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

Mode:

The class with the highest frequency is 35–45 (f = 23).
So, Modal class = 35–45, l = 35, f₁ = 23, f₀ = 21, f₂ = 14, h = 10.

$$\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h = 35 + \left(\frac{23 - 21}{2(23) - 21 - 14}\right) \times 10$$

$$= 35 + \frac{2}{11} \times 10 = 35 + 1.8 = \textbf{36.8 years}$$

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Mean (Direct Method): Taking class marks $x_i$:

| Age | $f_i$ | $x_i$ | $f_i x_i$ |
|---|---|---|---|
| 5–15 | 6 | 10 | 60 |
| 15–25 | 11 | 20 | 220 |
| 25–35 | 21 | 30 | 630 |
| 35–45 | 23 | 40 | 920 |
| 45–55 | 14 | 50 | 700 |
| 55–65 | 5 | 60 | 300 |
| Total | 80 | | 2830 |

$$\bar{x} = \frac{\Sigma f_i x_i}{\Sigma f_i} = \frac{2830}{80} = \textbf{35.375 years}$$

Interpretation: The mode (36.8 years) is the age group most common among patients. The mean (35.375 years) is the average age of all patients admitted.

Source: Chapter 13, Exercise 13.2, Q.1

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• Modal class = class with highest frequency (23), not just the highest frequency value.
• Apply the standard mode formula; examiners check correct substitution of f₀ (preceding class freq.) and f₂ (succeeding class freq.).
• For mean, the direct method is simplest here since x_i values are manageable. Use class marks (midpoints) as representatives.
• The interpretation line is required by the question ("Compare and interpret") — don't skip it even in a 5-mark answer.
• Σf must equal total patients = 80; verify this before computing mean.