GCSE MATHS • HIGHER TIER
Median and Quartiles from Graphs Made Simple
Learn how to estimate the median, lower quartile, upper quartile, interquartile range and percentiles from cumulative frequency graphs.
Clear method
Exam practice
Worked proofs
Median and Quartiles from Graphs in Four Moves
1
Total
Find the total cumulative frequency.
2
Position
Calculate the required median, quartile or percentile position.
3
Read
Draw across to the curve and then down to the data axis.
4
Interpret
Record the estimated value and use it to describe or compare the data.
What You’ll Learn
Find the Median
Locate the 50% point of a cumulative frequency graph.
Find Quartiles
Use the 25% and 75% positions to estimate Q1 and Q3.
Calculate the IQR
Use Q3 − Q1 to measure the spread of the middle 50%.
Find Percentiles
Convert a percentage into a cumulative frequency position.
Compare Data Sets
Use quartiles and IQR to compare distributions.
Key Rules
Lower Quartile (Q1) = Total Frequency ÷ 4
Median (Q2) = Total Frequency ÷ 2
Upper Quartile (Q3) = 3 × Total Frequency ÷ 4
Interquartile Range (IQR) = Q3 − Q1
Worked Proof: Median and Quartiles from Graphs
Step-by-Step Method
- 1. Find the total cumulative frequency.
- 2. Calculate the required position: 25%, 50%, 75% or another percentile.
- 3. Locate that cumulative frequency on the vertical axis.
- 4. Draw a horizontal line to the cumulative frequency curve.
- 5. Draw a vertical line down to the horizontal axis.
- 6. Read and record the corresponding data value.
More Worked Examples
1
Finding the Lower Quartile
A cumulative frequency graph represents 120 values.
Step 1: Calculate 25% of 120.
120 ÷ 4 = 30.
Step 2: Locate cumulative frequency 30.
Step 3: Read the corresponding value from the graph.
120 ÷ 4 = 30.
Step 2: Locate cumulative frequency 30.
Step 3: Read the corresponding value from the graph.
2
Finding the Upper Quartile
A cumulative frequency graph represents 120 values.
Step 1: Calculate 75% of 120.
120 × 3 ÷ 4 = 90.
Step 2: Locate cumulative frequency 90.
Step 3: Read the corresponding value from the graph.
120 × 3 ÷ 4 = 90.
Step 2: Locate cumulative frequency 90.
Step 3: Read the corresponding value from the graph.
3
Finding the Interquartile Range
Suppose Q1 = 18 and Q3 = 46.
IQR = Q3 − Q1.
46 − 18 = 28.
46 − 18 = 28.
4
Finding a Percentile
A cumulative frequency graph contains 200 values. Find the 90th percentile.
Step 1: Calculate 90% of 200.
0.9 × 200 = 180.
Step 2: Locate cumulative frequency 180.
Step 3: Read the corresponding value from the graph.
0.9 × 200 = 180.
Step 2: Locate cumulative frequency 180.
Step 3: Read the corresponding value from the graph.
- Common GCSE Mistakes
- Median position: Use half the total frequency, not the total.
- Quartile positions: Use 25% for Q1 and 75% for Q3.
- Graph reading: Draw construction lines carefully before reading values.
- Class boundaries: Do not confuse quartiles with class boundaries.
- Exam Tips
- Find the total first: Use the maximum cumulative frequency.
- Show construction lines: Draw clearly across and down from the curve.
- Know the key positions: Q1 = 25%, median = 50%, Q3 = 75%.
- Compare spread with IQR: A smaller IQR means less spread.
- Show working: Write the cumulative frequency position before reading the graph.
Skills Used in Median and Quartiles from Graphs
Cumulative Frequency Graphs
Box Plots
Histograms
Interquartile Range
Statistics
Data Handling
Ready to Practise?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Median and Quartiles from Graphs Tutorial
Step-by-step walkthroughs of Median and Quartiles from Graphs with clear methods and exam tips.
Frequently Asked Questions
Find half the total frequency, locate that value on the cumulative frequency axis, draw across to the curve and then down to the data axis.
The lower quartile is at 25% of the total frequency.
Subtract the lower quartile from the upper quartile: IQR = Q3 − Q1.
A smaller IQR means the middle 50% of the data is less spread out.
Calculate the required percentage of the total frequency, locate it on the cumulative frequency axis and read the corresponding value from the graph.
The arc is the curved boundary of a sector and uses the same central angle.
