Data Representation

CIE A-Level Mathematics· Unit 4: Probability & Statistics 1· 15 min read

1. Frequency Distributions★☆☆☆☆⏱ 5 min

A frequency distribution organises raw data into non-overlapping classes, showing how many observations (frequency) fall into each class. For ungrouped discrete or categorical data, each value/category is its own class. For grouped continuous data, we use class boundaries to eliminate gaps between adjacent classes.

📘 Definition

Class Boundaries

The actual upper and lower limits of a grouped data class, adjusted to avoid gaps between classes

Example:

A class written as $10 - 19$ for whole-number data has boundaries $9.5$ and $19.5$

📐 Worked Example

The following are heights (cm) of 20 students: 152, 156, 161, 162, 164, 165, 167, 168, 169, 170, 171, 172, 173, 175, 176, 178, 180, 181, 183, 185. Construct a grouped frequency distribution with 4 equal-width classes.

1
Calculate the range of the data and find class width:
2
$185 - 152 = 33, \quad \text{round up to width } 9$
3
Define classes and their boundaries, then count frequencies:
4
Stated Class Class Boundaries Frequency
150-158 149.5-158.5 2
159-167 158.5-167.5 5
168-176 167.5-176.5 8
177-185 176.5-185.5 5

Stated Class	Class Boundaries	Frequency
150-158	149.5-158.5	2
159-167	158.5-167.5	5
168-176	167.5-176.5	8
177-185	176.5-185.5	5

Exam tip:

Always write down class boundaries even if not asked, to avoid mistakes in later histogram calculations.

2. Histograms★★☆☆☆⏱ 6 min

Unlike bar charts, the area of each bar in a histogram equals frequency, not the height. This is critical for unequal class widths. The height of each bar (frequency density) is calculated with the standard formula:

\text{Frequency density} = \frac{\text{Frequency}}{\text{Class width}}

📘 Definition

Frequency Density

The height of a histogram bar, scaled so that area of the bar equals the class frequency

📐 Worked Example

A grouped frequency distribution has a class $20 - 30$ with frequency 12, and a class $31 - 50$ with frequency 28. Calculate the height of each bar for a histogram.

1
Find class boundaries and widths for each class:
2
$20-30: \text{ width } = 30.5 - 19.5 = 11 \\ 31-50: \text{ width } = 50.5 - 30.5 = 20$
3
Calculate frequency density (height) for each bar:
4
$\text{Height}_1 = \frac{12}{11} \approx 1.09, \quad \text{Height}_2 = \frac{28}{20} = 1.4$
5
The areas check out: $1.09 \times 11 = 12$ and $1.4 \times 20 = 28$ , matching the class frequencies.

Exam tip:

CIE almost always uses unequal class widths for histogram questions, so always check class width before calculating frequency density.

3. Stem-and-Leaf Plots★☆☆☆☆⏱ 4 min

A stem-and-leaf plot is a simple visual display that retains all original data values, making it easy to find the median and quartiles directly from the plot. Back-to-back stem-and-leaf plots are used to compare two distributions.

📐 Worked Example

Draw an ordered stem-and-leaf plot for the following test scores out of 50: 12, 45, 23, 17, 31, 25, 29, 35, 41, 19, 22, 32.

1
Use the tens digit as the stem, so stems are 1, 2, 3, 4. Sort the units digits (leaves) for each stem in ascending order:
2
Stem Leaves
1 2 7 9
2 2 3 5 9
3 1 2 5
4 1 5
3
Add a key to explain the plot: Key: $1∣2 = 12$ marks. The median can be read directly as the average of the 6th and 7th values: $\frac{25 + 29}{2} = 27$ .

Stem	Leaves
1	2 7 9
2	2 3 5 9
3	1 2 5
4	1 5

Exam tip:

You will always lose an accuracy mark if you forget to add a key to your stem-and-leaf plot.

