Locate a grouped median and modal class without inventing exact values
Find a median class from cumulative counts · Identify a modal class for equal-width classes · Interpret boundary-based range estimates honestly
Section Paper 4 · Historical Extended · calculator
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Grouped frequency tables
- Median and mode
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Find a median class from cumulative counts
- Identify a modal class for equal-width classes
- Interpret boundary-based range estimates honestly
Class information rather than exact observations
For the illustrated ten times, the cumulative counts are 2,7,10. Both the fifth and sixth observations lie in 10≤t<20, so that is the median class. It does not reveal their exact values or exact median. Under a uniform-within-class interpolation model, using cumulative count N/2=5 estimates median 10+(5−2)/5×10=16.
Mode and range have grouping limits
For these equal-width classes the largest frequency is 5, making 10≤t<20 the modal class. The exact most frequent individual time is unknown. For unequal-width histogram classes, compare frequency densities to locate the highest-density class instead of comparing raw frequencies alone.
The outer boundaries span 30−0=30 minutes, a grouped boundary-based range estimate. They are not the observed minimum and maximum; the exact range is unknown and, with these half-open bounds, is less than 30. Do not label 0 and 30 as observed extremes.
Sources for this lesson
Pause and explain
Does the first lower and last upper boundary give the exact observed range?
Worked example
Find the median class and modal class for the illustrated table.
Show the worked solution
- Cumulative counts place positions 5 and 6 in 10≤t<20.
- That equal-width class also has the greatest frequency.
Answer Both are 10≤t<20; exact median and individual mode are unknown.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
In the illustrated equal-width table, identify the modal class and its frequency.
Give me a hint
Find the largest frequency.
Compare my reasoning
- The frequencies are 2,5,3.
- The greatest is 5 in 10≤t<20.
Modal class 10≤t<20; frequency 5.
Look for these in your work
- Report an interval for the class
- Keep the frequency distinct
Use uniform-within-class interpolation to estimate the illustrated median from cumulative count N/2.
Give me a hint
N=10 and the median count 5 lies between cumulative 2 and 7.
Compare my reasoning
- Start at 10 minutes and move (5−2)/5 of the ten-minute class width.
- Estimate=10+6=16.
Estimated median 16 min under the stated model.
Look for these in your work
- Locate the correct class
- State the interpolation assumption
The table has occupied classes from 0≤t<10 to 20≤t<30. Can you say the actual minimum is 0 and maximum 30? State the range information.
Give me a hint
Only interval membership is known.
Compare my reasoning
- The maximum is below 30; the minimum is at least 0.
- The exact range is unknown; 30 minutes is the outer-boundary span, not an observed exact range.
No; exact extrema and range are unknown. Boundary-span estimate: 30 min.
Look for these in your work
- Respect half-open class bounds
- Do not invent individual values
Common mistakes
- A modal class is an interval, not its frequency or midpoint.
- Class boundaries are not automatically observed extrema.
What can you explain now?
Does the first lower and last upper boundary give the exact observed range?
Compare with the explanation
No; it gives a boundary-span estimate.
The actual minimum and maximum inside those classes are not supplied.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Does the first lower and last upper boundary give the exact observed range?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.