Cumulative Frequency Diagrams (Copy)
1.
The table shows marks of 40 students. Complete the cumulative frequency column.
| Marks | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|——-|——-|——–|——–|——–|
| Frequency | 4 | 6 | 10 | 12 | 8 |
2.
Plot the cumulative frequency curve for Q1.
3.
From Q1, estimate the median mark.
4.
From Q1, estimate the lower quartile (Q₁).
5.
From Q1, estimate the upper quartile (Q₃).
6.
From Q1, find the interquartile range (IQR).
7.
The times (minutes) taken by 50 students to finish a race are given:
| Time | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|——|——-|——–|——–|——–|
| Frequency | 5 | 8 | 15 | 12 | 10 |
Draw the cumulative frequency table.
8.
Plot the cumulative frequency diagram for Q7.
9.
From Q7, estimate the median time.
10.
From Q7, estimate the interquartile range.
11.
The weights of 60 students are grouped:
| Weight (kg) | 40–50 | 50–60 | 60–70 | 70–80 | 80–90 |
|————-|——–|——–|——–|——–|
| Frequency | 6 | 12 | 18 | 14 | 10 |
Construct the cumulative frequency table.
12.
Plot the cumulative frequency graph for Q11.
13.
From Q11, estimate the median weight.
14.
From Q11, estimate the lower quartile.
15.
From Q11, estimate the upper quartile.
16.
From Q11, calculate the IQR.
17.
The table shows heights of 80 students:
| Height (cm) | 140–150 | 150–160 | 160–170 | 170–180 | 180–190 |
|————-|———-|———-|———-|———-|
| Frequency | 10 | 20 | 25 | 15 | 10 |
Construct the cumulative frequency distribution.
18.
Plot the cumulative frequency curve for Q17.
19.
From Q17, estimate the median height.
20.
From Q17, estimate the 90th percentile.
21.
From Q17, estimate the lower quartile.
22.
From Q17, estimate the upper quartile.
23.
From Q17, calculate the interquartile range.
24.
The ages of 100 workers in a company are recorded:
| Age | 20–29 | 30–39 | 40–49 | 50–59 | 60–69 |
|—–|——–|——–|——–|——–|
| Frequency | 12 | 28 | 30 | 20 | 10 |
Complete the cumulative frequency table.
25.
Plot the cumulative frequency diagram for Q24.
26.
From Q24, estimate the median age.
27.
From Q24, estimate the lower quartile.
28.
From Q24, estimate the upper quartile.
29.
From Q24, calculate the interquartile range.
30.
Explain why cumulative frequency diagrams are more useful than bar charts when comparing two sets of data.
1.
Frequencies: 4, 6, 10, 12, 8.
Cumulative: 4, 10, 20, 32, 40.
2.
Points plotted at class upper bounds: (10,4), (20,10), (30,20), (40,32), (50,40). Smooth curve drawn.
3.
Median position = 40 ÷ 2 = 20th value.
At 20 → boundary ≈ 30. Median ≈ 30 marks.
4.
Q₁ = ¼ × 40 = 10th value.
At 10 → boundary ≈ 20. Q₁ ≈ 20 marks.
5.
Q₃ = ¾ × 40 = 30th value.
At cumulative 30 → boundary ≈ 37–38. Q₃ ≈ 37.5 marks.
6.
IQR = Q₃ − Q₁ = 37.5 − 20 = 17.5 marks.
7.
Frequencies: 5, 8, 15, 12, 10.
Cumulative: 5, 13, 28, 40, 50.
8.
Points plotted: (10,5), (20,13), (30,28), (40,40), (50,50). Smooth curve drawn.
9.
Median = 50 ÷ 2 = 25th value.
From graph, 25th ≈ 28 mins.
10.
Q₁ = 12.5th ≈ 17 mins.
Q₃ = 37.5th ≈ 38 mins.
IQR ≈ 21.
11.
Cumulative: 6, 18, 36, 50, 60.
12.
Points: (50,6), (60,18), (70,36), (80,50), (90,60).
13.
Median = 60 ÷ 2 = 30th value.
From graph → ≈ 67 kg.
14.
Q₁ = 15th → ≈ 58 kg.
15.
Q₃ = 45th → ≈ 76 kg.
16.
IQR = 76 − 58 = 18 kg.
17.
Cumulative: 10, 30, 55, 70, 80.
18.
Points: (150,10), (160,30), (170,55), (180,70), (190,80).
19.
Median = 80 ÷ 2 = 40th value.
From curve → ≈ 165 cm.
20.
90th percentile = 0.9 × 80 = 72nd.
From curve → ≈ 181 cm.
21.
Q₁ = 20th → ≈ 155 cm.
22.
Q₃ = 60th → ≈ 174 cm.
23.
IQR = 174 − 155 = 19 cm.
24.
Cumulative: 12, 40, 70, 90, 100.
25.
Points: (29,12), (39,40), (49,70), (59,90), (69,100).
26.
Median = 100 ÷ 2 = 50th.
From curve → ≈ 44 years.
27.
Q₁ = 25th → ≈ 34 years.
28.
Q₃ = 75th → ≈ 54 years.
29.
IQR = 54 − 34 = 20 years.
30.
Cumulative frequency diagrams allow visual comparison of medians, quartiles, and spread directly, which bar charts cannot. They show distributions, not just frequencies.