Question 1: Palette 1/112

JEE Mains 2026 MCQ

The mean and variance of 10 observations are 9 and 34.2 , respectively. If 8 of these observations are $2,3,5,10,11,13,15,21$, then the mean deviation about the median of all the 10 observations is

A

7

B

4

C

5

D

6

Question 2: Palette 2/112

JEE Mains 2026 MCQ

Let $\mathrm{X}=\{x \in \mathrm{~N}: 1 \leq x \leq 19\}$ and for some $a, b \in \mathbb{R}, \mathrm{Y}=\{a x+b: x \in \mathrm{X}\}$. If the mean and variance of the elements of Y are 30 and 750 , respectively, then the sum of all possible values of $b$ is

A

20

B

100

C

80

D

60

Question 3: Palette 3/112

JEE Mains 2026 MCQ

The mean and variance of a data of 10 observations are 10 and 2 , respectively. If an observations $\alpha$ in this data is replaced by $\beta$, then the mean and variance become 10.1 and 1.99 , respectively. Then $\alpha+\beta$ equals

A

15

B

10

C

5

D

20

Question 4: Palette 4/112

JEE Mains 2026 MCQ

If the mean and the variance of the data

$ \begin{array}{|c|c|c|c|c|} \hline \text { Class } & 4-8 & 8-12 & 12-16 & 16-20 \\ \hline \text { Frequency } & 3 & \lambda & 4 & 7 \\ \hline \end{array} $

are $\mu$ and 19 respectively, then the value of $\lambda+\mu$ is :

A

21

B

19

C

18

D

20

Question 5: Palette 5/112

JEE Mains 2026 MCQ

Let the mean and variance of 8 numbers $-10,-7,-1, x, y, 9,2,16$ be $\frac{7}{2}$ and $\frac{293}{4}$, respectively.

Then the mean of 4 numbers $x, y, x+y+1,|x-y|$ is :

A

11

B

9

C

10

D

12

Question 6: Palette 6/112

JEE Mains 2026 MCQ

If the mean deviation about the median of the numbers $\mathrm{k}, 2 \mathrm{k}, 3 \mathrm{k}, \ldots ., 1000 \mathrm{k}$ is 500 , then $\mathrm{k}^2$ is equal to :

A

1

B

16

C

9

D

4

Question 7: Palette 7/112

JEE Mains 2026 MCQ

A random variable X takes values 0, 1, 2, 3 with probabilities $\frac{2a+1}{30}$, $\frac{8a-1}{30}$, $\frac{4a+1}{30}$, $b$ respectively, where $a, b \in \mathbb{R}$.

Let $\mu$ and $\sigma$ respectively be the mean and standard deviation of $X$ such that $\sigma^2 + \mu^2 = 2$.

Then $\frac{a}{b}$ is equal to:

A

12

B

60

C

30

D

3

Question 8: Palette 8/112

JEE Mains 2026 MCQ

A set of four observations has mean 1 and variance 13. Another set of six observations has mean 2 and variance 1 . Then, the variance of all these 10 observations is equal to :

A

5.96

B

6.14

C

6.04

D

6.24

Question 9: Palette 9/112

JEE Mains 2026 MCQ

Let the mean and the variance of seven observations $2,4, \alpha, 8, \beta, 12,14, \alpha<\beta$, be 8 and 16 respectively. Then the quadratic equation whose roots are $3 \alpha+2$ and $2 \beta+1$ is :

A

$x^2-35 x+306=0$

B

$x^2-41 x+420=0$

C

$x^2-45 x+506=0$

D

$x^2-37 x+342=0$

Question 10: Palette 10/112

JEE Mains 2026 MCQ

A data consists of 20 observations $x_1, x_2, \ldots, x_{20}$. If $\sum\limits_{i=1}^{20}\left(x_i+5\right)^2=2500$ and $\sum\limits_{i=1}^{20}\left(x_i-5\right)^2=100$, then the ratio of mean to standard deviation of this data is :

A

2 : 1

B

3 : 1

C

3 : 2

D

4 : 1

Question 11: Palette 11/112

JEE Mains 2026 MCQ

A variable $X$ takes values $0,0,2,6,12,20, \ldots, n(n-1)$ with frequencies ${ }^n C_0,{ }^n C_1,{ }^n C_2,{ }^n C_3,{ }^n C_4,{ }^n C_5, \ldots,{ }^n C_n$, respectively. If the mean of this data is 60 , then its median is :

