The daily expenditure of 100 families are given below. Calculate f 1 and f 2 , if the mean daily expenditure is ₹ 188 . Expenditure (in ₹ ) 140 - 160 160 - 180 180 - 200 200 - 220 220 - 240 No. of families 5 25 f 1 f 2 5

The daily expenditures of 100 families are given in the form of a table For finding the mean of daily expenditures, let us follow the direct method. So, we have Expenditures (in ₹ ) Frequencies f i Class Mark x i f i x i 140 - 160 5 150 750 160 - 180 25 170 4250 180 - 200 f 1 190 190 f 1 200 - 220 f 2 210 210 f 2 220 - 240 5 230 1150 Total 35 + f 1 + f 2 6150 + 190 f 1 + 210 f 2 It is given that mean of daily expenditures = ₹ 188 ⇒ 6150 + 190 f 1 + 210 f 2 35 + f 1 + f 2 = 188 ⇒ 6150 + 190 f 1 + 210 f 2 = 188 35 + f 1 + f 2 ⇒ 6150 + 190 f 1 + 210 f 2 = 6580 + 188 f 1 + 188 f 2 ⇒ 190 f 1 - 188 f 1 + 210 f 2 - 188 f 2 = 6580 - 6150 ⇒ 2 f 1 + 22 f 2 = 430 ⇒ f 1 + 11 f 2 = 215 . . . . i Also, according to the given question, we have ∑ f i = 100 ⇒ 35 + f 1 + f 2 = 100 ⇒ f 1 + f 2 = 65 . . . . i i Subtracting equation i i from equation i , we have 10 f 2 = 215 - 65 = 150 ⇒ f 2 = 15 Substituting the value of f 2 in equation i i , we have f 1 = 65 - 15 = 50 Hence, f 1 = 50 and f 2 = 15

Calculate the mean, the median and the mode of the following distribution: Age in years 12 13 14 15 16 17 18 No. of students 2 3 5 6 4 3 2

Age in years ( x i ) d i = x i - a ( a = 15 ) No. of students ( f i ) f i d i 12 - 3 2 - 6 13 - 2 3 - 6 14 - 1 5 - 5 15 0 6 0 16 1 4 4 17 2 3 6 18 3 2 6 Mean age = a + ∑ f i d i ∑ f i = 15 + - 1 25 = 15 - 0 . 04 = 14 . 96 years Age in years No. of students Cumulative frequency 12 2 2 13 3 5 14 5 10 15 6 16 16 4 20 17 3 23 18 2 25 Here, n (number of students) = 25 , which is odd. ∴ Median age = n + 1 2 th observation = 13 th observation = 15 years ( ∵ the observations from 11 th to 16 th are equal, = 15 each) Now, ∵ The age with the value 15 has more frequency than any other age value in the given data, ∴ Modal age = 15 years

The daily wages of 30 employees in an establishment are distributed as follows: Daily wages (in ₹ ) 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 No. of employees 1 8 10 5 4 2 Estimate the modal daily wages for this distribution by a graphical method.

Steps of construction: ( i ) Take the scale along x - axis as 2 cm = 10 units , and along y - axis as 2 cm = 2 units . ( i i ) Construct the rectangles according to the data given in the question. The required histrogram is as shown in the figure above. ( i i i ) In the highest rectangle, draw two straight lines AB and CD from the corners of the rectangles on either side of the highest rectangle to the opposite corners of the highest rectangle. ( i v ) Let M be the point of intersection of the lines AB and CD . ( v ) Through the point M , draw a vertical line that meets the x - axis at point L . The abscissa of the point L represents the value 23 . ∴ The required modal wages = ₹ 23

Draw a cumulative frequency curve for the following data: Marks obtained 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 No. of students 8 10 22 40 20 Hence, determine the median.

The cumulative frequency distribution for the given data is as follows: Marks obtained No. of students Cumulative frequency 0 - 10 8 8 10 - 20 10 18 20 - 30 22 40 30 - 40 40 80 40 - 50 20 100 To draw the cumulative frequency curve: Plot the points ( 10 , 8 ) , ( 20 , 18 ) , ( 30 , 40 ) , ( 40 , 80 ) , ( 50 , 100 ) and ( 0 , 0 ) . The cumulative frequency curve for the given data is as follows: Here, n (total number of students) = 100 , which is even. To find the Median: ( i ) Let A be the point on y - axis representing frequency = n 2 = 50 ( i i ) Through A , we draw a horizontal line to meet the cumulative frequency curve at P . ( i i i ) Through P , draw a vertical line to meet the x - axis at Q . The abscissa of the point Q represents 32 . 5 marks . ∴ The required median marks = 32 . 5 .

Draw a cumulative frequency curve for the following data: Marks obtained 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 No. of students 8 10 22 40 20 Hence, determine the pass marks, if 85 % of the students pass.

