Case Study India meteorological department observes seasonal and annual rainfall every year in different sub-divisions of our country. It helps them to compare and analyse the results. The table given below shows sub-division wise seasonal (monsoon) rainfall (mm) in 2018: Rainfall (mm) Number of Sub-divisions --- --- 200-400 2 400-600 4 600-800 7 800-1000 4 1000-1200 2 1200-1400 3 1400-1600 1 1600-1800 1 Based on the above information, answer the following questions: (I) Write the modal class. (II) Find the median of the given data. (II) Find the mean rainfall in this season. (III) If sub-division having at least 1000 mm rainfall during monsoon season, is considered good rainfall sub-division, then how many sub-divisions had good rainfall?

(I) 600–800; (II)(a) median = 5400/7 771.43 mm; (II)(b) mean = 850 mm; (III) 7 sub-divisions

MathsCBSE Class 10 · Standard · Board exam 2023, set 30/1/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2023: question 37

Set in the CBSE Class 10 board exam of 2023, paper 30/1/1, 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

Case Study

India meteorological department observes seasonal and annual rainfall every year in different sub-divisions of our country. It helps them to compare and analyse the results. The table given below shows sub-division wise seasonal (monsoon) rainfall (mm) in 2018:

Rainfall (mm)	Number of Sub-divisions
200-400	2
400-600	4
600-800	7
800-1000	4
1000-1200	2
1200-1400	3
1400-1600	1
1600-1800	1

Based on the above information, answer the following questions:

(I)
Write the modal class.
1 mark
(II)
Find the median of the given data.
2 marks
(II)
Find the mean rainfall in this season.
2 marks
(III)
If sub-division having at least 1000 mm rainfall during monsoon season, is considered good rainfall sub-division, then how many sub-divisions had good rainfall?
1 mark

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/1/1.

Hints

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

Hint 1

The modal class is the class with the largest frequency.

Hint 2

For the median, build cumulative frequencies and find the class where $N /2$ falls. For the mean, use class marks and $\sum f_{i} x_{i} / \sum f_{i}$ .

Hint 3

'At least 1000 mm' means every class starting from 1000 upward. Add their frequencies.

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 highest frequency is 7, for the class 600–800. Modal class = 600–800.
1 mark
(II)(a) Cumulative frequencies: 200–400: 2; 400–600: 6; 600–800: 13; 800–1000: 17; 1000–1200: 19; 1200–1400: 22; 1400–1600: 23; 1600–1800: 24.
$N = 24$ , so $\frac{N}{2} = 12$ . The first cf that is at least 12 is 13, so the median class is 600–800.
1 mark
(II)(a) Take $l = 600$ , $c f = 6$ , $f = 7$ , $h = 200$ .
Median $= l + (\frac{\frac{N}{2} - c f}{f}) h = 600 + \frac{12 - 6}{7} \times 200 = 600 + \frac{1200}{7} = \frac{5400}{7} \approx 771.43$ mm
1 mark
(II)(b) Class marks $x_{i}$ : 300, 500, 700, 900, 1100, 1300, 1500, 1700.
The products $f_{i} x_{i}$ are 600, 2000, 4900, 3600, 2200, 3900, 1500, 1700.
So $\sum f_{i} = 24$ and $\sum f_{i} x_{i} = 20400$ .
1 mark
(II)(b) Mean $= \frac{\sum f _{i} x _{i}}{\sum f _{i}} = \frac{20400}{24} = 850$ mm
1 mark
(III) 'At least 1000 mm' covers the classes 1000–1200, 1200–1400, 1400–1600 and 1600–1800.
Number of good-rainfall sub-divisions $= 2 + 3 + 1 + 1 = 7$ .
1 mark

Answer: (I) 600–800; (II)(a) median $= \frac{5400}{7} \approx 771.43$ mm; (II)(b) mean $= 850$ mm; (III) 7 sub-divisions

Where the marks are

(I) Modal class 600–800 (1 mark)
(II)(a) Cumulative frequency table and median class 600–800 from $N /2 = 12$ (1 mark)
(II)(a) Correct substitution in the median formula, giving $\approx 771.4$ mm (1 mark)
(II)(b) Correct table of $x_{i}$ and $f_{i} x_{i}$ with $\sum f_{i} x_{i} = 20400$ (1 mark)
(II)(b) Mean $= 850$ mm (1 mark)
(III) Counts $2 + 3 + 1 + 1 = 7$ (1 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 paper2023 board paper, set 30/1/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2023: question 37

Hints

Step-by-step solution

Revise the chapter

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