A

56

B

42

C

72

D

90

Question 12: Palette 12/112

JEE Mains 2026 MCQ

The mean deviation about the mean for the data

$ \begin{array}{|c|c|c|c|c|c|c|} \hline x_i & 5 & 7 & 9 & 10 & 12 & 15 \\ \hline f_i & 8 & 6 & 2 & 2 & 2 & 6 \\ \hline \end{array} $

$ \text { is equal to: } $

A

$ \frac{40}{13} $

B

$ \frac{42}{13} $

C

$ \frac{44}{13} $

D

$ \frac{46}{13} $

Question 13: Palette 13/112

JEE Mains 2026 MCQ

For 10 observations $x_1, x_2, \ldots, x_{10}$, if $\sum\limits_{i=1}^{10}\left(x_i+2\right)^2=180$ and $\sum\limits_{i=1}^{10}\left(x_i-1\right)^2=90$, then their standard deviation is :

A

2

B

$ \sqrt{3} $

C

$ 2\sqrt{2} $

D

3

Question 14: Palette 14/112

JEE Mains 2026 MCQ

Suppose that the mean and median of the non-negative numbers $21,8,17, a, 51,103, b, 13,67,(a>b)$, are 40 and 21 , respectively. If the mean deviation about the median is 26 , then $2 a$ is equal to :

A

109

B

117

C

161

D

131

Question 15: Palette 15/112

JEE Mains 2026 MCQ

The mean and variance of n observations are 8 and 16, respectively. If the sum of the first (n − 1) observations is 48 and the sum of squares of the first (n − 1) observations is 496, then the value of n is :

A

21

B

16

C

13

D

7

Question 16: Palette 16/112

JEE Mains 2026 MCQ

If the mean of the data

Class	5-10	10-15	15-20	20-25	25-30	30-35
Frequency	2	k	28	54	k + 1	5

is 21, then $k$ is one of the roots of the equation:

A

$2x^2 - 23x - 10 = 0$

B

$4x^2 - 35x + 24 = 0$

C

$2x^2 - 19x - 10 = 0$

D

$2x^2 - 35x + 98 = 0$

Question 17: Palette 17/112

JEE Mains 2025 MCQ

The mean and standard deviation of 100 observations are 40 and 5.1 , respectively. By mistake one observation is taken as 50 instead of 40 . If the correct mean and the correct standard deviation are $\mu$ and $\sigma$ respectively, then $10(\mu+\sigma)$ is equal to

A

447

B

445

C

449

D

451

Question 18: Palette 18/112

JEE Mains 2025 MCQ

Let the mean and the standard deviation of the observation $2,3,3,4,5,7, a, b$ be 4 and $\sqrt{2}$ respectively. Then the mean deviation about the mode of these observations is :

A

$\frac{1}{2}$

B

$\frac{3}{4}$

C

1

D

2

Question 19: Palette 19/112

JEE Mains 2025 MCQ

Let the Mean and Variance of five observations $x_1=1, x_2=3, x_3=a, x_4=7$ and $x_5=\mathrm{b}, a>\mathrm{b}$, be 5 and 10 respectively. Then the Variance of the observations $n+x_n, n=1,2, \ldots, 5$ is

A

17

B

16

C

16.4

D

17.4

Question 20: Palette 20/112

JEE Mains 2025 MCQ

If the mean and the variance of $6,4, a, 8, b, 12,10,13$ are 9 and 9.25 respectively, then $a+b+a b$ is equal to :

A

103

B

106

C

100

D

105

Question 21: Palette 21/112

JEE Mains 2025 MCQ

Let $x_1, x_2, ..., x_{10}$ be ten observations such that $\sum\limits_{i=1}^{10} (x_i - 2) = 30$, $\sum\limits_{i=1}^{10} (x_i - \beta)^2 = 98$, $\beta > 2$, and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^2$ are respectively the mean and the variance of $2(x_1 - 1) + 4\beta, 2(x_2 - 1) + 4\beta, ..., 2(x_{10} - 1) + 4\beta$, then $\frac{\beta\mu}{\sigma^2}$ is equal to :