The cumulative frequency distribution for the given data is as follows: Marks obtained No. of students Cumulative frequency 0 - 10 8 8 10 - 20 10 18 20 - 30 22 40 30 - 40 40 80 40 - 50 20 100 To draw the cumulative frequency curve: Plot the points ( 10 , 8 ) , ( 20 , 18 ) , ( 30 , 40 ) , ( 40 , 80 ) , ( 50 , 100 ) and ( 0 , 0 ) . The cumulative frequency curve for the given data is as follows: Here, n (total number of students) = 100 , which is even. To find the pass marks: ( i ) Let D be the point on y - axis representing frequency = 15 % of 100 = 15 ( i i ) Through D , we draw a horizontal line to meet the cumulative frequency curve at E . ( i i i ) Through E , draw a vertical line to meet the x - axis at F . The abscissa of the point F represents 18 marks . ∴ The required pass marks = 18 marks .

Draw a cumulative frequency curve for the following data: Marks obtained 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 No. of students 8 10 22 40 20 Hence, determine the marks which 45 % of the students exceed.

The cumulative frequency distribution for the given data is as follows: Marks obtained No. of students Cumulative frequency 0 - 10 8 8 10 - 20 10 18 20 - 30 22 40 30 - 40 40 80 40 - 50 20 100 To draw the cumulative frequency curve: Plot the points ( 10 , 8 ) , ( 20 , 18 ) , ( 30 , 40 ) , ( 40 , 80 ) , ( 50 , 100 ) and ( 0 , 0 ) . The cumulative frequency curve for the given data is as follows: Here, n (total number of students) = 100 . ∵ 45 % of total students exceed a certain value of marks, it implies that ( 100 - 45 ) % = 55 % of students are either below that value, or equals to that value of marks. To find the required marks: ( i ) Let B be the point on y - axis representing frequency = 55 % of 100 = 55 ( i i ) Through B , we draw a horizontal line to meet the cumulative frequency curve at M . ( i i i ) Through M , draw a vertical line to meet the x - axis at N . The abscissa of the point N represents 34 marks . ∴ The required marks = 34 marks .

M L Aggarwal Solutions for Exercise 8: Chapter Test

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Exercise 8: Chapter Test

Attempt the practice questions from Exercise 8: Chapter Test with hints and solutions to strengthen your understanding. Understanding ICSE Mathematics Class 10 solutions are prepared by Experienced Embibe Experts.

Questions from M L Aggarwal Solutions for Exercise 8: Chapter Test with Hints & Solutions

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Chapter Test>Q 2

The average height of $30$ students is $150 cm$ . It was detected later that one value of $165 cm$ was wrongly copied as $135 cm$ for the computation of mean. Find the correct mean.

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Chapter Test>Q 8

The daily expenditure of $100$ families are given below. Calculate $f_{1}$ and $f_{2}$ , if the mean daily expenditure is $₹ 188$ .

$\begin{array}{lccccc} Expenditure (in ₹) & 140 - 160 & 160 - 180 & 180 - 200 & 200 - 220 & 220 - 240 \\ No. of families & 5 & 25 & f_{1} & f_{2} & 5 \end{array}$

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Chapter Test>Q 11

The marks scored by $16$ students in a class test are: $3, 6, 8, 13, 15, 5, 21, 23, 17, 10, 9, 1, 20, 21, 18, 12$ . Find: inter quartile range.

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Chapter Test>Q 12

Calculate the mean, the median and the mode of the following distribution:

Age in years	$12$	$13$	$14$	$15$	$16$	$17$	$18$
No. of students	$2$	$3$	$5$	$6$	$4$	$3$	$2$

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Chapter Test>Q 13

The daily wages of $30$ employees in an establishment are distributed as follows:

Daily wages (in $₹$ )	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$
No. of employees	$1$	$8$	$10$	$5$	$4$	$2$

Estimate the modal daily wages for this distribution by a graphical method.

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Chapter Test>Q 14

Draw a cumulative frequency curve for the following data:

Marks obtained	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$
No. of students	$8$	$10$	$22$	$40$	$20$

Hence, determine the median.

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Chapter Test>Q 14

Draw a cumulative frequency curve for the following data:

Marks obtained	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$
No. of students	$8$	$10$	$22$	$40$	$20$

Hence, determine the pass marks, if $85 %$ of the students pass.

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Chapter Test>Q 14

Draw a cumulative frequency curve for the following data:

Marks obtained	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$
No. of students	$8$	$10$	$22$	$40$	$20$

Hence, determine the marks which $45 %$ of the students exceed.

M L Aggarwal Solutions for Exercise 8: Chapter Test

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Exercise 8: Chapter Test

Questions from M L Aggarwal Solutions for Exercise 8: Chapter Test with Hints & Solutions

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