Data representation: histograms, box plots, cumulative frequency

IB Mathematics: Analysis and Approaches SL· Unit 4: Statistics and Probability, Topic 4.2· 10 min read

1. Histograms for Grouped Data★★☆☆☆⏱ 5 min

Histograms are used to visualise grouped continuous (or grouped discrete) data. Unlike bar charts for categorical data, the area of each bar in a histogram is proportional to the frequency of the class, not the height. For bins of equal width, height equals frequency. For unequal bin widths, we calculate frequency density to get the correct bar height.

📘 Definition

Frequency Density

$f d = \frac{f}{w}$

The frequency per unit of class width, which gives the height of a histogram bar. This ensures the area of the bar equals the class frequency $f$ , where $w$ is the class width.

Example:

For a class 0-10 with frequency 15, frequency density = $\frac{15}{10} = 1.5$

📐 Worked Example

A data set of 50 plant heights has these classes: 10-15 (f=8), 15-20 (f=12), 20-30 (f=20), 30-50 (f=10). Calculate the height of each histogram bar.

1
Calculate class width and frequency density for each class:
2
10-15: width = 5, $f d = \frac{8}{5} =$
$1.6$
3
15-20: width = 5, $f d = \frac{12}{5} =$
$2.4$
4
20-30: width = 10, $f d = \frac{20}{10} =$
$2.0$
5
30-50: width = 20, $f d = \frac{10}{20} =$
$0.5$
6
The heights of the bars are the frequency density values calculated above.

Exam tip:

Examiners regularly test unequal width bins. Always check class widths before drawing a histogram.

2. Box Plots and Outliers★★☆☆☆⏱ 4 min

Box plots (box-and-whisker plots) summarise a distribution using its five-number summary, making it easy to identify spread, centre, and outliers. They are also commonly used to compare two or more distributions in exam questions.

📘 Definition

1.5×IQR Outlier Rule

A value is classified as an outlier if it is less than $Q_{1} - 1.5 \times I QR$ or greater than $Q_{3} + 1.5 \times I QR$ , where $I QR = Q_{3} - Q_{1}$ .

📐 Worked Example

A data set has five-number summary: min = 12, Q1 = 20, median = 25, Q3 = 30, max = 48. Identify outliers and describe the box plot.

1
Calculate the interquartile range:
$IQR = 30 - 20 = 10$
2
Calculate outlier bounds:
3
Lower bound = $20 - 1.5 (10) = 5$ , Upper bound = $30 + 1.5 (10) = 45$
4
Check for outliers: 48 > 45, so 48 is an outlier.
5
The box plot will have a box from Q1=20 to Q3=30, a line at median 25, a lower whisker to min=12, an upper whisker to the highest non-outlier 45, and 48 marked as a separate point.

Exam tip:

When comparing two box plots, always comment on both centre (difference in median) and spread (difference in IQR) for full marks.

3. Cumulative Frequency Graphs★★★☆☆⏱ 6 min

A cumulative frequency graph plots the running total (cumulative frequency) of observations against the upper boundary of each class. This graph can be used to estimate the median, quartiles, and any percentile for grouped data, where we cannot calculate these values exactly.

📘 Definition

Cumulative Frequency

The total number of observations with a value less than or equal to the upper boundary of the current class. It ranges from 0 to the total number of observations $n$ .

📐 Worked Example

Test scores out of 50 are grouped as: 0-10 (f=5), 10-20 (f=12), 20-30 (f=18), 30-40 (f=10), 40-50 (f=5). Estimate the median from a cumulative frequency graph.

1
Calculate cumulative frequency for each upper boundary:
2
Upper 10: cf=5, Upper 20: cf=17, Upper 30: cf=35, Upper 40: cf=45, Upper 50: cf=50
3
Total $n = 50$ , so the median is at $\frac{n}{2} = 25$ on the cumulative frequency axis.
4
Plot points $(upper boundary, c f)$ and connect with a smooth curve. Read across from $c f = 25$ to the curve, then down to the x-axis.
5
The estimated median is approximately 24.

Exam tip:

Always plot cumulative frequency at the upper class boundary, not the midpoint of the class. This is a very common exam mistake.

4. Common Pitfalls

Wrong move:

Using frequency as bar height for histograms with unequal bin widths

Why:

The area of each bar must equal frequency, so unadjusted heights for wider bins overrepresent their frequency

Correct move:

Calculate frequency density = frequency / class width, and use this as the bar height

Wrong move:

Extending box plot whiskers to include outliers

Why:

Outliers are extreme values that do not represent the main distribution

Correct move:

Mark outliers as separate points, and draw whiskers only to the nearest non-outlier value

Wrong move:

Plotting cumulative frequency at the midpoint of each class

Why:

Cumulative frequency counts all values below the upper boundary, so midpoint plotting shifts the curve left

Correct move:

Plot each cumulative frequency value at the upper boundary of its class

Wrong move:

Using $(n + 1) /2$ for the median on a cumulative frequency graph

Why:

This formula is for discrete ungrouped data, not estimated medians for grouped data

Correct move:

Use $n /2$ for the median, $n /4$ for Q1, and $3 n /4$ for Q3 for cumulative frequency estimates

5. Quick Reference Cheatsheet

Graph Type	Key Purpose	Core Rule
Histogram	Summarise grouped continuous data	Area = frequency; height = frequency/bin width for unequal bins
Box Plot	Show 5-number summary, find outliers, compare distributions	Outlier = outside $Q_{1} - 1.5 I QR$ to $Q_{3} + 1.5 I QR$
Cumulative Frequency Graph	Estimate median, quartiles and percentiles	Plot cf against upper class boundary; median at $n /2$

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.

2021 · Paper 1
Cumulative frequency percentile calculation
2022 · Paper 2
Histogram with unequal bin widths
2023 · Paper 1
Box plot outlier identification

Going deeper

syllabusIB AA SL Official Syllabus ExtractTopic 4.2 Descriptive statistics

What's Next

Mastering these data representation techniques is the foundation of all descriptive and inferential statistics in IB AA SL. These graphs regularly appear as the first part of multi-part statistical questions on both Paper 1 and Paper 2, and they are used to interpret results for more advanced topics like correlation, regression, and hypothesis testing. Being able to correctly construct and interpret these graphs will help you avoid losing easy method marks in the exam, and prepare you for university-level statistics if you choose to study quantitative subjects.