JEE Main Previous Year Statistics Questions and Solutions

Q 1 :

Let M denote the median of the following frequency distribution

Class 0 - 4 4 - 8 8 - 12 12 - 16 16 - 20

Frequency 3 9 10 8 6

Then 20 M is equal to [2024]

416
52
208
104

1

2

3

4

(3)

Class	Frequency	Cumulative Frequency
0 - 4	3	3
4 - 8	9	12
8 - 12	10	22
12 - 16	8	30
16 - 20	6	36
	36

Where, $h = 4, l = 8, c . f . = 12, f = 10, n = 18$

$M = l + [\frac{\frac{n}{2} - c . f .}{f}] \times h = 8 + \frac{18 - 12}{10} \times 4$

$= 8 + \frac{6 \times 4}{10} = 10.4$

Now, $20 M = 20 \times 10.4 = 208$

Q 2 :

Marks obtains by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is [2025]

48
52
40
44

1

2

3

4

(4)

In the given question, l = 12, cf = 18, median = 14, f = 12 and h = 6

$∴ Median = l + (\frac{\frac{N}{2} - c f}{f}) \times h$

$\Rightarrow 14 = 12 + (\frac{\frac{N}{2} - 18}{12}) \times 6$

$\Rightarrow 2 = \frac{\frac{N}{2} - 18}{2} \Rightarrow 4 = \frac{N}{2} - 18$

$\Rightarrow \frac{N}{2} = 22 \Rightarrow N = 44$

So, total number of students = 44.

Q 3 :

Let $x_{1}, x_{2}, \dots, x_{100}$ be in an arithmetic progression with $x_{1} = 2$ and their mean equal to 200. If $y_{i} = i (x_{i} - i),$ $1 \leq i \leq 100,$ then the mean of $y_{1}, y_{2}, \dots, y_{100}$ is [2023]

10101.50
10051.50
10049.50
10100

1

2

3

4

(3)

$Mean = 200 \Rightarrow \frac{\frac{100}{2} (2 \times 2 + 99 d)}{100} = 200$

$\Rightarrow 4 + 99 d = 400 \Rightarrow d = 4$

$y_{i} = i (x_{i} - i) = i (2 + (i - 1) 4 - i) = 3 i^{2} - 2 i$

$Mean = \frac{\sum y_{i}}{100} = \frac{1}{100} \sum_{i = 1}^{100} (3 i^{2} - 2 i)$

$= \frac{1}{100} {\frac{3 \times 100 \times 101 \times 201}{6} - \frac{2 \times 100 \times 101}{2}}$

$= 101 {\frac{201}{2} - 1} = 101 \times 99.5 = 10049.50$

Q 4 :

The mean of the coefficients of $x, x^{2}, \dots, x^{7}$ in the binomial expansion of ${(2 + x)}^{9}$ is __________. [2023]

(2736)

${(2 + x)}^{9} = {}^{9}C_{0} 2^{9} + {}^{9}C_{1} 2^{8} x^{1} + {}^{9}C_{2} 2^{7} x^{2} + {}^{9}C_{3} 2^{6} x^{3} + {}^{9}C_{4} 2^{5} x^{4} + {}^{9}C_{5} 2^{4} x^{5} + {}^{9}C_{6} 2^{3} x^{6} + {}^{9}C_{7} 2^{2} x^{7} + {}^{9}C_{8} 2 x^{8} + {}^{9}C_{9} x^{9}$

$Coefficient of x = {}^{9}C_{1} 2^{8}$

$Coefficient of x^{2} = {}^{9}C_{2} 2^{7}, \dots, coefficient of x^{7} = {}^{9}C_{7} 2^{2}$

$Mean = \frac{{}^{9}C_{1} \cdot 2^{8} + {}^{9}C_{2} \cdot 2^{7} + \dots + {}^{9}C_{7} \cdot 2^{2}}{7}$

$= \frac{{(1 + 2)}^{9} - {}^{9}C_{0} \cdot 2^{9} - {}^{9}C_{8} \cdot 2^{1} - {}^{9}C_{9}}{7}$

$= \frac{3^{9} - 2^{9} - 18 - 1}{7} = \frac{19152}{7} = 2736$

Q 5 :

A fair $n (n > 1)$ faces die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac{n}{9},$ then $n$ is equal to _______ . [2023]

(10)

$Mean = 1 \cdot \frac{n - 1}{n} + 2 \frac{1}{n} (\frac{n - 1}{n}) + 3 {(\frac{1}{n})}^{2} (\frac{n - 1}{n}) + \dots$

$∴$ $\frac{n}{9} = (\frac{n - 1}{n}) (1 + 2 (\frac{1}{n}) + 3 {(\frac{1}{n})}^{2} + \dots)$

$\Rightarrow \frac{n}{9} = (\frac{n - 1}{n}) {(1 - \frac{1}{n})}^{- 2} = (\frac{n - 1}{n}) \cdot \frac{n^{2}}{{(n - 1)}^{2}}$

$\Rightarrow \frac{n}{9} = \frac{n}{n - 1} \Rightarrow n = 10$

Q 6 :

If the mean deviation about the median of the numbers k, 2k, 3k, ....., 1000k is 500, then $k ²$ is equal to : [2026]

9
4
1
16

1

2

3

4

(2)

$∵$ $median = \frac{1001 k}{2} = X_{M}$

$∴$ $mean deviation about median = \frac{\sum | X_{i} - X_{M} |}{n}$

$= \frac{2 (\frac{k}{2} + \frac{3 k}{2} + \frac{5 k}{2} + \dots 500 terms)}{1000}$

