MathsCBSE Class 10 · Standard · Sample paper, session 2023-24 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard sample paper 2023-24: question 31

From the official CBSE sample question paper for 2023-24, section C, question 31, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31 · 3 marks · Short answerStatistics

The length of 40 leaves of a plant are measured correct to nearest millimetre, and the data obtained is represented in the following table. Find the mean length of the leaves.

Length [in mm]	Number of leaves
118 – 126	3
127 – 135	5
136 – 144	9
145 – 153	12
154 – 162	5
163 – 171	4
172 – 180	2

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

Find the class mark of each class. Take 149, the class mark of the class with the highest frequency, as the assumed mean.

Hint 2

Use Mean = a + Σfd/Σf. Check that Σfd comes to −81.

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 the assumed mean $a = 149$ and $d_{i} = x_{i} - 149$ . The class marks are the midpoints, for example $(118 + 126) /2 = 122$ .
Length (mm) $f_{i}$ $x_{i}$ $d_{i}$ $f_{i} d_{i}$
118–126 3 122 −27 −81
127–135 5 131 −18 −90
136–144 9 140 −9 −81
145–153 12 149 0 0
154–162 5 158 9 45
163–171 4 167 18 72
172–180 2 176 27 54
Total 40 −81
So $\sum f_{i} = 40$ and $\sum f_{i} d_{i} = - 252 + 171 = - 81$ .
2 marks
$Mean = a + \frac{\sum f _{i} d _{i}}{\sum f _{i}} = 149 + \frac{- 81}{40}$
½ mark
$= 149 - 2.025 = 146.975$
The mean length of the leaves is 146.975 mm.
½ mark

Length (mm)	$f_{i}$	$x_{i}$	$d_{i}$	$f_{i} d_{i}$
118–126	3	122	−27	−81
127–135	5	131	−18	−90
136–144	9	140	−9	−81
145–153	12	149	0	0
154–162	5	158	9	45
163–171	4	167	18	72
172–180	2	176	27	54
Total	40			−81

Answer: Mean length = 146.975 mm

Where the marks are

Correct table of class marks, deviations and f·d, with Σf = 40 and Σfd = −81 (2 marks)
Applies the assumed-mean formula correctly (1/2 mark)
Gets the mean as 146.975 mm (1/2 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 paperSample paper 2023-24 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2023-24: question 31

Hints

Step-by-step solution

Revise the chapter

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