Find Q 1 , Q 3 , D 6 and P 45 for the following data: x i 18 19 20 21 22 23 24 25 26 27 f i 15 18 25 27 40 25 19 16 8 7

We construct the following table : x i 18 19 20 21 22 23 24 25 26 27 f i 15 18 25 27 40 25 19 16 8 7 Cumulative frequency 15 33 58 85 125 150 169 185 193 200 Here, n = ∑ f i = 200 Q 1 = n + 1 4 th value = 200 + 1 4 th value = 50 . 25 th value = 50 th value + 0 . 25 51 th - 50 th value = 20 + 0 . 25 20 - 20 = 20 Q 3 = 3 n + 1 4 th value = 3 200 + 1 4 th value = 150 . 75 th value = 120 th value + 0 . 75 151 th - 150 th value = 23 + 0 . 75 24 - 23 = 23 . 75 D 6 = 6 n + 1 10 th value = 6 200 + 1 10 th value = 120 . 6 th value = 120 th value + 0 . 6 121 th - 120 th value = 22 + 0 . 6 22 - 22 = 22 P 45 = 45 n + 1 100 th value = 45 200 + 1 100 th value = 90 . 45 th value = 90 th value + 0 . 45 91 th - 90 th value = 22 + 0 . 45 22 - 22 = 22 Hence, the partition values of Q 1 , Q 3 , D 6 and P 45 are 20 , 23 . 75 , 22 and 22 .

For the following data, compute lower and upper quartiles, P 90 and P 5 . Marks Below 10 10 - 20 20 - 40 40 - 60 60 - 80 Above 80 Number of students 8 10 22 25 10 5

We construct the following table : Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 Number of students 8 10 11 11 12 . 5 12 . 5 5 5 5 Cumulative frequency 8 18 29 40 52 . 5 65 70 75 80 Here, n = ∑ f i = 80 Lower quartile is the n 4 th value i.e 20 th value which lies in the class 20 - 30 . ∴ l = 20 ; c = 18 ; f = 11 ; i = 10 Thus, Q 1 = l + n 4 - c f × i = 20 + 20 - 18 11 × 10 = 20 + 20 11 ⇒ Q 1 = 21 . 82 Upper quartile Q 3 is the 3 n 4 th value i.e 60 th value which lies in the class 50 - 60 . ∴ l = 50 ; c = 52 . 5 ; f = 12 . 5 ; i = 10 Thus, Q 3 = l + 3 n 4 - c f × i = 50 + 60 - 52 . 5 12 . 5 × 10 = 20 + 7 . 5 × 10 12 . 5 = 56 P 90 is the 90 n 100 th value i.e 90 × 80 100 th value i.e 72 th value which lies in the class 70 - 80 . ∴ l = 70 ; c = 70 ; f = 5 ; i = 10 Thus, P 90 = l + 90 n 100 - c f × i = 70 + 72 - 70 5 × 10 = 74 P 5 is the 5 n 100 th value i.e 5 × 80 100 th value i.e 4 th value which lies in the class 0 - 10 . ∴ l = 0 ; c = 0 ; f = 8 ; i = 10 Thus, P 5 = l + 5 n 100 - c f × i = 0 + 4 - 0 5 × 10 = 8

Calculate the interquartile range for the following data : Daily wages (in ₹) 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 No. of workers 22 38 46 35 20

We construct the table is as under : Daily wages (in ₹) 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 No. of workers 22 38 46 35 20 Cumulative frequency 22 60 106 141 161 Here, n = ∑ f i = 161 (odd) Lower quartile Q 1 is the n 4 th value i.e 161 4 th = 40 . 25 th value which lies in the class 40 - 50 . ∴ l = 40 ; c = 38 ; f = 22 ; i = 10 Thus, Q 1 = l + n 4 - c f × i = 40 + 40 . 25 - 22 38 × 10 = 40 + 4 . 8026 = 44 . 8026 Upper quartile Q 3 is the 3 n 4 th value i.e 3 × 161 4 th = 120 . 75 th value which lies in the class 60 - 70 . ∴ l = 60 ; c = 35 ; f = 106 ; i = 10 Thus, Q 3 = l + 3 n 4 - c f × i = 60 + 120 . 75 - 106 35 × 10 = 60 + 4 . 2143 = 64 . 2143 Hence, interquartile range = Q 3 - Q 1 = 64 . 2143 - 44 . 8026 = 19 . 41

Calculate all three quartiles for the following frequency distribution : Height ( in inches) 58 59 60 61 62 63 64 65 66 No. of students 15 20 32 35 35 22 20 10 8

We construct the cumulative frequency distribution table is as under : Height ( in inches) 58 59 60 61 62 63 64 65 66 No. of students 15 20 32 35 35 22 20 10 8 Cumulative frequency 15 35 67 102 137 159 179 189 197 ∴ Q 1 is the n + 1 4 th value i.e 197 + 1 4 th = 49 . 5 th value Thus, Q 1 = 60 [ since 49 th and 50 th values are same i.e 60 . ] Q 2 be the median = 2 n + 1 4 th value = 197 + 1 2 th value = 99 th value = 61 And Q 3 = 3 n + 1 4 th value = 3 197 + 1 4 th value 148 . 5 th = 63 . Hence, all the three quartiles are 60 , 61 and 63 .

