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Mathematics · 2026 · 5 marks
CBSE 2026 · Region 4 · Set 2 · Q33
Find mean and mode of the following frequency distribution : Class : $\displaystyle 5$-$\displaystyle 15$ $\displaystyle 15$-$\displaystyle 25$ $\displaystyle 25$-$\displaystyle 35$ $\displaystyle 35$-$\displaystyle 45$ $\displaystyle 45$-$\displaystyle 55$ $\displaystyle 55$-$\displaystyle 65$ Frequency : $\displaystyle 11$ $\displaystyle 20$ $\displaystyle 25$ $\displaystyle 22$ $\displaystyle 12$ $\displaystyle 10$
The median of the following data is $\displaystyle 32.5$, find the missing frequencies $\displaystyle x$ and y : Class : $\displaystyle 0$-$\displaystyle 10$ $\displaystyle 10$-$\displaystyle 20$ $\displaystyle 20$-$\displaystyle 30$ $\displaystyle 30$-$\displaystyle 40$ $\displaystyle 40$-$\displaystyle 50$ $\displaystyle 50$-$\displaystyle 60$ $\displaystyle 60$-$\displaystyle 70$ Total Frequency : $\displaystyle x$ $\displaystyle 5$ $\displaystyle 9$ $\displaystyle 12$ y $\displaystyle 3$ $\displaystyle 2$ $\displaystyle 40$
Find mean and mode of the following frequency distribution :
| Class : | $\displaystyle 5$-$\displaystyle 15$ | $\displaystyle 15$-$\displaystyle 25$ | $\displaystyle 25$-$\displaystyle 35$ | $\displaystyle 35$-$\displaystyle 45$ | $\displaystyle 45$-$\displaystyle 55$ | $\displaystyle 55$-$\displaystyle 65$ |
| Frequency : | $\displaystyle 11$ | $\displaystyle 20$ | $\displaystyle 25$ | $\displaystyle 22$ | $\displaystyle 12$ | $\displaystyle 10$ |
The median of the following data is $\displaystyle 32.5$, find the missing frequencies $\displaystyle x$ and y :
| Class : | $\displaystyle 0$-$\displaystyle 10$ | $\displaystyle 10$-$\displaystyle 20$ | $\displaystyle 20$-$\displaystyle 30$ | $\displaystyle 30$-$\displaystyle 40$ | $\displaystyle 40$-$\displaystyle 50$ | $\displaystyle 50$-$\displaystyle 60$ | $\displaystyle 60$-$\displaystyle 70$ | Total |
| Frequency : | $\displaystyle x$ | $\displaystyle 5$ | $\displaystyle 9$ | $\displaystyle 12$ | y | $\displaystyle 3$ | $\displaystyle 2$ | $\displaystyle 40$ |
Marking-scheme solution
| Class | Frequency | \(\displaystyle \boldsymbol{x}_{\boldsymbol{i}}\) | \(\displaystyle \boldsymbol{u}_{\boldsymbol{i}}=\frac{\boldsymbol{x}_{\boldsymbol{i}}-30}{\mathbf{1 0}}\) | \(\displaystyle \boldsymbol{f}_{\boldsymbol{i}} \boldsymbol{u}_{\boldsymbol{i}}\) |
| $\displaystyle 5$-$\displaystyle 15$ | $\displaystyle 11$ | $\displaystyle 10$ | -$\displaystyle 2$ | -$\displaystyle 22$ |
| $\displaystyle 15$-$\displaystyle 25$ | $\displaystyle 20$ | $\displaystyle 20$ | -$\displaystyle 1$ | -$\displaystyle 20$ |
| $\displaystyle 25$-$\displaystyle 35$ | $\displaystyle 25$ | $\displaystyle 30$ | $\displaystyle 0$ | $\displaystyle 0$ |
| $\displaystyle 35$-$\displaystyle 45$ | $\displaystyle 22$ | $\displaystyle 40$ | $\displaystyle 1$ | $\displaystyle 22$ |
| $\displaystyle 45$-$\displaystyle 55$ | $\displaystyle 12$ | $\displaystyle 50$ | $\displaystyle 2$ | $\displaystyle 24$ |
| $\displaystyle 55$-$\displaystyle 65$ | $\displaystyle 10$ | $\displaystyle 60$ | $\displaystyle 3$ | $\displaystyle 30$ |
| Total | $\displaystyle 100$ | $\displaystyle 34$ |
\[\begin{aligned}
\therefore \text { Mean } & =\bar{x}=30+10 \times \frac{34}{100} \\
= & 33.4
\end{aligned}
\]
Modal Class \(\displaystyle =25-35\)
\[\begin{aligned}
\therefore \text { Mode } & =25+\frac{25-20}{2 \times 25-20-22} \times 10 \\
& =25+\frac{50}{8} \\
& =31.25
\end{aligned}
\]
| Class | Frequency | cf |
| $\displaystyle 0$-$\displaystyle 10$ | \(\displaystyle x\) | \(\displaystyle x\) |
| $\displaystyle 10$-$\displaystyle 20$ | $\displaystyle 5$ | \(\displaystyle x+5\) |
| $\displaystyle 20$-$\displaystyle 30$ | $\displaystyle 9$ | \(\displaystyle x+14\) |
| $\displaystyle 30$-$\displaystyle 40$ | $\displaystyle 12$ | \(\displaystyle x+26\) |
| $\displaystyle 40$-$\displaystyle 50$ | \(\displaystyle y\) | \(\displaystyle x+y+26\) |
| $\displaystyle 50$-$\displaystyle 60$ | $\displaystyle 3$ | \(\displaystyle x+y+29\) |
| $\displaystyle 60$-$\displaystyle 70$ | $\displaystyle 2$ | \(\displaystyle x+y+31\) |
| Total | $\displaystyle 40$ |
Median Class \(\displaystyle =30-40\)
\[\therefore 32.5=30+\frac{10}{12}\left(\frac{40}{2}-(x+14)\right)
\]
\(\displaystyle \Rightarrow x=3\)
\[x+y+31=40
\]
\(\displaystyle \Rightarrow 3+y=9\)
\[\therefore y=9-3=6
\]
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