A

100

B

90

C

120

D

110

Question 22: Palette 22/112

JEE Mains 2025 MCQ

For a statistical data $\mathrm{x}_1, \mathrm{x}_2, \ldots, \mathrm{x}_{10}$ of 10 values, a student obtained the mean as 5.5 and $\sum_{i=1}^{10} x_i^2=371$. He later found that he had noted two values in the data incorrectly as 4 and 5 , instead of the correct values 6 and 8 , respectively. The variance of the corrected data is

A

5

B

7

C

9

D

4

Question 23: Palette 23/112

JEE Mains 2025 MCQ

Marks obtains by all the students of class 12 are presented in a freqency 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 :

A

52

B

44

C

40

D

48

Question 24: Palette 24/112

JEE Mains 2024 MCQ

If the variance of the frequency distribution

$x$	$c$	$2c$	$3c$	$4c$	$5c$	$6c$
$f$	2	1	1	1	1	1

is 160, then the value of $c\in N$ is

A

5

B

8

C

6

D

7

Question 25: Palette 25/112

JEE Mains 2024 MCQ

The frequency distribution of the age of students in a class of 40 students is given below.

Age	15	16	17	18	19	20
No of Students	5	8	5	12	$x$	$y$

If the mean deviation about the median is 1.25, then $4x+5y$ is equal to :

A

43

B

46

C

44

D

47

Question 26: Palette 26/112

JEE Mains 2024 MCQ

The mean and standard deviation of 20 observations are found to be 10 and 2 , respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is

A

1.94

B

$\sqrt{3.96}$

C

$\sqrt{3.86}$

D

1.8

Question 27: Palette 27/112

JEE Mains 2024 MCQ

Let $\alpha, \beta \in \mathbf{R}$. Let the mean and the variance of 6 observations $-3,4,7,-6, \alpha, \beta$ be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is :

A

$\frac{16}{3}$

B

$\frac{11}{3}$

C

$\frac{14}{3}$

D

$\frac{13}{3}$

Question 28: Palette 28/112

JEE Mains 2024 MCQ

Consider 10 observations $x_1, x_2, \ldots, x_{10}$ such that $\sum\limits_{i=1}^{10}\left(x_i-\alpha\right)=2$ and $\sum\limits_{i=1}^{10}\left(x_i-\beta\right)^2=40$, where $\alpha, \beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$ respectively. Then $\frac{\beta}{\alpha}$ is equal to :

A

2

B

1

C

$\frac{5}{2}$

D

$\frac{3}{2}$

Question 29: Palette 29/112

JEE Mains 2024 MCQ

Let the median and the mean deviation about the median of 7 observation $170,125,230,190,210$, a, b be 170 and $\frac{205}{7}$ respectively. Then the mean deviation about the mean of these 7 observations is :

A

31

B

28

C

30

D

32

Question 30: Palette 30/112

JEE Mains 2024 MCQ

Let the mean and the variance of 6 observations $a, b, 68,44,48,60$ be $55$ and $194$, respectively. If $a>b$, then $a+3 b$ is

A

180

B

210

C

190

D

200

Question 31: Palette 31/112

JEE Mains 2024 MCQ

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 20M is equal to :

A

104

B

52

C

208

D

416

Question 32: Palette 32/112

JEE Mains 2024 MCQ

If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively and the mean of the first four observations is $\frac{7}{2}$, then the variance of the first four observations in equal to

A

$\frac{5}{4}$

B

$\frac{4}{5}$

C

$\frac{105}{4}$

D

$\frac{77}{12}$

Question 33: Palette 33/112

JEE Mains 2024 MCQ

Let $\mathrm{a}_1, \mathrm{a}_2, \ldots \mathrm{a}_{10}$ be 10 observations such that $\sum\limits_{\mathrm{k}=1}^{10} \mathrm{a}_{\mathrm{k}}=50$ and $\sum\limits_{\forall \mathrm{k} < \mathrm{j}} \mathrm{a}_{\mathrm{k}} \cdot \mathrm{a}_{\mathrm{j}}=1100$. Then the standard deviation of $\mathrm{a}_1, \mathrm{a}_2, \ldots, \mathrm{a}_{10}$ is equal to :

