Question
A frequency distribution table is given below. Find the median marks scored by students.
| Marks | Number of Students |
|---|---|
| 0 – 10 | 5 |
| 10 – 20 | 8 |
| 20 – 30 | 20 |
| 30 – 40 | 15 |
| 40 – 50 | 7 |
| Total | 55 |
Solution — Step by Step
Count all frequencies: .
Now compute . We need to find which class contains the 27.5th observation.
Cumulative frequency (cf) means: how many observations fall at or below the upper boundary of each class.
| Marks | Frequency | Cumulative Frequency |
|---|---|---|
| 0–10 | 5 | 5 |
| 10–20 | 8 | 13 |
| 20–30 | 20 | 33 |
| 30–40 | 15 | 48 |
| 40–50 | 7 | 55 |
We need the class where cumulative frequency first crosses .
After 10–20, cf = 13 (not enough). After 20–30, cf = 33 (crosses 27.5). So 20–30 is the median class.
From the median class 20–30:
- (lower class boundary)
- (frequency of the median class)
- (cumulative frequency of the class before the median class)
- (class width)
Substituting:
The median is 27.25 marks.
Why This Works
In a grouped frequency distribution, we don’t have individual values — we have ranges. The median formula essentially asks: within the median class, how far do we need to go to reach the middle observation?
We’ve already “used up” 13 observations before entering the 20–30 class. We still need more observations. Since there are 20 observations spread evenly across a width of 10, we travel units into the class.
This is linear interpolation — we assume observations are uniformly distributed within each class. It’s an approximation, but a very good one for most real distributions.
Alternative Method — Ogive (Cumulative Frequency Curve)
For CBSE boards, you can also find the median graphically using a cumulative frequency curve (ogive).
Plot upper class boundaries on the x-axis and cumulative frequencies on the y-axis. Draw a smooth curve. From on the y-axis, draw a horizontal line to the ogive, then drop a vertical line to the x-axis — that x-value is your median.
The graphical method is often asked as a 3-mark question in CBSE Class 10 boards (“draw a cumulative frequency curve and find the median”). Plot both “less than” and “more than” ogives — their intersection point also gives the median. This has appeared consistently in CBSE sample papers.
The algebraic and graphical methods should give the same answer (approximately). If they differ significantly, recheck your cumulative frequency column.
Common Mistake
The most common error here is taking as the cumulative frequency of the median class (33) instead of the class before it (13). Students see the median class is 20–30, look up cf = 33, and substitute that. The formula needs the “already accounted for” observations — meaning the cf of everything before you entered the median class. Using cf = 33 gives Median = 20 + [(27.5 − 33)/20] × 10, which produces a negative addition and a wrong answer of 17.25.
A quick sanity check: your median must always lie inside the median class. Here, 27.25 lies between 20 and 30 — we’re good. If your answer falls outside the class boundaries, cf is almost certainly wrong.