# Cumulative Frequency & Histograms

> Mathematics · 0580 2025-2027
> Source: https://www.owlsprep.com/study/cie-0580-u9-cumulative-frequency-histograms/

This Extended-tier guide teaches you to build cumulative frequency curves, estimate median/quartiles/IQR, calculate frequency density for unequal class widths, and interpret histograms for CIE IGCSE Maths 0580 Paper 2 and 4 exams.

**Prerequisites:** [Grouped frequency tables for continuous data](https://www.owlsprep.com/study/cie-0580-u9-grouped-frequency-tables/); [Measures of central tendency and spread for grouped data](https://www.owlsprep.com/study/cie-0580-u9-averages-spread-grouped-data/)

## Learning objectives

- Construct cumulative frequency tables and smooth curves from grouped continuous data
- Estimate median, quartiles, percentiles and interquartile range from cumulative frequency curves
- Calculate frequency density for unequal class widths to draw valid histograms
- Interpret histograms and use area proportionality to solve for unknown frequencies

## Constructing Cumulative Frequency Tables & Curves

**Cumulative Frequency** — The running total of frequencies up to the upper boundary of a given class interval, representing the number of data points less than or equal to the boundary value.

*Example:* If classes are $0 \leq t < 10$ (freq=8) and $10 \leq t < 20$ (freq=12), cumulative frequency at 20 is $8+12=20$.

To build a cumulative frequency table, add a column where you sum frequencies sequentially from the first class to each subsequent class. Always use upper class boundaries for cumulative frequency values, as they represent the maximum value included in the running total. Plot cumulative frequency on the y-axis against upper class boundaries on the x-axis, then join points with a smooth S-shaped curve.

**Worked example:** The grouped frequency table shows time taken by 80 students to complete a puzzle. Construct the cumulative frequency table and state the coordinates of the first two points you would plot on the cumulative frequency curve.

| Time (t minutes) | Frequency |
|------------------|-----------|
| $0 \leq t < 5$   | 7         |
| $5 \leq t < 10$  | 18        |
| $10 \leq t < 15$ | 29        |
| $15 \leq t < 20$ | 22        |
| $20 \leq t < 25$ | 4         |

1. Calculate running totals for cumulative frequency for each upper class boundary:
2. $$cf\textrm{ at } 5 = 7$$
3. $$cf\textrm{ at } 10 = 7 + 18 = 25$$
4. $$cf\textrm{ at } 15 = 25 + 29 = 54$$
5. $$cf\textrm{ at } 20 = 54 + 22 = 76$$
6. $$cf\textrm{ at } 25 = 76 + 4 = 80$$
7. Plot points at (upper class boundary, cumulative frequency). The first two points are (0, 0) [anchor point] and (5,7).

> **Exam tip:** Always include the point (lower boundary of the first class, 0) when plotting cumulative frequency curves to anchor the curve at the origin for accurate value reading.

*Calculator:* allowed

## Estimating Statistics from Cumulative Frequency Curves

**Interquartile Range (IQR)** — The difference between the upper quartile ($Q_3$, 75th percentile) and lower quartile ($Q_1$, 25th percentile), representing the spread of the middle 50% of data, and not affected by extreme outlier values.

*Notation:* IQR = Q_3 - Q_1

To estimate statistics from a cumulative frequency curve with total frequency $n$:
- Median: Read the x-value at $y = \frac{n}{2}$
- Lower quartile ($Q_1$): Read the x-value at $y = \frac{n}{4}$
- Upper quartile ($Q_3$): Read the x-value at $y = \frac{3n}{4}$
- k-th percentile: Read the x-value at $y = \frac{k}{100} \times n$

**Worked example:** Use the cumulative frequency curve for the 80 puzzle completion times to estimate the median, lower quartile, upper quartile and IQR.

1. $$n = 80$$
2. Median: $y = 80/2 = 40$. Read x-value: 12 minutes.
3. Lower quartile $Q_1$: $y = 80/4 = 20$. Read x-value: 8.5 minutes.
4. Upper quartile $Q_3$: $y = 3 \times 80 / 4 = 60$. Read x-value: 16 minutes.
5. $$IQR = Q_3 - Q_1 = 16 - 8.5 = 7.5\textrm{ minutes}$$

> **tip**
>
> To estimate how many values are above/below a given threshold, draw a vertical line from the x-value to the curve, then read the cumulative frequency on the y-axis to find the count of values below the threshold.

*Calculator:* allowed

## Calculating Frequency Density & Drawing Histograms

**Frequency Density** — A measure of frequency per unit class width, used for histograms with unequal class widths to ensure the area of each bar is proportional to the frequency of the class interval.