A

5

B

$\sqrt{115}$

C

10

D

$\sqrt{5}$

Question 34: Palette 34/112

JEE Mains 2023 MCQ

The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is :

A

11

B

12

C

13

D

14

Question 35: Palette 35/112

JEE Mains 2023 MCQ

Let the mean of 6 observations $1,2,4,5, \mathrm{x}$ and $\mathrm{y}$ be 5 and their variance be 10 . Then their mean deviation about the mean is equal to :

A

$\frac{10}{3}$

B

$\frac{8}{3}$

C

$\frac{7}{3}$

D

3

Question 36: Palette 36/112

JEE Mains 2023 MCQ

Let sets A and B have 5 elements each. Let the mean of the elements in sets A and B be 5 and 8 respectively and the variance of the elements in sets A and B be 12 and 20 respectively. A new set C of 10 elements is formed by subtracting 3 from each element of $\mathrm{A}$ and adding 2 to each element of $\mathrm{B}$. Then the sum of the mean and variance of the elements of $\mathrm{C}$ is ___________.

A

36

B

40

C

38

D

32

Question 37: Palette 37/112

JEE Mains 2023 MCQ

Let $\mu$ be the mean and $\sigma$ be the standard deviation of the distribution

${x_i}$	0	1	2	3	4	5
${f_i}$	$k + 2$	$2k$	${k^2} - 1$	${k^2} - 1$	${k^2} + 1$	$k - 3$

where $\sum f_{i}=62$. If $[x]$ denotes the greatest integer $\leq x$, then $\left[\mu^{2}+\sigma^{2}\right]$ is equal to :

A

9

B

8

C

6

D

7

Question 38: Palette 38/112

JEE Mains 2023 MCQ

Let the mean and variance of 12 observations be $\frac{9}{2}$ and 4 respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is $\frac{m}{n}$, where $\mathrm{m}$ and $\mathrm{n}$ are coprime, then $\mathrm{m}+\mathrm{n}$ is equal to :

A

317

B

316

C

314

D

315

Question 39: Palette 39/112

JEE Mains 2023 MCQ

The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of another set of 15 numbers are 14 and $\sigma^{2}$ respectively. If the variance of all the 30 numbers in the two sets is 13 , then $\sigma^{2}$ is equal to :

A

12

B

11

C

10

D

9

Question 40: Palette 40/112

JEE Mains 2023 MCQ

Let $9=x_{1} < x_{2} < \ldots < x_{7}$ be in an A.P. with common difference d. If the standard deviation of $x_{1}, x_{2}..., x_{7}$ is 4 and the mean is $\bar{x}$, then $\bar{x}+x_{6}$ is equal to :

A

$2\left(9+\frac{8}{\sqrt{7}}\right)$

B

25

C

$18\left(1+\frac{1}{\sqrt{3}}\right)$

D

34

Question 41: Palette 41/112

JEE Mains 2023 MCQ

The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5, then the sum of cubes of the remaining two observations is :

A

1792

B

1216

C

1456

D

1072

Question 42: Palette 42/112

JEE Mains 2023 MCQ

Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and $\alpha(>$ 0 ), and the mean and standard deviation of marks of class $B$ of $n$ students be respectively 55 and 30 $-\alpha$. If the mean and variance of the marks of the combined class of $100+\mathrm{n}$ studants are respectively 50 and 350 , then the sum of variances of classes $A$ and $B$ is :

A

450

B

900

C

650

D

500

Question 43: Palette 43/112

JEE Mains 2023 MCQ

Let $S$ be the set of all values of $a_1$ for which the mean deviation about the mean of 100 consecutive positive integers $a_1, a_2, a_3, \ldots ., a_{100}$ is 25 . Then $S$ is :

A

$\{9\}$

B

$\phi$

C

$\{99\}$

D

N

Question 44: Palette 44/112

JEE Mains 2023 MCQ

Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $\mu$ and $\sigma^2$ represent mean and variance of X, respectively, then $10(\mu^2+\sigma^2)$ is equal to :

A

20

B

30

C

250

D

25

Question 45: Palette 45/112

JEE Mains 2023 MCQ

The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new variance is equal to :

