Cumulative Frequency

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Notes

Cumulative Frequency Basics

**Cumulative frequency** is a running total of frequencies as you go through the groups.
In a grouped frequency table, add frequencies sequentially to get cumulative frequencies.
Cumulative frequency can be shown in an extra column or in a separate table with upper bounds.
To find individual frequency from cumulative, subtract previous cumulative from current.
The final cumulative frequency equals the total number of data values.

Drawing Cumulative Frequency Diagrams

Plot cumulative frequency against the **upper bound** of each class interval.
Always include a starting point at the lower bound of the first interval with cumulative frequency 0.
Join the points with a **smooth curve** (not straight line segments).
The curve typically has a stretched 'S' shape and never decreases.
Label axes: horizontal for the variable, vertical for cumulative frequency.

Finding Median from Cumulative Frequency Diagram

Median position = $\frac{n}{2}$ where $n$ is total data values.
Draw a horizontal line from $\frac{n}{2}$ on the cumulative frequency axis to the curve.
From the intersection, draw a vertical line down to the horizontal axis.
Read the value on the horizontal axis as the **median** estimate.

Finding Quartiles and Interquartile Range

Lower quartile (LQ) position = $\frac{n}{4}$ .
Upper quartile (UQ) position = $\frac{3n}{4}$ .
Use the same horizontal-to-vertical method as for median to find LQ and UQ.
**Interquartile range (IQR)** $= UQ - LQ$ .
IQR measures the spread of the middle 50% of data.

Finding Percentiles

The $p$ th percentile position = $\frac{np}{100}$ .
For example, 60th percentile position = $\frac{60n}{100}$ .
Draw horizontal line from that position to the curve, then vertical to axis.
The 25th and 75th percentiles are the same as LQ and UQ; 50th is median.

Estimating Frequencies Above or Below a Value

To find number of data values **less than** a given value: draw vertical line up from that value to curve, then horizontal to cumulative frequency axis, read off.
To find number **greater than** a value: subtract the cumulative frequency from total $n$ .
For percentages, divide the count by $n$ and multiply by 100.

Interpreting Cumulative Frequency Diagrams

Cumulative frequency diagrams are used with **grouped continuous data**.
All estimates (median, quartiles, percentiles) are approximations because raw data is unknown.
The curve assumes data is smoothly distributed within each interval.
Use the diagram to compare distributions (e.g., medians and IQRs).

Cumulative Frequency Table and Graph Construction

Finding Median and Quartiles from Cumulative Frequency Curve

Estimating Number Above a Value

Cumulative Frequency Curve Shape

Practice questions

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1.In a cumulative frequency table, what does the cumulative frequency represent?
Easy
- AThe frequency of a single class interval
- BThe running total of frequencies up to the end of each class interval
- CThe total number of data values in the whole data set
- DThe average of the frequencies
2.When drawing a cumulative frequency diagram, the cumulative frequency is plotted against which value?
Easy
- AThe lower bound of the class interval
- BThe midpoint of the class interval
- CThe upper bound of the class interval
- DThe class width
3.For 80 data values, what is the position of the median on a cumulative frequency diagram?
Easy
- A40
- B80
- C20
- D60
4.The cumulative frequency diagram for 100 items shows that at 50 seconds the cumulative frequency is 70. How many items took more than 50 seconds?
Medium
- A70
- B30
- C50
- D20
5.From a cumulative frequency diagram of 80 items, the lower quartile is found at 12 minutes and the upper quartile at 28 minutes. What is the interquartile range?
Medium
- A16 minutes
- B40 minutes
- C8 minutes
- D20 minutes
6.A cumulative frequency table has entries: $0 < t \le 10: CF=5, 0 < t \le 20: CF=18, 0 < t \le 30: CF=40$ . What is the frequency of the interval $10 < t \le 20$ ?
Medium
- A13
- B18
- C5
- D23
7.For 120 data values, what is the position of the 60th percentile?
Medium
- A60
- B72
- C120
- D6
8.The cumulative frequency diagram for 90 bags of sweets shows that the upper quartile is 55 grams. Approximately how many bags weigh more than 55 grams?
Hard
- A22.5
- B45
- C67.5
- D90

Summary

## Cumulative Frequency (IGCSE Extended) ### What is Cumulative Frequency? - **Cumulative frequency** is the running total of frequencies up to the end of each class interval. - In a table, add frequencies as you go down: e.g., for intervals 0–20 (freq 14), 20–40 (freq 25), cumulative frequencies are 14, 39 (14+25), etc. - Alternatively, a table may show cumulative frequencies directly (e.g., x < 20, x < 40, …). To find individual frequency, subtract: e.g., frequency of 20–40 = cumulative at 40 minus cumulative at 20. ### Drawing a Cumulative Frequency Diagram 1. Complete the cumulative frequency table. 2. Plot points: **x-coordinate = upper bound** of interval, **y-coordinate = cumulative frequency**. 3. Add a starting point at the lowest lower bound with cumulative frequency 0. 4. Join points with a **smooth curve** (S-shape). Never goes back down. ### Interpreting the Diagram - **Median**: value at position n/2 (n = total frequency). Draw horizontal line from n/2 on y-axis to curve, then vertical down to x-axis. - **Lower Quartile (Q1)**: position n/4. - **Upper Quartile (Q3)**: position 3n/4. - **Interquartile Range (IQR)** = Q3 – Q1. - **Percentiles**: pth percentile at position (n × p)/100. E.g., 60th percentile: (60/100)×n. - **Number above a value**: read cumulative frequency at that value, subtract from total. ### Key Points - Always plot cumulative frequency against **upper class boundaries**. - Estimates are approximate because raw data is unknown. - Use the graph to find medians, quartiles, percentiles, and counts above/below thresholds.

