To estimate a median from grouped data, build the cumulative frequency, plot each total at the upper boundary of its class, then read across from half of the total frequency to the curve and down to the axis. The same method gives quartiles and the interquartile range.
This skill belongs to data displays and cumulative reasoning. It builds on the class frequencies you found in reading a histogram, and it is one of the most reliable sources of marks once the method is automatic.
What does cumulative frequency actually count?
Cumulative frequency is a running total: how many values are less than or equal to a given boundary. The final cumulative frequency equals the total number of values.
That is why the plotting point for each class is its upper boundary. The value “at most 30 minutes” is only fully counted once you reach 30.
How to estimate a median, step by step
- Add a running total down the frequency column to make the cumulative frequency column.
- Plot points as (upper boundary, cumulative frequency) and join them with a smooth curve or straight segments, as the question asks. Start from the lower boundary of the first class with cumulative frequency 0.
- Find the position: n/2 for the median, n/4 for the lower quartile, 3n/4 for the upper quartile.
- Read across from that height on the vertical axis to the curve, then down to the horizontal axis.
- Answer with the units and say estimate. If you need a number between two plotted points, linear interpolation gives the same idea without a graph.
Worked example
The times, in minutes, that 80 students take to travel to school are grouped as follows.
| Time t (minutes) | Frequency | Upper boundary | Cumulative frequency |
|---|---|---|---|
| 0 < t ≤ 10 | 5 | 10 | 5 |
| 10 < t ≤ 20 | 12 | 20 | 17 |
| 20 < t ≤ 30 | 24 | 30 | 41 |
| 30 < t ≤ 40 | 22 | 40 | 63 |
| 40 < t ≤ 50 | 12 | 50 | 75 |
| 50 < t ≤ 60 | 5 | 60 | 80 |
Step 1, check the total: the last cumulative frequency is 80, which matches 80 students.
Step 2, median position: 80 ÷ 2 = 40. The 40th value lies between the points (20, 17) and (30, 41).
Step 3, interpolate: from 17 to 41 the curve rises by 24. We need to rise from 17 to 40, which is 23, so the fraction of the way across is 23/24 ≈ 0.958. The time is 20 + 10 × 0.958 ≈ 29.6.
Median ≈ 29.6 minutes.
Step 4, quartiles: n/4 = 20, between (20, 17) and (30, 41): 20 + 10 × 3/24 = 21.25. 3n/4 = 60, between (30, 41) and (40, 63): 30 + 10 × 19/22 ≈ 38.64.
Interquartile range ≈ 38.64 − 21.25 ≈ 17.4 minutes.
Step 5, extra question: how many students took more than 35 minutes? At 35, the cumulative frequency is 41 + 22 × 0.5 = 52. So 80 − 52 = 28 students.
The mistake to watch for
The common slip is to plot at the midpoint of each class, or to read the median off the frequency instead of off the cumulative frequency.
Mistaken answer: “The median is 24, because 24 is the largest frequency.”
The student used a frequency value as if it were a position. The median is a time, found by reading across at height 40 on the cumulative frequency axis.
The correction is to ask two questions in order: “What position do I need?” (n/2) and “What value sits at that position?” (read across to the curve, then down). Every answer should be on the horizontal axis.
Check yourself
Try these, then open each answer.
1. A cumulative frequency curve is drawn for 60 values. At which cumulative frequency should you read across to find the median, and the upper quartile?
Show answer
Median: 60 ÷ 2 = 30. Upper quartile: 3 × 60 ÷ 4 = 45.
2. A test is scored out of 100. The cumulative frequencies at 20, 40, 60, 80 and 100 marks are 4, 14, 38, 54 and 60. Estimate the median mark.
Show answer
n = 60, so we need the 30th value. It lies between (40, 14) and (60, 38). The rise needed is 30 − 14 = 16 out of 24, so the mark is 40 + 20 × 16/24 = 40 + 13.33 ≈ 53.3.
3. Using the same data, estimate how many students scored 70 marks or less.
Show answer
70 is halfway between 60 and 80, where the cumulative frequencies are 38 and 54. Halfway gives 38 + 16 × 0.5 = 46. About 46 students scored 70 or less.
Where this leads next
The median from grouped data is often compared between two groups, which is the focus of comparing two distributions with compatible scales. You can revisit the lesson on histograms and frequency density to see how the same grouped data looks in a different display. The non-calculator working trainer is handy for the fraction steps.
Some students can draw a neat curve and still lose marks reading it. A teacher who sees where the reading line starts and stops can fix that quickly, which is part of how we work in online one-to-one Mathematics tuition.