A

3.92

B

4.08

C

3.96

D

4.04

Question 46: Palette 46/112

JEE Mains 2023 MCQ

Let the six numbers $\mathrm{a_1,a_2,a_3,a_4,a_5,a_6}$, be in A.P. and $\mathrm{a_1+a_3=10}$. If the mean of these six numbers is $\frac{19}{2}$ and their variance is $\sigma^2$, then 8$\sigma^2$ is equal to :

A

220

B

210

C

105

D

200

Question 47: Palette 47/112

JEE Mains 2022 MCQ

If the mean deviation about median for the numbers 3, 5, 7, 2k, 12, 16, 21, 24, arranged in the ascending order, is 6 then the median is :

A

11.5

B

10.5

C

12

D

11

Question 48: Palette 48/112

JEE Mains 2022 MCQ

The number of values of a $\in$ N such that the variance of 3, 7, 12, a, 43 $-$ a is a natural number is :

A

0

B

2

C

5

D

infinite

Question 49: Palette 49/112

JEE Mains 2022 MCQ

Let the mean and the variance of 5 observations x₁, x₂, x₃, x₄, x₅ be ${24 \over 5}$ and ${194 \over 25}$ respectively. If the mean and variance of the first 4 observation are ${7 \over 2}$ and a respectively, then (4a + x₅) is equal to:

A

13

B

15

C

17

D

18

Question 50: Palette 50/112

JEE Mains 2022 MCQ

The mean and variance of the data 4, 5, 6, 6, 7, 8, x, y, where x < y, are 6 and ${9 \over 4}$ respectively. Then ${x^4} + {y^2}$ is equal to :

A

162

B

320

C

674

D

420

Question 51: Palette 51/112

JEE Mains 2022 MCQ

The mean and standard deviation of 50 observations are 15 and 2 respectively. It was found that one incorrect observation was taken such that the sum of correct and incorrect observations is 70. If the correct mean is 16, then the correct variance is equal to :

A

10

B

36

C

43

D

60

Question 52: Palette 52/112

JEE Mains 2022 MCQ

The mean of the numbers a, b, 8, 5, 10 is 6 and their variance is 6.8. If M is the mean deviation of the numbers about the mean, then 25 M is equal to :

A

60

B

55

C

50

D

45

Question 53: Palette 53/112

JEE Mains 2021 MCQ

The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is :

A

${{92} \over 5}$

B

${{134} \over 5}$

C

${{536} \over {25}}$

D

${{112} \over 5}$

Question 54: Palette 54/112

JEE Mains 2021 MCQ

The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. if $\alpha$ and $\sqrt \beta $ are the mean and standard deviation respectively for correct data, then ($\alpha$, $\beta$) is :

A

(11, 26)

B

(10.5, 25)

C

(11, 25)

D

(10.5, 26)

Question 55: Palette 55/112

JEE Mains 2021 MCQ

Let the mean and variance of the frequency distribution

$\matrix{ {x:} & {{x_1} = 2} & {{x_2} = 6} & {{x_3} = 8} & {{x_4} = 9} \cr {f:} & 4 & 4 & \alpha & \beta \cr } $

be 6 and 6.8 respectively. If x₃ is changed from 8 to 7, then the mean for the new data will be :

A

4

B

5

C

${{17} \over 3}$

D

${{16} \over 3}$

Question 56: Palette 56/112

JEE Mains 2021 MCQ

If the mean and variance of the following data : 6, 10, 7, 13, a, 12, b, 12 are 9 and ${{37} \over 4}$

respectively, then (a $-$ b)² is equal to :

A

24

B

12

C

32

D

16

Question 57: Palette 57/112

JEE Mains 2021 MCQ

The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation $\sqrt {13.44} $, then the standard deviation of the second sample is :

A

8

B

6

C

4

D

5

Question 58: Palette 58/112

JEE Mains 2021 MCQ

If the mean and variance of six observations 7, 10, 11, 15, a, b are 10 and ${{20} \over 3}$, respectively, then the value of | a $-$ b | is equal to :

A

9

B

11

C

7

D

1

Question 59: Palette 59/112

JEE Mains 2021 MCQ

The mean of 6 distinct observations is 6.5 and their variance is 10.25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are :

A

10, 11

B

3, 18

C

8, 13

D

1, 20

Question 60: Palette 60/112

JEE Mains 2021 MCQ

Let in a series of 2n observations, half of them are equal to a and remaining half are equal to $-$a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of a² + b² is equal to :

A

425

B

250

C

925

D

650

Question 61: Palette 61/112

JEE Mains 2021 MCQ

Consider three observations a, b, and c such that b = a + c. If the standard deviation of a + 2, b + 2, c + 2 is d, then which of the following is true?