Calculate the mean and median for the following frequency distribution : Wages (in ₹) 15 - 19 20 - 24 25 - 29 30 - 34 35 - 39 40 - 44 45 - 49 50 - 54 55 - 59 60 - 64 No. of workers 31 47 59 78 104 113 81 60 52 25 Also calculate Q 1 , Q 3 , P 10 , P 90 .

Since the given distribution is discontinuous but equal class interval is of equal width. To make it continuous we use adjustment factor = 20 - 19 2 = 0 . 5 Now we subtract 0 . 5 from lower limit and add 0 . 5 to upper limit of each class intervals. So we construct the following table of values is as under : Classes Class mark x i f i Cumulative frequency d i = x i - A A = 37 u i = d c c = 5 f i u i 14 . 5 - 19 . 5 19 . 5 - 24 . 5 24 . 5 - 29 . 5 29 . 5 - 34 . 5 34 . 5 - 39 . 5 39 . 5 - 44 . 5 44 . 5 - 49 . 5 49 . 5 - 54 . 5 54 . 5 - 59 . 5 59 . 5 - 64 . 5 17 22 27 32 37 42 47 52 57 62 31 47 59 78 104 113 81 60 52 25 31 78 137 215 319 432 513 573 625 650 - 20 - 15 - 10 - 5 0 5 10 15 20 25 - 4 - 3 - 2 - 1 0 1 2 3 4 5 - 124 - 141 - 118 - 78 0 113 162 180 208 125 ∑ f i = 650 ∑ f i u i = 327 Thus by step deviation method, Mean x ¯ = A + ∑ f i u i ∑ f i × c = 37 + 327 650 × 5 = 39 . 52 Here n = ∑ f i = 650 even ∴ Median = n 2 th value = 650 2 th value = 325 th value Which lies in the class 39 . 5 - 44 . 5 l = 39 . 5 ; f = 113 ; c = 319 ; i = 5 ∴ Median = l + n 2 - c f × i = 39 . 5 + 325 - 319 113 × 5 = 39 . 5 + 30 113 = 39 . 765 Q 1 = n 4 th value = 650 4 th value i . e 162 . 5 th value Which lies in the class 29 . 5 - 34 . 5 Here l = 29 . 5 ; f = 78 ; c = 137 ; i = 5 Q 1 = l + n 4 - c f × i = 29 . 5 + 162 . 5 - 137 78 × 5 = 29 . 5 + 1 . 6346 = 31 . 13 Q 3 = 3 n 4 th value = 3 × 650 4 th value i . e 487 . 5 th value Which lies in the class 44 . 5 - 49 . 5 Here l = 44 . 5 ; f = 81 ; c = 432 ; i = 5 Q 3 = l + 3 n 4 - c f × i = 44 . 5 + 487 . 5 - 432 81 × 5 = 44 . 5 + 3 . 4259 = 47 . 93 P 10 = 10 n 100 th value = 10 × 650 100 th value i . e 65 th value Which lies in the class 19 . 5 - 24 . 5 Here l = 19 . 5 ; f = 47 ; c = 31 ; i = 5 P 10 = l + 10 n 100 - c f × i = 19 . 5 + 657 . 5 - 31 47 × 5 = 19 . 5 + 3 . 6170 = 23 . 1170 P 90 = 90 n 100 th value = 90 × 650 100 th value i . e 585 th value Which lies in the class 54 . 5 - 59 . 5 Here, l = 54 . 5 ; f = 52 ; c = 573 ; i = 5 P 90 = l + 90 n 100 - c f × i = 54 . 5 + 585 - 573 52 × 5 = 54 . 5 + 1 . 1538 = 55 . 65 Hence, the required value of Q 1 , Q 3 , P 10 , P 90 are 31 . 13 , 47 . 93 , 23 . 12 , 55 . 65

The distribution of masses, to the nearest kg, of 90 men is shown in the following table : Mass (kg) 50 - 58 59 - 67 68 - 76 77 - 85 86 - 94 95 - 103 Frequency 15 15 23 20 9 8 Find the lower and upper quartile.

Here the given distribution is discontimous but equal class interval is of equal width. To make it continuous we use adjustment factor = 59 - 58 2 = 0 . 5 Now we subtract 0 . 5 from lower limit and add 0 . 5 to upper limit of cach class intervals. So we construct the following table of values is as under : Mass (kg) 49 . 5 - 58 . 5 58 . 5 - 67 . 5 67 . 5 - 76 . 5 76 . 5 - 85 . 5 85 . 5 - 94 . 5 94 . 5 - 103 . 5 Frequency 15 15 23 20 9 8 Cumulative frequency 15 30 53 73 82 90 Here n = ∑ f i = 90 Here lower quartile Q 1 = n 4 th value = 22 . 5 th value Which lies in 58 . 5 - 67 . 5 Here l = 58 . 5 ; f = 15 ; c = 15 ; i = 9 Q 1 = l + n 4 - c f × i = 58 . 5 + 22 . 5 . 5 - 15 15 × 9 = 58 . 5 + 4 . 5 = 6 . 3 Upper quartile Q 3 = 3 n 4 th value = 3 × 90 4 th value i . e 67 . 5 th value Which lies in the class 76 . 5 - 85 . 5 Here l = 85 . 5 . 5 ; f = 20 ; c = 53 ; i = 9 Q 3 = l + 3 n 4 - c f × i = 85 . 5 + 67 . 5 - 53 20 × 9 = 85 . 5 + 6 . 525 = 92 . 025

