Heart Rate: The heart rate is one of the 'vital signs' of health in the human body. It measures the number of times per minute that the heart contracts or beats. While a normal heart rate does not guarantee that a person is free of health problems, it is a useful benchmark for identifying a range of health issues. Thirty women were examined by doctors of AIIMS and the number of heart beats per minute were recorded and summarized as follows: Number of heart beats per minute Number of Women --- --- 65–68 2 68–71 4 71–74 3 74–77 8 77–80 7 80–83 4 83–86 2 Based on the above information, answer the following questions: (i) How many women are having heart beat in the range 68 – 77? (ii) What is the median class of heart beats per minute for these women? (iii) (a) Find the modal value of heart beats per minute for these women. (iii) (b) Find the median value of heart beats per minute for these women.

MathsCBSE Class 10 · Basic · Board exam 2024, set 430/2/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2024: question 37

Set in the CBSE Class 10 board exam of 2024, paper 430/2/1, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedStatistics

Heart Rate: The heart rate is one of the 'vital signs' of health in the human body. It measures the number of times per minute that the heart contracts or beats. While a normal heart rate does not guarantee that a person is free of health problems, it is a useful benchmark for identifying a range of health issues.

Thirty women were examined by doctors of AIIMS and the number of heart beats per minute were recorded and summarized as follows:

Number of heart beats per minute	Number of Women
65–68	2
68–71	4
71–74	3
74–77	8
77–80	7
80–83	4
83–86	2

Based on the above information, answer the following questions:

(i)
How many women are having heart beat in the range 68 – 77?
1 mark
(ii)
What is the median class of heart beats per minute for these women?
1 mark
(iii) (a)
Find the modal value of heart beats per minute for these women.
2 marks
OR
(iii) (b)
Find the median value of heart beats per minute for these women.
2 marks

Figure in the paper: Two ECG-style heart rhythm traces drawn on a grid: each shows repeated sharp spikes (QRS complexes) with smaller bumps before and after, one trace above the other. The graph is illustrative only. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/2/1.

Hints

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

Hint 1

For (i), list which class intervals lie between 68 and 77 and add their frequencies.

Hint 2

For the median class, make a cumulative frequency column and look for the class where the cf first crosses $N /2$ .

Hint 3

Modal class is the one with the highest frequency; use $l + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h$ for mode and $l + \frac{N /2 - c f}{f} \times h$ for median.

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.

(i) The range 68–77 covers the classes 68–71, 71–74 and 74–77.
Number of women $= 4 + 3 + 8 = 15$
1 mark
(ii) $N = 30$ , so $\frac{N}{2} = 15$ . Cumulative frequencies: 2, 6, 9, 17, 24, 28, 30. The first cf greater than 15 is 17, so the median class is 74–77.
1 mark
(iii)(a) Highest frequency is 8, so modal class is 74–77: $l = 74$ , $f_{1} = 8$ , $f_{0} = 3$ , $f_{2} = 7$ , $h = 3$ .
Mode $= l + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \times h = 74 + (\frac{8 - 3}{16 - 3 - 7}) \times 3$
1½ marks
$= 74 + \frac{5}{6} \times 3 = 74 + 2.5 = 76.5$ beats per minute
½ mark
(iii)(b) Cumulative frequency table:
Class f cf
65–68 2 2
68–71 4 6
71–74 3 9
74–77 8 17
77–80 7 24
80–83 4 28
83–86 2 30
$\frac{N}{2} = 15$ , median class 74–77: $l = 74$ , $c f = 9$ , $f = 8$ , $h = 3$ .
½ mark
Median $= l + (\frac{\frac{N}{2} - c f}{f}) \times h = 74 + \frac{15 - 9}{8} \times 3$
1 mark
$= 74 + 2.25 = 76.25$ beats per minute
½ mark

Class	f	cf
65–68	2	2
68–71	4	6
71–74	3	9
74–77	8	17
77–80	7	24
80–83	4	28
83–86	2	30

Answer: (i) 15 women; (ii) 74–77; (iii)(a) Mode = 76.5; (iii)(b) Median = 76.25

Where the marks are

(i) $4 + 3 + 8 = 15$ women (1 mark)
(ii) Median class 74–77 (1 mark)
(iii)(a) Correct values substituted in the mode formula (1½ marks)
(iii)(a) Mode $= 76.5$ (½ mark)
(iii)(b) Correct cumulative frequency table (½ mark)
(iii)(b) Correct substitution in the median formula (1 mark)
(iii)(b) Median $= 76.25$ (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 paper2024 board paper, set 430/2/1 (Basic): 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 Basic 2024: question 37

Hints

Step-by-step solution

Revise the chapter

More questions from Statistics