Cumulative Frequency

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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.

Cumulative frequency diagram for quiz times

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.

Reading the median from the curve

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.

Reading the quartiles from the curve

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.

Reading a percentile from the curve

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.

Reading a frequency from the curve

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).

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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 ≤ 10: CF=5, 0 < t ≤ 20: CF=18, 0 < t ≤ 30: CF=40. What is the frequency of the interval 10 < t ≤ 20?
Medium
- A13
- B18
- C5
- D23
7.For 120 data values, what is the position of the 60th percentile?
Easy
- 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?
Medium
- A22.5
- B45
- C67.5
- D90

Thêm câu hỏi

9.A cumulative frequency diagram shows that 40 out of 200 students scored less than 60 marks. What percentage of students scored 60 marks or more?
Medium
- A20%
- B80%
- C40%
- D60%
10.In a cumulative frequency diagram, what does the point (25, 0) represent?
Easy
- AThe first data value is 25
- BNo data values are below 25
- CThe cumulative frequency at 25 is 0
- DThe class interval starts at 25
11.From a cumulative frequency diagram of 80 swimmers, the median time is 45 seconds. How many swimmers took less than 45 seconds?
Easy
- A40
- B80
- C20
- D60
12.The cumulative frequency graph for 100 male runners shows that 30 runners took less than 4 hours. What is the estimated probability that a randomly chosen male runner took less than 4 hours?
Medium
- A0.3
- B0.7
- C0.03
- D30
13.What is the first point that should be plotted on a cumulative frequency diagram?
Easy
- A(lowest upper bound, 0)
- B(lowest lower bound, 0)
- C(0, 0)
- D(highest upper bound, total frequency)
14.For 200 data values, the lower quartile is at 15 and the upper quartile at 35. What is the interquartile range?
Medium
- A20
- B50
- C10
- D30
15.The cumulative frequency diagram for 180 children's heights shows that at 106 cm the cumulative frequency is 144. How many children are taller than 106 cm?
Medium
- A36
- B144
- C106
- D180
16.In a cumulative frequency table, the cumulative frequency for the interval 0 < w ≤ 4 is 63 and for 0 < w ≤ 5 is 73. What is the frequency of the interval 4 < w ≤ 5?
Medium
- A10
- B63
- C73
- D136
17.The cumulative frequency diagram for 80 data values shows that at 50 seconds the cumulative frequency is 70. How many items took more than 50 seconds?
Medium
- A10
- B70
- C80
- D50
18.A grouped frequency table shows times (t minutes) of 80 office workers: 0<t≤20:5, 20<t≤40:30, 40<t≤60:20, 60<t≤80:15, 80<t≤100:8, 100<t≤120:2. What is the cumulative frequency for t ≤ 60?
Medium
- A55
- B35
- C20
- D80
19.The cumulative frequency diagram for 100 children shows that at 3 hours the cumulative frequency is 70. How many children spent more than 3 hours exercising?
Medium
- A30
- B70
- C100
- D3
20.A cumulative frequency diagram for 120 plants shows the median height is 35 cm. What is the position of the median?
Easy
- A60
- B30
- C90
- D120
21.The cumulative frequency diagram for 90 bags of sweets shows that 30 bags weigh less than 50 g. The total number of bags is 90. How many bags weigh at least 50 g?
Medium
- A60
- B30
- C90
- D50
22.The table shows the masses of 80 parcels. | Mass (w kg) | Frequency | |---|---| | 0 < w ≤ 1 | 19 | | 1 < w ≤ 2 | 17 | | 2 < w ≤ 3 | 15 | | 3 < w ≤ 4 | 12 | | 4 < w ≤ 5 | 10 | | 5 < w ≤ 6 | 7 | The cumulative frequency for mass ≤ 4 kg is 63. How many parcels have mass between 3 kg and 4 kg?
Medium
- A12
- B15
- C17
- D19
23.The cumulative frequency diagram for 50 students' test scores shows that the 80th percentile is 72 marks. How many students scored 72 marks or less?
Easy
- A40
- B10
- C72
- D50
24.The table shows the weights of 80 parcels. | Weight (w kg) | Frequency | |---|---| | 0 < w ≤ 1 | 19 | | 1 < w ≤ 2 | 17 | | 2 < w ≤ 3 | 15 | | 3 < w ≤ 4 | 12 | | 4 < w ≤ 5 | 10 | | 5 < w ≤ 6 | 7 | Complete the cumulative frequency table. What is the cumulative frequency for 0 < w ≤ 3?
