The following table shows the number of patients of different age group who were discharged from the hospital in a particular month : Age (in years) Number of Patients Discharged --- --- 5-15 6 15-25 11 25-35 21 35-45 23 45-55 14 55-65 5 Total 80 Find the 'mean' and the 'mode' of the above data.

MathsCBSE Class 10 · Standard · Board exam 2025, set 30/4/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2025: question 35

Set in the CBSE Class 10 board exam of 2025, paper 30/4/1, section D, question 35, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35 · 5 marks · Long answerStatistics

The following table shows the number of patients of different age group who were discharged from the hospital in a particular month :

Age (in years)	Number of Patients Discharged
5-15	6
15-25	11
25-35	21
35-45	23
45-55	14
55-65	5
Total	80

Find the 'mean' and the 'mode' of the above data.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/4/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

For the mean of grouped data, replace each class by its mid-point (class mark) and multiply by its frequency.

Hint 2

For the mode, first pick the class with the largest frequency, then note the frequencies of the classes just before and just after it.

Hint 3

Use Mode $= l + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Find the class marks $x_{i} = \frac{lower limit + upper limit}{2}$ and the products $f_{i} x_{i}$ :
Age (years) $f_{i}$ $x_{i}$ $f_{i} x_{i}$
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 $\sum f_{i} = 80$ $\sum f_{i} x_{i} = 2830$
1½ marks
Mean (direct method): $\overset{x}{ˉ} = \frac{\sum f _{i} x _{i}}{\sum f _{i}} = \frac{2830}{80}$
1 mark
$\overset{x}{ˉ} = \frac{283}{8} = 35.375 \approx 35.38$ years
½ mark
The highest frequency is 23, so the modal class is 35-45.
Hence $l = 35$ , $f_{1} = 23$ , $f_{0} = 21$ , $f_{2} = 14$ and $h = 10$ .
½ mark
Mode $= l + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \times h = 35 + \frac{23 - 21}{46 - 21 - 14} \times 10$
1 mark
$= 35 + \frac{2}{11} \times 10 = 35 + \frac{20}{11} = \frac{405}{11} \approx 36.82$ years
½ mark

Age (years)	$f_{i}$	$x_{i}$	$f_{i} x_{i}$
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	$\sum f_{i} = 80$		$\sum f_{i} x_{i} = 2830$

Answer: Mean = 283/8 ≈ 35.38 years; Mode = 405/11 ≈ 36.82 years

Where the marks are

Correct table of class marks $x_{i}$ and $f_{i} x_{i}$ with $\sum f_{i} x_{i} = 2830$ (1½ marks)
Uses the mean formula $\frac{\sum f _{i} x _{i}}{\sum f _{i}} = \frac{2830}{80}$ (1 mark)
Mean ≈ 35.38 years (½ mark)
Identifies the modal class as 35-45 (½ mark)
Correct substitution in the mode formula (1 mark)
Mode ≈ 36.82 years (½ mark)

Revise the chapter

Chapter: Statistics (notes, formulas, where marks are lost and every past question)
Topic: Mean of Grouped Data: Direct Method

Practise this chapterBoard-style questions from Statistics, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 30/4/1 (Standard): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Standard 2025: question 35

Hints

Step-by-step solution

Revise the chapter

More questions from Statistics