The Air Quality Index (AQI) is a scale from 0 to 500 that indicates air quality, with higher numbers signifying more pollution and greater health concerns. Mansi collected the daily data of AQI of her city for a month and presented it as given below : AQI Range 1 – 100 101 – 200 201 – 300 301 – 400 401 – 500 --- --- --- --- --- --- Number of Days 3 9 12 4 2 Based on the above, answer the following questions : (i) Convert the data to continuous frequency distribution. (ii) What is the quality of air in most of the days of the month ? (iii) (a) Using table formed in part (i), find mode of the data. (iii) (b) Using table formed in part (i), find median of the data.

MathsCBSE Class 10 · Basic · Board exam 2026, set 430/4/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2026: question 37

Set in the CBSE Class 10 board exam of 2026, paper 430/4/1, 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

The Air Quality Index (AQI) is a scale from 0 to 500 that indicates air quality, with higher numbers signifying more pollution and greater health concerns. Mansi collected the daily data of AQI of her city for a month and presented it as given below :

AQI Range	1 – 100	101 – 200	201 – 300	301 – 400	401 – 500
Number of Days	3	9	12	4	2

Based on the above, answer the following questions :

(i)
Convert the data to continuous frequency distribution.
1 mark
(ii)
What is the quality of air in most of the days of the month ?
1 mark
(iii)
(a) Using table formed in part (i), find mode of the data.
2 marks
OR
(iii)
(b) Using table formed in part (i), find median of the data.
2 marks

Figure in the paper: Table titled 'Central Pollution Control Board's Air Quality Standards': AQI 0-50 Good; 51-100 Satisfactory; 101-200 Moderate; 201-300 Poor; 301-400 Very Poor; 401-500 Severe. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 430/4/1.

Hints

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

Hint 1

Look at the gap between one class's upper limit and the next class's lower limit; split it equally on both sides.

Hint 2

For the mode, use the class with the highest frequency; for the median, find $N /2$ and the class where the cumulative frequency first reaches it.

Hint 3

Use Mode $= l + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h$ and Median $= l + \frac{N /2 - c f}{f} \times h$ with $h = 100$ .

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 gap between consecutive classes is $101 - 100 = 1$ . Subtract $0.5$ from each lower limit and add $0.5$ to each upper limit.
AQI (class) Number of days
0.5 – 100.5 3
100.5 – 200.5 9
200.5 – 300.5 12
300.5 – 400.5 4
400.5 – 500.5 2
1 mark
(ii) The highest number of days (12) falls in the range 201–300. By the CPCB scale this range is 'Poor', so the air quality was Poor on most days.
1 mark
(iii)(a) The modal class is $200.5 - 300.5$ , since its frequency 12 is the highest.
Here $l = 200.5$ , $f_{1} = 12$ , $f_{0} = 9$ , $f_{2} = 4$ and $h = 100$ .
Mode $= l + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \times h = 200.5 + (\frac{12 - 9}{24 - 9 - 4}) \times 100 = 200.5 + \frac{300}{11}$
1½ marks
$= 200.5 + 27.27 = 227.77$ (approx.)
½ mark
(iii)(b) Cumulative frequency table:
Class f cf
0.5 – 100.5 3 3
100.5 – 200.5 9 12
200.5 – 300.5 12 24
300.5 – 400.5 4 28
400.5 – 500.5 2 30
$N = 30$ , so $\frac{N}{2} = 15$ . The cf first exceeds 15 in the class $200.5 - 300.5$ , so this is the median class.
½ mark
Here $l = 200.5$ , $c f = 12$ , $f = 12$ and $h = 100$ .
Median $= l + (\frac{\frac{N}{2} - c f}{f}) \times h = 200.5 + (\frac{15 - 12}{12}) \times 100$
1 mark
$= 200.5 + 25 = 225.5$
½ mark

AQI (class)	Number of days
0.5 – 100.5	3
100.5 – 200.5	9
200.5 – 300.5	12
300.5 – 400.5	4
400.5 – 500.5	2

Class	f	cf
0.5 – 100.5	3	3
100.5 – 200.5	9	12
200.5 – 300.5	12	24
300.5 – 400.5	4	28
400.5 – 500.5	2	30

Answer: (i) Classes 0.5–100.5, 100.5–200.5, 200.5–300.5, 300.5–400.5, 400.5–500.5 with frequencies 3, 9, 12, 4, 2; (ii) Poor; (iii)(a) Mode ≈ 227.77; (iii)(b) Median = 225.5

Where the marks are

Writes the correct continuous class intervals with their frequencies (1 mark)
States that the air quality was Poor on most days (1 mark)
Identifies the modal class and substitutes correctly into the mode formula (1½ marks)
Gets mode ≈ 227.77 (½ mark)
Makes the correct cumulative frequency table and median class (½ mark)
Substitutes correctly into the median formula (1 mark)
Gets median = 225.5 (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 paper2026 board paper, set 430/4/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2026: question 37

Hints

Step-by-step solution

Revise the chapter

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