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Mathematics · 2026 · 5 marks
CBSE 2026 · Region 4 · Set 1 · Q35
(a) The median of the following data is 50 and sum of all frequencies is 90: Find the values of p and q.
The median of the following data is $\displaystyle 50$ and sum of all frequencies is $\displaystyle 90$ :
Find the values of p and q.
| Class : | $\displaystyle 20$-$\displaystyle 30$ | $\displaystyle 30$-$\displaystyle 40$ | $\displaystyle 40$-$\displaystyle 50$ | $\displaystyle 50$-$\displaystyle 60$ | $\displaystyle 60$-$\displaystyle 70$ | $\displaystyle 70$-$\displaystyle 80$ | $\displaystyle 80$-$\displaystyle 90$ |
| Frequency : | p | $\displaystyle 15$ | $\displaystyle 25$ | $\displaystyle 20$ | q | $\displaystyle 8$ | $\displaystyle 10$ |
Find mean and mode of the following distribution :
| Class : | $\displaystyle 0$-$\displaystyle 15$ | $\displaystyle 15$-$\displaystyle 30$ | $\displaystyle 30$-$\displaystyle 45$ | $\displaystyle 45$-$\displaystyle 60$ | $\displaystyle 60$-$\displaystyle 75$ | $\displaystyle 75$-$\displaystyle 90$ | $\displaystyle 90$-$\displaystyle 105$ |
| Frequency : | $\displaystyle 4$ | $\displaystyle 8$ | $\displaystyle 11$ | $\displaystyle 14$ | $\displaystyle 10$ | $\displaystyle 7$ | $\displaystyle 6$ |
Marking-scheme solution
| Class | Frequency | cf |
| $\displaystyle 20$-$\displaystyle 30$ | p | p |
| $\displaystyle 30$-$\displaystyle 40$ | $\displaystyle 15$ | \(\displaystyle \mathrm{p}+15\) |
| $\displaystyle 40$-$\displaystyle 50$ | $\displaystyle 25$ | \(\displaystyle \mathrm{p}+40\) |
| $\displaystyle 50$-$\displaystyle 60$ | $\displaystyle 20$ | \(\displaystyle \mathrm{p}+60\) |
| $\displaystyle 60$-$\displaystyle 70$ | q | \(\displaystyle \mathrm{p}+\mathrm{q}+60\) |
| $\displaystyle 70$-$\displaystyle 80$ | $\displaystyle 8$ | \(\displaystyle \mathrm{p}+\mathrm{q}+68\) |
| $\displaystyle 80$-$\displaystyle 90$ | $\displaystyle 10$ | \(\displaystyle \mathrm{p}+\mathrm{q}+78\) |
Median Class \(\displaystyle =50-60\)
\[\therefore 50=50+\frac{10}{20}\left(\frac{90}{2}-p-40\right)
\]
\[\Longrightarrow p=5
\]
\[\text { As, } p+q+78=90
\]
\[\Rightarrow 5+q=12
\]
\[\therefore q=7
\]
| Class | Frequency | \(\displaystyle \boldsymbol{x}_{\boldsymbol{i}}\) | \(\displaystyle \boldsymbol{u}_{\boldsymbol{i}}=\frac{\boldsymbol{x}_{\boldsymbol{i}}-\mathbf{5 2 . 5}}{\mathbf{1 5}}\) | \(\displaystyle \boldsymbol{f}_{\boldsymbol{i}} \boldsymbol{u}_{\boldsymbol{i}}\) |
| $\displaystyle 0$-$\displaystyle 15$ | $\displaystyle 4$ | $\displaystyle 7.5$ | -$\displaystyle 3$ | -$\displaystyle 12$ |
| $\displaystyle 15$-$\displaystyle 30$ | $\displaystyle 8$ | $\displaystyle 22.5$ | -$\displaystyle 2$ | -$\displaystyle 16$ |
| $\displaystyle 30$-$\displaystyle 45$ | $\displaystyle 11$ | $\displaystyle 37.5$ | -$\displaystyle 1$ | -$\displaystyle 11$ |
| $\displaystyle 45$-$\displaystyle 60$ | $\displaystyle 14$ | $\displaystyle 52.5$ | $\displaystyle 0$ | $\displaystyle 0$ |
| $\displaystyle 60$-$\displaystyle 75$ | $\displaystyle 10$ | $\displaystyle 67.5$ | $\displaystyle 1$ | $\displaystyle 10$ |
| $\displaystyle 75$-$\displaystyle 90$ | $\displaystyle 7$ | $\displaystyle 82.5$ | $\displaystyle 2$ | $\displaystyle 14$ |
| $\displaystyle 90$-$\displaystyle 105$ | $\displaystyle 6$ | $\displaystyle 97.5$ | $\displaystyle 3$ | $\displaystyle 18$ |
| Total | $\displaystyle 60$ | $\displaystyle 3$ |
∴ Mean \(\displaystyle =\bar{x}=52.5+15 \times \frac{3}{60}\)
\[=53.25
\]
Modal Class \(\displaystyle =45-60\)
∴ Mode \(\displaystyle =45+\frac{14-11}{28-11-10} \times 15\)
\[=45+\frac{3 \times 15}{7}
\]
= $\displaystyle 51.4$ (approx.)
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