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

Given: Mean = 35, $\Sigma f_i = 25$

Step 1: From the sum of frequencies:

$$1 + x + 5 + 7 + y + 3 + 1 = 25$$
$$17 + x + y = 25$$
$$\boxed{x + y = 8} \quad \text{...(i)}$$

Step 2: Find class marks and compute $\Sigma f_i x_i$:

| Class | $f_i$ | $x_i$ | $f_i x_i$ |
|-------|--------|--------|------------|
| 0–10 | 1 | 5 | 5 |
| 10–20 | $x$ | 15 | $15x$ |
| 20–30 | 5 | 25 | 125 |
| 30–40 | 7 | 35 | 245 |
| 40–50 | $y$ | 45 | $45y$ |
| 50–60 | 3 | 55 | 165 |
| 60–70 | 1 | 65 | 65 |

$$\Sigma f_i x_i = 605 + 15x + 45y$$

Step 3: Apply mean formula:

$$\bar{x} = \frac{\Sigma f_i x_i}{\Sigma f_i} \Rightarrow 35 = \frac{605 + 15x + 45y}{25}$$

$$875 = 605 + 15x + 45y$$
$$15x + 45y = 270$$
$$\boxed{x + 3y = 18} \quad \text{...(ii)}$$

Step 4: Solve equations (i) and (ii):

Subtracting (i) from (ii):
$$2y = 10 \Rightarrow y = 5$$
$$x = 8 - 5 = 3$$

$$\therefore x = 3, \quad y = 5$$

Source: Statistics, Section 13.2 Mean of Grouped Data, Chapter 13

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• Two equations, two unknowns — examiners expect you to set up both equations clearly: one from $\Sigma f_i = 25$ and one from the mean formula.
• Always use class marks (midpoints) as $x_i$ for grouped data mean.
• Show the table of $f_i x_i$ neatly — it earns method marks even if arithmetic slips.
• Label your equations (i) and (ii) and show the subtraction step clearly.
• Final answer must state both values of $x$ and $y$ explicitly.