Data representation: histograms, box plots, cumulative frequency

IB Mathematics: Applications and Interpretation SL· Unit 4: Statistics and Probability, Topic 2· 15 min read

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

📘 Definition

Histogram

Vertical axis = frequency density, Horizontal axis = class boundaries

A graphical representation of grouped continuous data, where the area of each bar is proportional to the frequency of the class interval.

Example:

For a class 10-20 with frequency 15, frequency density = $15/ (20 - 10) = 1.5$

Unlike bar charts for categorical data, histograms have no gaps between bars because the underlying data is continuous. When class widths are unequal, the height of each bar (frequency density) is not equal to frequency: only the area of the bar represents total frequency.

📐 Worked Example

Calculate frequency density for the grouped mass data of 50 apples, and describe how to draw the histogram. Classes: 50-60 (f=8), 60-80 (f=22), 80-100 (f=20)

1
Calculate class width for each interval:
2
$50\text{-}60: 60 - 50 = 10; \quad 60\text{-}80: 80 - 60 = 20; \quad 80\text{-}100: 100 - 80 = 20$
3
Calculate frequency density = frequency / class width for each interval:
4
$50\text{-}60: 8/10 = 0.8; \quad 60\text{-}80: 22/20 = 1.1; \quad 80\text{-}100: 20/20 = 1.0$
5
Draw axes: mass on the horizontal axis, frequency density on the vertical axis. Draw each bar from the lower to upper class boundary, with height equal to the calculated frequency density.

Exam tip:

IB exam questions almost always use unequal class widths to test your understanding that area equals frequency, not height. Always calculate frequency density first for unequal classes.

2. Box Plots and Five-Number Summaries★★☆☆☆⏱ 5 min

📘 Definition

Box Plot (Box-and-Whisker Plot)

Box spans Q1 to Q3, line at median, whiskers extend to min/max

A visual display of numerical data based on the five-number summary, used to compare distributions and identify skewness and outliers.

Example:

A symmetric distribution will have the median line in the centre of the box, and equal-length whiskers.

Box plots are the most common tool in IB exams for comparing two or more distributions. To get full marks, you must comment on both the centre (measured by the median) and the spread (measured by the interquartile range or range) of the distributions.

📐 Worked Example

Find the five-number summary for the raw data set: $[2, 3, 5, 7, 8, 10, 14, 15, 18, 20, 22]$

1
Confirm the data is sorted in ascending order (it is already sorted here, with $n = 11$ values)
2
Find the median (Q2): position = $\frac{n + 1}{2} = 6^{th}$ term, so median = $10$
3
Find Q1: median of the lower half of data (first 5 terms): $3^{rd}$ term = $5$
4
Find Q3: median of the upper half of data (last 5 terms): $9^{th}$ term = $18$
5
Five-number summary: minimum = $2$ , Q1 = $5$ , median = $10$ , Q3 = $18$ , maximum = $22$

Exam tip:

When comparing two box plots, always make two separate comments: one about average/centre (median) and one about variation/spread (IQR) to earn full marks.

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

📘 Definition

Cumulative Frequency Diagram

Vertical axis = cumulative frequency, Horizontal axis = upper class boundary

A plot of the running total of frequencies against upper class boundaries, used to estimate percentiles, quartiles and medians for grouped data.

To find any percentile, draw a horizontal line from the required cumulative frequency to the curve, then drop a vertical line down to the horizontal axis to read the estimated value. For $n$ total data points, median is at $\frac{n}{2}$ , Q1 at $\frac{n}{4}$ , Q3 at $\frac{3 n}{4}$ .

📐 Worked Example

Estimate the median height for 100 students from the cumulative frequency data: Upper bound (cm): 150 (cf=12), 160 (cf=35), 170 (cf=68), 180 (cf=92), 190 (cf=100)

1
Median for 100 students is the 50th value, so we need the height at cumulative frequency = 50
2
Plot all (upper bound, cumulative frequency) points and join with a smooth curve
3
50 falls between cf=35 (160 cm) and cf=68 (170 cm). Interpolate to estimate:
4
$\text{Estimated median} = 160 + (170-160) \times \frac{50-35}{68-35} \approx 164.5 \text{ cm}$

4. Common Pitfalls

Wrong move:

Treating histogram bar height as frequency for unequal class widths

Why:

Confusion between histograms and bar charts leads to incorrect frequency calculations

Correct move:

Always calculate frequency density = frequency / class width for unequal classes, remember area = frequency

Wrong move:

Calculating quartiles for raw data without sorting first

Why:

Rushing through the question leads to incorrect positions and values for quartiles

Correct move:

Always sort raw data in ascending order before finding any percentiles or the five-number summary

Wrong move:

Plotting cumulative frequency against class midpoints

Why:

Confusion with histogram plotting conventions leads to incorrect percentile estimates

Correct move:

Always plot cumulative frequency against upper class boundaries, as it counts all values up to that boundary

Wrong move:

Only commenting on average when comparing box plots

Why:

Forgetting that comparison questions require comment on both centre and spread for full marks

Correct move:

Always make one comment about centre (median) and one comment about spread (IQR/range)

5. Quick Reference Cheatsheet

Graph Type	Key Features	Common Exam Uses	Quick Tip
Histogram	Area = frequency, frequency density (y-axis), no gaps	Display grouped continuous data	Check for unequal class widths
Box Plot	5-number summary, shows outliers	Compare distributions, identify skewness	Compare median AND IQR
Cumulative Frequency	Plot against upper class boundaries	Estimate quartiles/percentiles	Median = n/2, Q1 = n/4, Q3 = 3n/4

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.

2025 · Paper 1
Interpret box plot and compare distributions
2024 · Paper 2
Estimate quartiles from cumulative frequency
2023 · Paper 1
Construct histogram for unequal classes

Going deeper

syllabusIB Mathematics AI SL SyllabusOfficial unit 4 reference

What's Next

This sub-topic is the foundation of all descriptive statistics in IB AI SL, assessed in both Paper 1 and Paper 2. Mastering these data representations is essential for progressing to bivariate data analysis, correlation, and probability distributions. These skills also frequently appear in extended response questions that combine multiple statistics topics, so building a strong understanding early will help you with more complex topics later.