*Notation:* \textrm{Frequency Density} = \frac{\textrm{Frequency}}{\textrm{Class Width}}

Histograms are used for continuous grouped data, with no gaps between bars. For equal class widths you can use frequency as bar height, but for unequal class widths you must use frequency density for bar height to follow the area-proportional-to-frequency rule. Always check if class intervals are equal or unequal before drawing a histogram.

**Worked example:** Calculate the frequency density for each class of the puzzle completion time data to draw a histogram.

| Time (t minutes) | Frequency |
|------------------|-----------|
| $0 \leq t < 5$   | 7         |
| $5 \leq t < 10$  | 18        |
| $10 \leq t < 15$ | 29        |
| $15 \leq t < 20$ | 22        |
| $20 \leq t < 25$ | 4         |

1. Class width for each interval is 5 minutes.
2. Calculate frequency density for each class using $fd = \frac{f}{w}$:
3. $$0-5: 7/5 = 1.4$$
4. $$5-10: 18/5 = 3.6$$
5. $$10-15: 29/5 = 5.8$$
6. $$15-20: 22/5 = 4.4$$
7. $$20-25: 4/5 = 0.8$$

> **Exam tip:** If a histogram has a scaling factor for area, use a known class to find the factor first before calculating unknown frequencies: $\textrm{Frequency} = \textrm{Area} \times \textrm{Scaling Factor}$.

*Calculator:* allowed

## Interpreting Histograms for Exam Questions

Most high-mark histogram questions ask you to find unknown frequencies or total number of data points from a given histogram. Always start by identifying the relationship between area and frequency for a known class, then apply that to all other bars.

**Worked example:** A histogram of exam marks has a class interval $40 \leq m < 50$ (class width 10) with bar height 2.4 frequency density, and frequency 24. Find the frequency of the class $70 \leq m < 85$ (class width 15) with bar height 1.6 frequency density.

1. Confirm area = frequency for this histogram: $2.4 \times 10 = 24$, which matches the given frequency, so no scaling factor is needed.
2. Calculate area of the 70-85 class bar: $1.6 \times 15 = 24$.
3. Frequency of the class is 24.

> **warning**
>
> Never use bar height alone to compare frequencies of classes with unequal widths: always use the area of the bar to compare frequencies.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Plotting cumulative frequency against class midpoints instead of upper class boundaries
  - Why it fails: Cumulative frequency counts all values up to the maximum value of the class (upper boundary), not the midpoint, so midpoint plotting gives incorrect curve values.
  - Correct: Always pair cumulative frequency values with upper class boundaries when plotting curves.
- **Wrong:** Using frequency instead of frequency density for bar height in histograms with unequal class widths
  - Why it fails: This makes wider bars appear to have higher frequency than they actually do, breaking the area-proportional-to-frequency rule.
  - Correct: Calculate frequency density = frequency / class width for all classes before drawing histograms with unequal widths.
- **Wrong:** Leaving gaps between bars in a histogram
  - Why it fails: Histograms represent continuous data with no gaps between class intervals, so gaps incorrectly imply discrete or categorical data.
  - Correct: Draw adjacent bars with no gaps between them for all histograms.
- **Wrong:** Calculating IQR as the difference between upper and lower boundaries of the median class
  - Why it fails: IQR is the difference between Q3 and Q1, estimated from the full cumulative frequency curve, not directly from the median class boundaries.
  - Correct: Read Q1 and Q3 values from the cumulative frequency curve first, then subtract Q1 from Q3 to get IQR.

## Cheatsheet

| Concept | Formula / Rule | Exam Use Case |
| --- | --- | --- |
| Cumulative Frequency Curve | Plot (upper class boundary, cumulative frequency), start at (0,0) | Construct curves to estimate statistics |
| Median from CF Curve | Read x at $y = n/2$ | Estimate typical value of data set |
| IQR from CF Curve | $IQR = Q_3 - Q_1$, $Q_1$ at $n/4$, $Q_3$ at $3n/4$ | Measure spread of middle 50% of data |
| Frequency Density | $fd = \frac{f}{w}$ | Draw histograms with unequal class widths |
| Histogram Frequency | $f = fd \times w$ | Find unknown frequencies from histograms |

## What's next

Now that you have mastered cumulative frequency curves and histograms, you are ready to tackle the 6-8 mark structured statistics questions that appear on almost every CIE IGCSE Maths 0580 Extended Paper 4. These topics are frequently tested together, so practice past paper questions to build speed and accuracy interpreting curves and histograms in context. Next, you can work through mixed statistics practice sets, then move on to probability topics to complete the full statistics and probability unit for your Extended tier exam preparation.

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/cie-0580-u9-cumulative-frequency-histograms/