From the following data, calculate the percentage of workers who earn more than ₹ 840 per week. Weekly wages (in ₹) 500 - 600 600 - 700 700 - 800 800 - 900 900 - 1000 1000 - 1100 No. of workers 12 17 22 29 12 6

Firstly we construct the cumulative frequency distribution table is as under : Weekly wages (in ₹) 500 - 600 600 - 700 700 - 800 800 - 900 900 - 1000 1000 - 1100 No. of workers 12 17 22 29 12 6 Cumulative frequency 12 29 51 80 92 98 Let x % workers earn upto ₹ 840 ∴ P x = 840 ; n = ∑ f i = 98 , which lies in 800 - 900 ∴ l = 800 ; i = 100 ; c = 51 ; f = 29 Thus, P x = l + x × n 100 - c f × i ⇒ 840 = 800 + x × 98 100 - 51 29 × 100 ⇒ 29 × 40 = 98 x 100 - 50 × 100 ⇒ 1160 100 = 98 x 100 - 5 ⇒ 11 . 6 + 5 = 98 x 100 ⇒ 62 . 5 × 100 98 = x ⇒ x = 63 . 88 Thus 62 . 86 % workers earn upto ₹ 840 . Hence 100 - 63 . 88 % i . e . 36 . 12 % workers earn more than ₹ 840 .

HARD

11th ICSE

IMPORTANT

Earn 100

Calculate the $3^{rd}$ quartile $6^{th}$ decile and $70^{th}$ percentile for the following data :

$28, 17, 12, 25, 26, 19, 13, 27, 21, 16$ .

Important Questions on Statistics

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 4

Find $Q_{1}, Q_{3}, D_{6}$ and $P_{45}$ for the following data:

$x_{i}$	$18$	$19$	$20$	$21$	$22$	$23$	$24$	$25$	$26$	$27$
$f_{i}$	$15$	$18$	$25$	$27$	$40$	$25$	$19$	$16$	$6^{th}$	$7$

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 5

For the following data, compute lower and upper quartiles, $P_{90}$ and $P_{5}$ .

Marks	Below $10$	$10 - 20$	$20 - 40$	$40 - 60$	$60 - 80$	Above $80$
Number of students	$8$	$10$	$22$	$25$	$10$	$5$

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 6

Calculate the interquartile range for the following data :

Daily wages (in ₹)	$30 - 40$	$40 - 50$	$50 - 60$	$60 - 70$	$70 - 80$
No. of workers	$22$	$38$	$46$	$35$	$20$

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 7

Calculate all three quartiles for the following frequency distribution :

Height ( in inches)	$58$	$59$	$60$	$61$	$62$	$63$	$64$	$65$	$66$
No. of students	$15$	$20$	$32$	$35$	$35$	$22$	$20$	$10$	$8$

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 8

Calculate the mean and median for the following frequency distribution :

Wages (in ₹)	$15 - 19$	$20 - 24$	$25 - 29$	$30 - 34$	$35 - 39$	$40 - 44$	$45 - 49$	$50 - 54$	$55 - 59$	$60 - 64$
No. of workers	$31$	$47$	$59$	$78$	$104$	$113$	$81$	$60$	$52$	$25$

Also calculate $Q_{1}, Q_{3}, P_{10}, P_{90}$ .

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 9

The distribution of masses, to the nearest kg, of $90$ men is shown in the following table :

Mass (kg)	$50 - 58$	$59 - 67$	$68 - 76$	$77 - 85$	$86 - 94$	$95 - 103$
Frequency	$15$	$15$	$23$	$20$	$9$	$8$

Find the lower and upper quartile.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 10

Given below is the distribution of marks of $60$ students in a test.

Marks	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50$ and above
No. of students	$3$	$7$	$11$	$12$	$23$	$4$

If $55 %$ of the students pass the test, find the minimum marks obtained by a pass candidate.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 11

From the following data, calculate the percentage of workers who earn more than $₹ 840$ per week.

Weekly wages (in ₹)	$500 - 600$	$600 - 700$	$700 - 800$	$800 - 900$	$900 - 1000$	$1000 - 1100$
No. of workers	$12$	$17$	$22$	$29$	$12$	$6$

Calculate the 3rd quartile 6th decile and 70th percentile for the following data : 28, 17, 12, 25, 26, 19, 13, 27, 21, 16.

Important Questions on Statistics

Learn and Prepare for any exam you want !

Calculate the $3^{rd}$ quartile $6^{th}$ decile and $70^{th}$ percentile for the following data :

$28, 17, 12, 25, 26, 19, 13, 27, 21, 16$ .