A

b² = 3(a² + c²) + 9d²

B

b² = 3(a² + c²) $-$ 9d²

C

b² = 3(a² + c² + d²)

D

b² = a² + c² + 3d²

Question 62: Palette 62/112

JEE Mains 2020 MCQ

If $\sum\limits_{i = 1}^n {\left( {{x_i} - a} \right)} = n$ and $\sum\limits_{i = 1}^n {{{\left( {{x_i} - a} \right)}^2}} = na$
(n, a > 1) then the standard deviation of n
observations x₁ , x₂ , ..., x_n is :

A

$a$ – 1

B

$n\sqrt {a - 1} $

C

$\sqrt {n\left( {a - 1} \right)} $

D

$\sqrt {a - 1} $

Question 63: Palette 63/112

JEE Mains 2020 MCQ

If the mean and the standard deviation of the
data 3, 5, 7, a, b are 5 and 2 respectively, then a and b are the roots of the equation :

A

x² – 20x + 18 = 0

B

2x² – 20x + 19 = 0

C

x² – 10x + 18 = 0

D

x² – 10x + 19 = 0

Question 64: Palette 64/112

JEE Mains 2020 MCQ

The mean and variance of 7 observations are 8 and 16, respectively. If five observations are 2, 4, 10, 12, 14, then the absolute difference of the remaining two observations is :

A

2

B

3

C

1

D

4

Question 65: Palette 65/112

JEE Mains 2020 MCQ

The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is :

A

5

B

3

C

7

D

9

Question 66: Palette 66/112

JEE Mains 2020 MCQ

Let x_i (1 $ \le $ i $ \le $ 10) be ten observations of a random variable X. If
$\sum\limits_{i = 1}^{10} {\left( {{x_i} - p} \right)} = 3$ and $\sum\limits_{i = 1}^{10} {{{\left( {{x_i} - p} \right)}^2}} = 9$
where 0 $ \ne $ p $ \in $ R, then the standard deviation of these observations is :

A

${7 \over {10}}$

B

${9 \over {10}}$

C

${4 \over 5}$

D

$\sqrt {{3 \over 5}} $

Question 67: Palette 67/112

JEE Mains 2020 MCQ

For the frequency distribution :
Variate (x) :      x₁   x₂   x₃ ....  x₁₅
Frequency (f) : f₁    f₂   f₃ ...... f₁₅
where 0 < x₁ < x₂ < x₃ < ... < x₁₅ = 10 and
$\sum\limits_{i = 1}^{15} {{f_i}} $ > 0, the standard deviation cannot be :

A

6

B

1

C

4

D

2

Question 68: Palette 68/112

JEE Mains 2020 MCQ

Let X = {x $ \in $ N : 1 $ \le $ x $ \le $ 17} and
Y = {ax + b: x $ \in $ X and a, b $ \in $ R, a > 0}. If mean
and variance of elements of Y are 17 and 216
respectively then a + b is equal to :

A

7

B

9

C

-7

D

-27

Question 69: Palette 69/112

JEE Mains 2020 MCQ

Let the observations x_i (1 $ \le $ i $ \le $ 10) satisfy the
equations, $\sum\limits_{i = 1}^{10} {\left( {{x_1} - 5} \right)} $ = 10 and $\sum\limits_{i = 1}^{10} {{{\left( {{x_1} - 5} \right)}^2}} $ = 40.
If $\mu $ and $\lambda $ are the mean and the variance of the
observations, x₁ – 3, x₂ – 3, ...., x₁₀ – 3, then
the ordered pair ($\mu $, $\lambda $) is equal to :

A

(6, 6)

B

(3, 3)

C

(3, 6)

