EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Cumulative frequency
Further statistics · Lesson 2 of 6
Warm-up
Answer each one, then check.
1. What is the median?
The middle value when the data are in order
2. What is the range?
Largest minus smallest
3. Running total of 4, 11, 20?
4, 15, 35
4. What is a grouped frequency table?
A table showing how many values fall in each class
5. Write the class 20 < t ≤ 30 in words.
More than 20 and up to and including 30
Learning Objectives
1. Work out cumulative frequencies from a grouped table.
2. Plot a cumulative frequency graph.
3. Estimate the median, lower quartile and upper quartile.
4. Find the interquartile range and answer questions from the graph.
Cumulative Frequency
Cumulative frequency is a running total of the frequencies. Plot each total against the upper class boundary.
Points are joined with a smooth curve that always rises.
Building the Table
Time taken (t minutes) to complete a puzzle, 60 students.
|
Time |
Frequency |
Cumulative frequency |
|---|---|---|
|
0 < t ≤ 10 |
4 |
4 |
|
10 < t ≤ 20 |
11 |
15 |
|
20 < t ≤ 30 |
20 |
35 |
|
30 < t ≤ 40 |
16 |
51 |
|
40 < t ≤ 50 |
7 |
58 |
|
50 < t ≤ 60 |
2 |
60 |
Reading the Curve
|
Read the quartiles at one quarter, one half and three quarters of the total. |
A cumulative frequency curve with the lower quartile, median and upper quartile marked by dashed lines.
Reading Quartiles
Use the total frequency n.
|
1 Median Go across at n/2, then down to the x-axis |
2 Lower quartile Go across at n/4, then down |
3 Upper quartile Go across at 3n/4, then down |
4 Interquartile range IQR = UQ − LQ |
Median and Quartiles
|
For the 60 students, estimate the median, the quartiles and the interquartile range. |
1. Total
n = 60
2. Median
At 30: about 27.5 minutes
3. Lower quartile
At 15: 20 minutes
4. Upper quartile
At 45: about 36 minutes
5. Interquartile range
36 − 20 = 16 minutes
Answer: Median ≈ 27.5 minutes, IQR ≈ 16 minutes.
How Many More Than...?
|
Use the graph to estimate how many of the 60 students took more than 40 minutes. |
1. Cumulative frequency at 40
51 students took 40 minutes or less
2. Subtract from the total
60 − 51 = 9
Answer: About 9 students took more than 40 minutes.
Common Mistakes
Watch for these.
▸ Wrong boundary. Plot at the upper class boundary, not the middle of the class.
▸ Straight lines. The graph is a smooth curve, not a polygon (unless the question says so).
▸ Reading frequency, not value. Go across from the cumulative frequency, then down to the data value.
▸ Forgetting the start. The curve starts at the lowest boundary with a cumulative frequency of 0.
Key Terms
|
Cumulative frequency A running total of frequencies. |
Upper class boundary The largest value in a class. |
|
Median The middle value; the value at n/2. |
Lower quartile The value one quarter of the way through the data. |
|
Upper quartile The value three quarters of the way through the data. |
Interquartile range Upper quartile minus lower quartile. |
Your Task: Build a Cumulative Frequency Table
12 minutes
|
Heights (h cm) of 40 plants: 0 < h ≤ 10: 3, 10 < h ≤ 20: 9, 20 < h ≤ 30: 16, 30 < h ≤ 40: 8, 40 < h ≤ 50: 4. (a) Complete the cumulative frequency column. (b) At what value of n do you read the median? 1. Add each frequency to the running total. 2. The last total equals the total frequency. |
A good answer shows: (a) 3, 12, 28, 36, 40. (b) 40/2 = 20, which lies in the class 20 < h ≤ 30.
Can I...?
☐ Find cumulative frequencies.
☐ Plot at upper class boundaries.
☐ Draw a smooth curve.
☐ Read the median.
☐ Read the quartiles.
☐ Find the IQR.
☐ Find how many are above or below a value.
☐ Avoid common mistakes.
Summary
✓ Plot cumulative frequency against upper class boundaries.
✓ Median at n/2, LQ at n/4, UQ at 3n/4.
✓ IQR = UQ − LQ.
✓ Values read from a graph are estimates.
|
EXAM FOCUS The cumulative frequency graph shows the times taken by 80 students. Use it to estimate the median and the interquartile range. (4 marks) Show your dashed lines on the graph and write the values down. |