The following table shows the ages of 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 7 10 21 22 15 5 Find 'mode' and 'median' of the above data.

MathsCBSE Class 10 · Basic · Board exam 2025, set 430/5/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2025: question 35(b)

Set in the CBSE Class 10 board exam of 2025, paper 430/5/1, section D, question 35(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35(b) · 5 marks · Long answerStatistics

The following table shows the ages of 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	7	10	21	22	15	5

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

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/5/1.

Hints

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

Hint 1

For the median you need cumulative frequencies. Find N/2 and locate the class whose cf first reaches it.

Hint 2

Median = l + ((N/2 − cf)/f) × h, where cf is the cumulative frequency of the class before the median class.

Hint 3

For the mode, take the class with the highest frequency. Use l + ((f1 − f0)/(2f1 − f0 − f2)) × 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.

Cumulative frequency table:
Age (years) 5-15 15-25 25-35 35-45 45-55 55-65
$f$ 7 10 21 22 15 5
$c f$ 7 17 38 60 75 80
The total is $N = 80$ .
1 mark
Median: $\frac{N}{2} = 40$ . The first cf to reach 40 is 60, so the median class is $35 - 45$ .
For this class, $l = 35$ , $c f = 38$ , $f = 22$ , $h = 10$ .
$Median = l + \frac{\frac{N}{2} - c f}{f} \times h = 35 + \frac{40 - 38}{22} \times 10$
1½ marks
$= 35 + \frac{20}{22} = 35 + 0.91 = 35.91$ years
½ mark
Mode: the highest frequency is 22, so the modal class is $35 - 45$ .
For this class, $l = 35$ , $f_{1} = 22$ , $f_{0} = 21$ , $f_{2} = 15$ , $h = 10$ .
$Mode = l + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h = 35 + \frac{22 - 21}{44 - 21 - 15} \times 10$
1½ marks
$= 35 + \frac{10}{8} = 35 + 1.25 = 36.25$ years
½ mark

Age (years)	5-15	15-25	25-35	35-45	45-55	55-65
$f$	7	10	21	22	15	5
$c f$	7	17	38	60	75	80

Answer: Median = 35.91 years; Mode = 36.25 years

Where the marks are

Correct cumulative frequency table with N = 80 (1 mark)
Identifies median class 35-45 and substitutes correctly in the median formula (1½ marks)
Median = 35.91 years (½ mark)
Identifies modal class 35-45 and substitutes correctly in the mode formula (1½ marks)
Mode = 36.25 years (½ mark)

Revise the chapter

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

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 430/5/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2025: question 35(b)

Hints

Step-by-step solution

Revise the chapter

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