$= \frac{2 \cdot \frac{k}{2} {(500)}^{2}}{1000}$

$= \frac{500 k}{2} = 500 (given)$

$∴$ $k = 2$

$∴$ $k^{2} = 4$

Q 7 :

The mean weight of 9 items is 15. If one more item is added to the series, the mean becomes 16. The value of 10th item is

(25)

$Let x_{1}, x_{2}, x_{3}, \dots, x_{9} are 9 items then their mean$ $\Rightarrow \frac{x_{1} + x_{2} + x_{3} + \dots + x_{9}}{9} = 15$

$∴$ $\sum_{i = 1}^{9} x_{i} = 135$ $If one more item x_{10} is added then$ $\sum_{i = 1}^{10} x_{i} = 135 + x_{10}$

$∴$ $mean \bar{x} = \frac{135 + x_{10}}{10} = 16$

$∴$ $x_{10} = 160 - 135 = 25$

Q 8 :

Consider the following frequency distribution:

Class: 0–6 6–12 12–18 18–24 24–30

Frequency: $a$ $b$ 12 9 5

If mean = $\frac{309}{22}$ and median = 14, then the value of ${(a - b)}^{2}$ is equal to _____.

2
6
4
8

1

2

3

4

(3)

$N = (26 + a + b), \sum f_{i} x_{i} = (504 + 3 a + 9 b)$

Topic Question Set

Q 1 :

Let M denote the median of the following frequency distribution

Class 0 - 4 4 - 8 8 - 12 12 - 16 16 - 20

Frequency 3 9 10 8 6

Then 20 M is equal to [2024]

Q 3 :

Let $x_{1}, x_{2}, \dots, x_{100}$ be in an arithmetic progression with $x_{1} = 2$ and their mean equal to 200. If $y_{i} = i (x_{i} - i),$ $1 \leq i \leq 100,$ then the mean of $y_{1}, y_{2}, \dots, y_{100}$ is [2023]

Q 4 :

The mean of the coefficients of $x, x^{2}, \dots, x^{7}$ in the binomial expansion of ${(2 + x)}^{9}$ is __________. [2023]

Q 5 :

A fair $n (n > 1)$ faces die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac{n}{9},$ then $n$ is equal to _______ . [2023]

Q 6 :

If the mean deviation about the median of the numbers k, 2k, 3k, ....., 1000k is 500, then $k ²$ is equal to : [2026]

Q 7 :

The mean weight of 9 items is 15. If one more item is added to the series, the mean becomes 16. The value of 10th item is

Q 8 :

Consider the following frequency distribution:

Class: 0–6 6–12 12–18 18–24 24–30

Frequency: $a$ $b$ 12 9 5

If mean = $\frac{309}{22}$ and median = 14, then the value of ${(a - b)}^{2}$ is equal to _____.

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Class:	0–6	6–12	12–18	18–24	24–30
Frequency:	$a$	$b$	12	9	5

Topic Question Set

Q 1 : Let M denote the median of the following frequency distribution Class 0 - 4 4 - 8 8 - 12 12 - 16 16 - 20 Frequency 3 9 10 8 6 Then 20 M is equal to [2024]

Q 3 : Let x1,x2,…,x100 be in an arithmetic progression with x1=2 and their mean equal to 200. If yi=i(xi-i), 1≤i≤100, then the mean of y1,y2,…,y100 is [2023]

Q 4 : The mean of the coefficients of x,x2,…,x7 in the binomial expansion of (2+x)9 is __________. [2023]

Q 5 : A fair n(n>1) faces die is rolled repeatedly until a number less than n appears. If the mean of the number of tosses required is n9, then n is equal to _______ . [2023]

Q 6 : If the mean deviation about the median of the numbers k, 2k, 3k, ....., 1000k is 500, then k² is equal to : [2026]

Q 7 : The mean weight of 9 items is 15. If one more item is added to the series, the mean becomes 16. The value of 10th item is

Q 8 : Consider the following frequency distribution: Class: 0–6 6–12 12–18 18–24 24–30 Frequency: a b 12 9 5 If mean = 30922 and median = 14, then the value of (a−b)2 is equal to _____.

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Q 1 :

Let M denote the median of the following frequency distribution

Class 0 - 4 4 - 8 8 - 12 12 - 16 16 - 20

Frequency 3 9 10 8 6

Then 20 M is equal to [2024]

Q 3 :

Let $x_{1}, x_{2}, \dots, x_{100}$ be in an arithmetic progression with $x_{1} = 2$ and their mean equal to 200. If $y_{i} = i (x_{i} - i),$ $1 \leq i \leq 100,$ then the mean of $y_{1}, y_{2}, \dots, y_{100}$ is [2023]

Q 4 :

The mean of the coefficients of $x, x^{2}, \dots, x^{7}$ in the binomial expansion of ${(2 + x)}^{9}$ is __________. [2023]

Q 5 :

A fair $n (n > 1)$ faces die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac{n}{9},$ then $n$ is equal to _______ . [2023]

Q 6 :

If the mean deviation about the median of the numbers k, 2k, 3k, ....., 1000k is 500, then $k ²$ is equal to : [2026]

Q 7 :

The mean weight of 9 items is 15. If one more item is added to the series, the mean becomes 16. The value of 10th item is

Q 8 :

Consider the following frequency distribution:

Class: 0–6 6–12 12–18 18–24 24–30

Frequency: $a$ $b$ 12 9 5

If mean = $\frac{309}{22}$ and median = 14, then the value of ${(a - b)}^{2}$ is equal to _____.