Easy
- A51
- B34
- C19
- D15
25.The cumulative frequency table shows the travel times of 80 office workers. | Time (t minutes) | Cumulative frequency | |---|---| | 0 < t ≤ 20 | 5 | | 0 < t ≤ 40 | 35 | | 0 < t ≤ 60 | 55 | | 0 < t ≤ 80 | 70 | | 0 < t ≤ 100 | 78 | | 0 < t ≤ 120 | 80 | What is the frequency of the interval 40 < t ≤ 60?
Medium
- A20
- B55
- C35
- D25
26.The cumulative frequency diagram for 100 children's exercise times shows the median is 2.5 hours. What is the position of the median on the cumulative frequency axis?
Easy
- A50
- B25
- C75
- D100
27.From the cumulative frequency graph of 80 swimmers, the lower quartile is 55 seconds and the upper quartile is 70 seconds. What is the interquartile range?
Easy
- A15 seconds
- B20 seconds
- C25 seconds
- D10 seconds
28.The cumulative frequency diagram for 120 plants shows that at height 40 cm the cumulative frequency is 72. How many plants have height greater than 40 cm?
Easy
- A48
- B72
- C40
- D120
29.In a cumulative frequency table, what does 'cumulative frequency' mean?
Easy
- AThe frequency of the most common group
- BThe running total of frequencies up to the end of each group
- CThe frequency of the last group only
- DThe average of all frequencies
30.When drawing a cumulative frequency diagram, the cumulative frequency is plotted against the lower bound of each class interval.
Easy
True or false?
31.In a cumulative frequency diagram, the point (25, 0) represents that 25 data items have a value of 0.
Medium
True or false?
32.Match each term with its definition in the context of cumulative frequency.
Medium
- Cumulative frequency
- Median
- Lower quartile
- Upper quartile
- Running total of frequencies
- Value at position n/2
- Value at position n/4
- Value at position 3n/4
33.Arrange the following steps for drawing a cumulative frequency diagram in the correct order.
Medium
- Add a starting point at the lowest bound with cumulative frequency 0
- Plot cumulative frequency against the upper bound of each class interval
- Calculate cumulative frequencies from the frequency table
- Join the points with a smooth curve
34.Which of the following statements about cumulative frequency diagrams are true? (Select all that apply)
Medium
- AThe cumulative frequency is plotted against the lower bound of each interval
- BThe cumulative frequency is plotted against the upper bound of each interval
- CA point at the start (lowest bound, 0) is always included
- DThe points are joined with straight lines
- EThe curve has a stretched-S shape
35.In a cumulative frequency diagram, the median is found at the position n/4 on the cumulative frequency axis.
Easy
True or false?
36.The cumulative frequency diagram for 200 students' test scores shows that at a score of 70, the cumulative frequency is 150. How many students scored above 70?
Easy
- A50
- B70
- C150
- D200
37.For a data set of 120 values, what is the position of the 75th percentile on a cumulative frequency diagram?
Easy
- A30
- B60
- C75
- D90
38.Order the steps to find the upper quartile from a cumulative frequency diagram for n data values.
Medium
- Compute 3n/4
- Draw a vertical line from the curve to the x-axis
- Draw a horizontal line from 3n/4 on the CF axis to the curve
- Read the value on the x-axis
39.For a data set with 80 values, what is the position of the median on a cumulative frequency diagram?
Easy
40.A cumulative frequency diagram for 80 data values shows that the lower quartile is 15 and the upper quartile is 35. What is the interquartile range?
Easy
41.Complete the sentence.
Easy
On a cumulative frequency diagram, the cumulative frequency is plotted against the ____ of each class interval.
42.Explain how to estimate the 60th percentile from a cumulative frequency diagram of 100 data values.
Medium
43.For a data set with 80 values, what is the position of the median on a cumulative frequency diagram?
Easy
- A36
- B44
- C40
- D48
44.A cumulative frequency diagram for 80 data values shows that the lower quartile is 15 and the upper quartile is 35. What is the interquartile range?
Easy
- A18
- B22
- C20
- D24

Ghi chú ôn tập

## 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.

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.

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