Cumulative Frequency & Histograms

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

Histograms are for continuous grouped data, have no gaps between bars, and bar area is proportional to frequency. Bar charts are for discrete/categorical data, have gaps between bars, and bar height represents frequency.

Q: How do I find the median from a cumulative frequency curve?

First find the total cumulative frequency . Draw a horizontal line from on the y-axis to the curve, then drop a vertical line to the x-axis: the value where it lands is the estimated median.

Mathematics· E9.6, E9.7· 20 min read

1. Constructing Cumulative Frequency Tables & Curves★★☆☆☆Extended only⏱ 5 min

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📘 Definition

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.

2. Estimating Statistics from Cumulative Frequency Curves★★★☆☆Extended only⏱ 6 min

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📘 Definition

Interquartile Range (IQR)

IQR = Q_3 - Q_1

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.

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 $y = \frac{3 n}{4}$
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
$IQR = Q_3 - Q_1 = 16 - 8.5 = 7.5\textrm{ minutes}$

3. Calculating Frequency Density & Drawing Histograms★★★☆☆Extended only⏱ 5 min

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📘 Definition

Frequency Density

\textrm{Frequency Density} = \frac{\textrm{Frequency}}{\textrm{Class Width}}

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.

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 $f d = \frac{f}{w}$ :
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: $Frequency = Area \times Scaling Factor$ .

4. Interpreting Histograms for Exam Questions★★★★☆Extended only⏱ 4 min

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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.

5. Common Pitfalls

Wrong move:

Plotting cumulative frequency against class midpoints instead of upper class boundaries

Why:

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 move:

Always pair cumulative frequency values with upper class boundaries when plotting curves.

Wrong move:

Using frequency instead of frequency density for bar height in histograms with unequal class widths

Why:

This makes wider bars appear to have higher frequency than they actually do, breaking the area-proportional-to-frequency rule.

Correct move:

Calculate frequency density = frequency / class width for all classes before drawing histograms with unequal widths.

Wrong move:

Leaving gaps between bars in a histogram

Why:

Histograms represent continuous data with no gaps between class intervals, so gaps incorrectly imply discrete or categorical data.

Correct move:

Draw adjacent bars with no gaps between them for all histograms.

Wrong move:

Calculating IQR as the difference between upper and lower boundaries of the median class

Why:

IQR is the difference between Q3 and Q1, estimated from the full cumulative frequency curve, not directly from the median class boundaries.

Correct move:

Read Q1 and Q3 values from the cumulative frequency curve first, then subtract Q1 from Q3 to get IQR.

6. Quick Reference 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	$I QR = Q_{3} - Q_{1}$ , $Q_{1}$ at $n /4$ , $Q_{3}$ at $3 n /4$	Measure spread of middle 50% of data
Frequency Density	$f d = \frac{f}{w}$	Draw histograms with unequal class widths
Histogram Frequency	$f = f d \times w$	Find unknown frequencies from histograms

7. Frequently Asked

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

Histograms are for continuous grouped data, have no gaps between bars, and bar area is proportional to frequency. Bar charts are for discrete/categorical data, have gaps between bars, and bar height represents frequency.

How do I find the median from a cumulative frequency curve?

First find the total cumulative frequency $n$ . Draw a horizontal line from $\frac{n}{2}$ on the y-axis to the curve, then drop a vertical line to the x-axis: the value where it lands is the estimated median.

Going deeper

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.