4. Box-and-Whisker Plots★★☆☆☆⏱ 5 min

A box plot displays the five-number summary of a data set: minimum, lower quartile ( $Q_{1}$ ), median ( $Q_{2}$ ), upper quartile ( $Q_{3}$ ), maximum. Outliers are defined as values less than $Q_{1} - 1.5 \times I QR$ or greater than $Q_{3} + 1.5 \times I QR$ , where $I QR = Q_{3} - Q_{1}$ . Outliers are plotted as separate crosses, and whiskers stop at the last non-outlier value.

📐 Worked Example

A data set has $Q_{1} = 20$ , median $= 28$ , $Q_{3} = 36$ , minimum $= 12$ , maximum $= 52$ . Draw the box plot and identify any outliers.

1
Calculate the interquartile range:
2
$IQR = Q_3 - Q_1 = 36 - 20 = 16$
3
Calculate outlier boundaries:
4
$\text{Lower} = 20 - (1.5 \times 16) = -4 \\ \text{Upper} = 36 + (1.5 \times 16) = 60$
5
All values fall between $- 4$ and $60$ , so there are no outliers. Draw a box from 20 to 36 with a line at 28, and whiskers to 12 and 52.

Exam tip:

When comparing two box plots, always make two comments: one about central tendency (median) and one about spread (IQR/range).

5. Common Pitfalls

Wrong move:

Treating the height of a histogram bar as equal to frequency for unequal class widths

Why:

This leads to incorrect bar areas, so any estimates from the histogram will be wrong

Correct move:

Always calculate height as frequency density = frequency / class width, regardless of class width equality

Wrong move:

Forgetting to add a key to a stem-and-leaf plot

Why:

CIE requires a key to confirm correct data representation, so you will lose an accuracy mark

Correct move:

Always add a simple key like $2∣3 = 23$ after drawing your plot

Wrong move:

Calculating class width from stated class limits instead of boundaries

Why:

For the class 10-19, the difference between 19 and 10 is 9, but the actual width is 10, leading to wrong frequency density

Correct move:

Always calculate width as upper boundary minus lower boundary

Wrong move:

Extending the whisker of a box plot to an outlier value

Why:

This misrepresents the distribution and shows incorrect understanding of outlier display

Correct move:

Plot outliers as separate crosses, and stop the whisker at the last non-outlier value

Wrong move:

Only commenting on one feature when comparing two box plots

Why:

Most comparison questions are worth two marks, one for each feature

Correct move:

Always comment on both the difference in median (centre) and difference in spread (IQR/range)

6. Quick Reference Cheatsheet

Display Type	Key Rule	Required Check
Grouped frequency distribution	Always calculate class boundaries first	No overlapping classes
Histogram	Area = frequency	Height = frequency ÷ class width
Stem-and-leaf plot	Retains all raw data values	Must include a clear key
Box plot	Five-number summary + outliers	Outlier: < Q1 - 1.5×IQR or > Q3 + 1.5×IQR
Comparing box plots	Two marks require two comments	Difference in median, 2. Difference in spread/IQR

7. Frequently Asked

What is the difference between a bar chart and a histogram?

Bar charts are for categorical/discrete data with gaps between bars, and height equals frequency. Histograms are for continuous grouped data, no gaps, and area equals frequency.

How do I calculate class width?

Class width = upper class boundary $-$ lower class boundary. Do not use the difference between stated class limits, which will give the wrong value for whole-number grouped data.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 12
Draw histogram from grouped data
2023 · 13
Compare two box-and-whisker plots
2021 · 11
Construct ordered stem-and-leaf plot

Going deeper

syllabusOfficial CIE 9709 SyllabusCheck latest Unit 4 requirements

What's Next

Data representation is the foundation for all further statistics topics in CIE A-Level, as you need to be able to organise and interpret raw data before you can calculate probabilities, analyse relationships or draw valid statistical conclusions. Mastery of construction rules for histograms, stem-and-leaf plots and box plots is required for both standalone questions and combined questions that ask you to interpret results alongside measures of spread or correlation. This sub-topic is almost always examined for 5-10 marks per paper, so consistent accuracy here translates directly to higher overall grades.