Cambridge O Level Statistics: Formation of Frequency Distributions (4040)

This guide covers Topic 3: Formation of data into ungrouped or grouped frequency distributions in Cambridge O Level Statistics (4040), following the site’s existing guides to Data and Its Collection and Summary Representation of Data. Where those two topics establish how data is collected and summarised in simple charts, Topic 3 turns to the specific structures – ungrouped and grouped frequency distributions – used to organise larger data sets before further analysis.

Where this fits in 4040

This topic is foundational for much of the rest of the syllabus: cumulative frequency distributions (Topic 4), several statistical measures (Topic 5), and later graphical techniques all build directly on correctly forming a frequency distribution first.

Syllabus coverage

CAMBRIDGE O LEVEL STATISTICS (4040) – TOPIC 3: FORMATION OF DATA INTO FREQUENCY DISTRIBUTIONS

• Ungrouped frequency distributions: tallying individual data values directly, appropriate when a data set has relatively few distinct values
• Grouped frequency distributions: organising data into class intervals, appropriate when a data set spans a wide range of values
• Class measures for grouped data: class limits, class boundaries, class midpoints, and class intervals
• Choosing an appropriate number and width of class intervals for a given data set
• Representing frequency distributions graphically through frequency polygons and histograms

How to approach it

The distinction between class limits and class boundaries is one of the most commonly tested, and most commonly confused, ideas in this topic – building genuine confidence in converting between the two, rather than memorising a single worked example, is the most reliable preparation.

Official syllabus

Cambridge O Level Statistics (4040) syllabus, for examination in 2025-2027 – cambridgeinternational.org.

Class limits vs class boundaries

Class limits are the stated lower and upper values of a class interval as written in the data table (for example, a class of “10-19”). Class boundaries adjust these stated limits to remove any gap between consecutive classes, typically by extending each limit by half a unit (so “10-19” becomes a boundary of 9.5 to 19.5) – this matters because boundaries, not limits, are what should be used when constructing a histogram, since a histogram’s bars must be continuous with no gaps between them. The class midpoint, used when estimating summary statistics like the mean from grouped data, is the value exactly halfway between the class boundaries (or equivalently, between the class limits).

Choosing sensible class intervals

A grouped frequency distribution with too few classes can hide meaningful patterns in the data by over-simplifying it, while too many classes can leave individual classes with very low frequencies, making the distribution’s overall shape hard to see. There is no single fixed rule for the “correct” number of classes, but exam questions often expect candidates to recognise when a given choice of class width is clearly too coarse or too fine for the data set described, and to suggest a more appropriate alternative with reasoning.

Ungrouped vs grouped: choosing the right approach

An ungrouped frequency distribution simply tallies how often each individual value occurs, which works well when a data set has only a small number of distinct values – for example, the number of siblings each student in a class has, where values might only range from 0 to 5. A grouped frequency distribution becomes necessary once a data set spans a wide range of possible values, since tallying every individual value separately would produce an unwieldy table with many classes containing only one or two observations each. Recognising which approach a given data set calls for, before starting to construct a table, is itself a skill this topic tests – a data set is not automatically grouped just because it is large, but because its range of individual values is wide relative to the number of observations.

Worked example: converting limits to boundaries

The routine below is an original model written for this resource, not a reproduction of any official past paper or mark scheme.

Given class: 20-29 (stated as whole-number class limits)

Step 1 - identify the gap between this class and the next:
the next class starts at 30, so there is a gap of 1 unit between
29 and 30.

Step 2 - split that gap in half:
half of 1 unit is 0.5.

Step 3 - extend the lower limit down by that half-gap, and the
upper limit up by that half-gap:
lower boundary = 20 - 0.5 = 19.5
upper boundary = 29 + 0.5 = 29.5

Step 4 - find the midpoint, if needed:
midpoint = (19.5 + 29.5) / 2 = 24.5

This half-unit adjustment specifically applies to data recorded in whole numbers; for data recorded to one decimal place, the same principle applies but the adjustment becomes half of one decimal place (0.05) instead of 0.5 – always check what precision the original data was recorded to before adjusting.

Common mistakes

Using class limits instead of class boundaries when constructing a histogram, leaving gaps between bars that should be continuous. Miscalculating the half-gap adjustment when precision is not whole numbers (for example, forgetting to halve 0.1 rather than 1 for data recorded to one decimal place). Choosing an unreasonably wide or narrow set of class intervals without justifying the choice. Confusing a frequency polygon (points plotted at class midpoints and joined by straight lines) with a histogram (bars drawn between class boundaries).

Quick revision checklist

• Practise converting class limits to class boundaries for data recorded at different levels of precision.
• Be able to calculate a class midpoint from either limits or boundaries.
• Know when a histogram requires class boundaries specifically, not class limits.
• Practise judging whether a given choice of class interval width is too coarse or too fine for a data set.
• Distinguish clearly between a frequency polygon and a histogram, both in construction and in when each is appropriate.