Mathematics — Standard (041) — CBSE Class 10 board question

Cumulative Frequency Table:

| Class | Frequency | Cumulative Frequency (cf) |
|-------|-----------|--------------------------|
| 0–10 | 3 | 3 |
| 10–20 | 5 | 8 |
| 20–30 | 11 | 19 |
| 30–40 | 10 | 29 |
| 40–50 | x | 29 + x |
| 50–60 | 3 | 32 + x |
| 60–70 | 2 | 34 + x |

Total $n = 34 + x$, so $\dfrac{n}{2} = \dfrac{34+x}{2}$

Median = 34.5 lies in class 30–40.
So: $l = 30,\ f = 10,\ cf = 19,\ h = 10$

Using the formula:
$$\text{Median} = l + \left(\frac{\dfrac{n}{2} - cf}{f}\right) \times h$$

$$34.5 = 30 + \left(\frac{\dfrac{34+x}{2} - 19}{10}\right) \times 10$$

$$4.5 = \frac{34+x}{2} - 19$$

$$4.5 + 19 = \frac{34+x}{2}$$

$$23.5 \times 2 = 34 + x$$

$$47 = 34 + x$$

$$\boxed{x = 13}$$

Source: Chapter 13, Section 13.4 Median of Grouped Data

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• First identify the median class: since median = 34.5, it lies in 30–40.
• Express $n$ in terms of $x$ and substitute into the median formula.
• The key step students miss: $cf$ before the median class is 19 (cumulative frequency up to 20–30), and $f = 10$ for the median class 30–40.
• Solve the linear equation carefully — $x = 13$ makes $n = 47$, so $n/2 = 23.5$, which is indeed greater than $cf = 19$ and less than $29$, confirming 30–40 is the correct median class.