 2.4 × 15 = 36

To find the modal class: Identify the tallest bar (highest frequency density), then read off its class interval from the horizontal axis.

To estimate the median or other values: You may need to calculate cumulative frequencies by working through the histogram systematically, then use these to locate specific statistical measures.

Dealing with different class boundary notations

CIE papers use various notations for class intervals:

Inequality notation: 10 < t ≤ 20

• Lower boundary: 10 (not included)
• Upper boundary: 20 (included)
• Width: 20 − 10 = 10

Range notation: 10-20

• Interpretation depends on context
• For continuous data, treat as 10 ≤ t < 20
• Width: 20 − 10 = 10

Decimal data: 1.5 < h ≤ 2.0

• Lower boundary: 1.5
• Upper boundary: 2.0
• Width: 2.0 − 1.5 = 0.5

Always read the question carefully to determine whether boundaries are included or excluded.

Working backwards from a histogram

Some questions provide a completed histogram and ask you to complete a frequency table. This tests understanding in reverse.

Method:

1. Read the frequency density from the bar height
2. Identify the class width from the horizontal axis span
3. Calculate frequency using: frequency = frequency density × class width
4. Round to the nearest whole number where appropriate

This skill often appears in Paper 4 (Extended) where students must demonstrate full understanding of the relationship between the three quantities.

Worked examples

Example 1: Constructing a histogram

The table shows the distances students travel to school.

Construct a histogram to represent this data.

Solution:

First, create a frequency density column:

Draw axes:

• Horizontal: Distance (km), scale 0 to 15
• Vertical: Frequency density, scale 0 to 13

Plot bars with heights 12, 12, 9, and 3 spanning the correct class widths with no gaps.

[2 marks for correct frequency densities; 2 marks for accurate histogram with labelled axes]

Example 2: Finding frequencies from a histogram

A histogram shows the ages of concert attendees. One bar spans from 20 to 30 years with a height of 3.5 on the frequency density axis. Another bar spans from 30 to 45 years with a height of 2.4.

Calculate the frequency for each age group.

Solution:

For 20 < age ≤ 30:

• Class width = 30 − 20 = 10
• Frequency density = 3.5
• Frequency = 3.5 × 10 = 35

For 30 < age ≤ 45:

• Class width = 45 − 30 = 15
• Frequency density = 2.4
• Frequency = 2.4 × 15 = 36

[1 mark for each correct frequency]

Example 3: Mixed histogram problem

The incomplete table and histogram show the times taken to complete a puzzle.

The bar for 5 < t ≤ 12 has a frequency density of 5.

(a) Complete the frequency table. (b) Calculate the frequency density for 12 < t ≤ 20. (c) Identify the modal class.

Solution:

(a) For 5 < t ≤ 12:

• Class width = 12 − 5 = 7
• Frequency density = 5
• Frequency = 5 × 7 = 35

(b) For 12 < t ≤ 20:

• Class width = 20 − 12 = 8
• Frequency = 32
• Frequency density = 32 ÷ 8 = 4

(c) Compare frequency densities:

• 0 < t ≤ 5: 20 ÷ 5 = 4
• 5 < t ≤ 12: 5
• 12 < t ≤ 20: 4
• 20 < t ≤ 35: 15 ÷ 15 = 1

Modal class is 5 < t ≤ 12 (highest frequency density)

[1 mark part (a); 1 mark part (b); 1 mark part (c)]

Common mistakes and how to avoid them

• Plotting frequency instead of frequency density on the vertical axis — Always check you're dividing frequency by class width before plotting. The vertical axis must be labelled "frequency density", not "frequency". This is the most common error costing multiple marks.

• Calculating class width incorrectly for inequality notation — For 10 < x ≤ 20, the width is 10, not 11. Don't treat continuous data like discrete counting. Subtract the boundaries directly without adjusting for integers.

• Leaving gaps between bars — Histograms represent continuous data, so bars must touch. Gaps are only appropriate for discrete data on bar charts. This error signals a fundamental misunderstanding to examiners.

• Forgetting to show working when calculating frequency density — Even if your final histogram is correct, marks are allocated for method. Always show the division calculation for at least one class interval explicitly.

• Misidentifying the modal class as the one with highest frequency — The modal class has the highest frequency density (tallest bar), not necessarily the highest frequency. A wide class can have high frequency but low density.

• Rounding frequency density too early — Keep at least 1 decimal place in frequency density calculations. Rounding to whole numbers before plotting creates inaccurate histograms and loses accuracy marks.

Exam technique for Statistics: Histograms with unequal class widths and frequency density

• Command word "construct" typically allocates 3-4 marks: 1 mark for calculating frequency densities, 2 marks for accurate plotting and scaling, 1 mark for complete labelling. Show your frequency density table even if not explicitly requested.

• Read scales carefully on provided histograms — Examiners deliberately use awkward scales (e.g., 2 small squares = 1 unit). Use a ruler to measure bar heights accurately. Estimation alone rarely earns full marks.

• When asked to "complete" a histogram, the question stem often provides one calculated frequency density. Use this as a check — calculate it yourself using the given frequency and class width to verify you understand the scale before completing remaining bars.

• Structure two-part questions methodically: complete the frequency table fully before attempting to draw the histogram. Errors in frequency density calculations propagate through to incorrect bar heights, losing marks in both parts.

Quick revision summary

Histograms display continuous data where area represents frequency. When class widths vary, calculate frequency density = frequency ÷ class width for each interval. Plot frequency density on the vertical axis against the data variable on the horizontal axis, drawing bars with no gaps spanning each class width. To find frequency from a histogram, multiply frequency density by class width. The modal class has the highest frequency density (tallest bar). Always label axes correctly and show calculation working for method marks.