The following table shows the ages of the patients admitted in a hospital during a year : Age (in years) 5 – 15 15 – 25 25 – 35 35 – 45 45 – 55 55 – 65 --- --- --- --- --- --- --- Number of patients 6 11 21 23 14 5 Find the mode and mean of the data given above.

MathsCBSE Class 10 · Standard · Board exam 2024, set 30/4/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2024: question 35

Set in the CBSE Class 10 board exam of 2024, 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 answerCompetency-basedStatistics

The following table shows the ages of the patients admitted in a hospital during a year :

Age (in years)	5 – 15	15 – 25	25 – 35	35 – 45	45 – 55	55 – 65
Number of patients	6	11	21	23	14	5

Find the mode and mean of the data given above.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/4/1.

Hints

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

Hint 1

For the mean you need the mid-point of each class; multiply it by the frequency and add the products.

Hint 2

For the mode, first find the class with the largest frequency, then use $l + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h$ .

Hint 3

For the mode, $f_{0}$ and $f_{2}$ are the frequencies of the classes just before and just after the modal class.

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.

Take class marks $x_{i} = \frac{lower limit + upper limit}{2}$ and form the table:
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 $Σ f_{i} = 80$ $Σ f_{i} x_{i} = 2830$
1½ marks
Mean $\overset{x}{ˉ} = \frac{Σ f _{i} x _{i}}{Σ f _{i}} = \frac{2830}{80}$
½ mark
$\overset{x}{ˉ} = 35.375$ years
½ mark
The highest frequency is 23, so the modal class is 35–45. Here $l = 35$ , $f_{1} = 23$ , $f_{0} = 21$ , $f_{2} = 14$ and $h = 10$ .
½ mark
Mode $= l + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \times h = 35 + (\frac{23 - 21}{46 - 21 - 14}) \times 10 = 35 + \frac{20}{11}$
1 mark
Mode $= 35 + 1.82 \approx 36.82$ years. So the mean age is 35.375 years and the modal age is about 36.82 years.
1 mark

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	$Σ f_{i} = 80$		$Σ f_{i} x_{i} = 2830$

Answer: Mean = 35.375 years; Mode ≈ 36.82 years

Where the marks are

Correct table of class marks and $f_{i} x_{i}$ with $Σ f_{i} = 80$ and $Σ f_{i} x_{i} = 2830$ (1½ marks)
Uses the mean formula $Σ f_{i} x_{i} /Σ f_{i}$ (½ mark)
Gets mean = 35.375 (½ mark)
Identifies modal class 35–45 (½ mark)
Substitutes correctly in the mode formula (1 mark)
Gets mode ≈ 36.82 (1 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 paper2024 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 2024: question 35

Hints

Step-by-step solution

Revise the chapter

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