Key terms

Cumulative frequency: The running total of frequencies up to the end of each class interval.
Cumulative frequency diagram: A graph plotting cumulative frequency against the upper bound of each class interval, used to estimate median, quartiles, and percentiles.
Median: The middle value; estimated from a cumulative frequency diagram at the value corresponding to n/2 on the cumulative frequency axis.
Lower quartile (Q1): The value below which 25% of the data lies; estimated at n/4 on the cumulative frequency axis.
Upper quartile (Q3): The value below which 75% of the data lies; estimated at 3n/4 on the cumulative frequency axis.
Interquartile range (IQR): The difference between the upper and lower quartiles (Q3 – Q1), measuring the spread of the middle 50% of data.
Percentile: The value below which a given percentage of data falls; the pth percentile is estimated at (n × p)/100 on the cumulative frequency axis.

Slides

What is Cumulative Frequency?

Cumulative frequency is a running total of frequencies as you go through the groups.
It tells you how many data values are less than or equal to a certain value.
For example, if 5 people are in 0-10 mins and 8 in 10-20 mins, the cumulative frequency for 20 mins is $5+8=13$ .

What does cumulative frequency mean?

The frequency of the most common group
The running total of frequencies up to the end of each group
The average of all frequencies
The difference between two frequencies

Creating a Cumulative Frequency Table

Start with a grouped frequency table. Add a column for cumulative frequency.
The first cumulative frequency equals the first frequency.
For each subsequent row, add the current frequency to the previous cumulative total.
Alternatively, you may see tables where each row shows the cumulative frequency for $'x <$ upper bound'.

In a cumulative frequency table, the first cumulative frequency $is ____ the$ first frequency.

Drawing a Cumulative Frequency Diagram

Plot cumulative frequency against the upper bound of each class interval.
Also plot a point at the lower bound of the first interval with cumulative frequency 0.
Join the points with a smooth curve (often S-shaped).
The curve never decreases and always goes from left to right.

When drawing a cumulative frequency diagram, what do you plot on the x-axis?

The lower bound of each interval
The upper bound of each interval
The midpoint of each interval
The frequency

Finding the Median from a Cumulative Frequency Diagram

For n data values, the median is at position n/2 on the cumulative frequency axis.
Draw a horizontal line from n/2 to the curve, then a vertical line down to the x-axis.
Read the value on the x-axis – this is the estimated median.

If there are 80 data values, at what cumulative frequency do you find the median?

Finding Quartiles from a Cumulative Frequency Diagram

Lower quartile (Q1) is at position n/4.
Upper quartile (Q3) is at position 3n/4.
Draw horizontal lines from these positions to the curve, then vertical lines to the x-axis.
Interquartile range $(IQR) = Q3 - Q1$ .

Which of these are true about quartiles?

The lower quartile is at position n/4.
The upper quartile is at position n/2.
The interquartile range is $Q3 - Q1$ .
The median is at position n/4.

Finding Percentiles from a Cumulative Frequency Diagram

The p-th percentile is at position $(n \times p)/100$ on the cumulative frequency axis.
For example, the 60th percentile with $n=100$ is at position 60.
Draw a horizontal line from that position to the curve, then vertical to the x-axis.
The 25th and 75th percentiles are the same as the lower and upper quartiles.

To find the 60th percentile with 100 data values, draw a horizontal line from cumulative frequency ____.

Interpreting Cumulative Frequency Diagrams

You can estimate how many data values are above or below a certain value.
Draw a vertical line from the value on the x-axis to the curve, then horizontal to the y-axis.
The reading on the y-axis is the number of data values less than or equal to that value.
Subtract from the total to find the number greater than that value.

From a cumulative frequency diagram of 80 calls, the curve shows that up to 10 minutes, 60 calls are accounted for. How many calls lasted more than 10 minutes?

Worked Example: Estimating Median and Quartiles

For 100 calls, median is at position 50, lower quartile at 25, upper quartile at 75.
From the diagram, median ≈ 6.2 minutes, Q1 ≈ 4.2 minutes, Q3 ≈ 8.2 minutes.
$IQR = 8.2 - 4.2 = 4.0$ minutes.

For 100 data values, what is the position of the upper quartile?

Key Points to Remember

Cumulative frequency is a running total.
Plot points at upper bounds (and start point).
Join with a smooth curve.
Use n/2 for median, n/4 for Q1, 3n/4 for Q3, and np/100 for percentiles.

Which of these are correct?

Cumulative frequency is plotted against the lower bound.
The median is at position n/2.
The curve should be a smooth S-shape.
The interquartile range is $Q1 - Q3$ .

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