D

(6, 3)

Question 70: Palette 70/112

JEE Mains 2020 MCQ

The mean and variance of 20 observations are found to be 10 and 4, respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11. Then the correct variance is

A

3.98

B

3.99

C

4.01

D

4.02

Question 71: Palette 71/112

JEE Mains 2020 MCQ

The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 resepectively. Each of these 10 observations is multiplied by p and then reduced by q, where p $ \ne $ 0 and q $ \ne $ 0. If the new mean and new s.d. become half of their original values, then q is equal to

A

10

B

-20

C

-10

D

-5

Question 72: Palette 72/112

JEE Mains 2019 MCQ

If the data x₁, x₂,......., x₁₀ is such that the mean of first four of these is 11, the mean of the remaining six is 16 and the sum of squares of all of these is 2,000 ; then the standard deviation of this data is :

A

$\sqrt 2 $

B

2

C

2$\sqrt 2 $

D

4

Question 73: Palette 73/112

JEE Mains 2019 MCQ

If both the mean and the standard deviation of 50 observations x₁, x₂,..., x₅₀ are equal to 16, then the mean of (x₁ – 4)² , (x₂ – 4)² ,....., (x₅₀ – 4)² is :

A

400

B

480

C

380

D

525

Question 74: Palette 74/112

JEE Mains 2019 MCQ

If for some x $ \in $ R, the frequency distribution of the marks obtained by 20 students in a test is :

Marks	2	3	5	7
Frequency	(x + 1)2	2x - 5	x2 - 3x	x

then the mean of the marks is

A

3.0

B

2.8

C

2.5

D

3.2

Question 75: Palette 75/112

JEE Mains 2019 MCQ

The mean and the median of the following ten numbers in increasing order 10, 22, 26, 29, 34, x, 42, 67, 70, y are 42 and 35 respectively, then ${y \over x}$ is equal to

A

${7 \over 2}$

B

${8 \over 3}$

C

${9 \over 4}$

D

${7 \over 3}$

Question 76: Palette 76/112

JEE Mains 2019 MCQ

If the standard deviation of the numbers –1, 0, 1, k is $\sqrt 5$ where k > 0, then k is equal to

A

2$\sqrt 6 $

B

$\sqrt 6 $

C

$2\sqrt {{{5} \over 6}} $

D

$2\sqrt {{{10} \over 3}} $

Question 77: Palette 77/112

JEE Mains 2019 MCQ

A student scores the following marks in five tests :

45, 54, 41, 57, 43.

His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is

A

$100 \over {\sqrt 3}$

B

$10 \over {\sqrt 3}$

C

$10 \over3$

D

$100 \over3$

Question 78: Palette 78/112

JEE Mains 2019 MCQ

The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the product of the remaining two observations is :

A

40

B

48

C

49

D

45

Question 79: Palette 79/112

JEE Mains 2019 MCQ

The mean and the variance of five observations are 4 and 5.20, respectively. If three of the observations are 3, 4 and 4 ; then the absolute value of the difference of the other two observations, is :

A

1

B

7

C

3

D

5

Question 80: Palette 80/112

JEE Mains 2019 MCQ

If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is :

A

31

B

50

C

51

D

30

Question 81: Palette 81/112

JEE Mains 2019 MCQ

The outcome of each of 30 items was observed; 10 items gave an outcome ${1 \over 2}$ – d each, 10 items gave outcome ${1 \over 2}$ each and the remaining 10 items gave outcome ${1 \over 2}$+ d each. If the variance of this outcome data is ${4 \over 3}$ then |d| equals :

A

${2 \over 3}$

B

${{\sqrt 5 } \over 2}$

C

${\sqrt 2 }$

D

2

Question 82: Palette 82/112

JEE Mains 2019 MCQ

If mean and standard deviation of 5 observations x₁, x₂, x₃, x₄, x₅ are 10 and 3, respectively, then the variance of 6 observations x₁, x₂, ….., x₅ and –50 is equal to

A

582.5

B

507.5

C

586.5

D

509.5

Question 83: Palette 83/112

JEE Mains 2019 MCQ

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is -

A

6 : 7

B

10 : 3

C

4 : 9

D

5 : 8

Question 84: Palette 84/112

JEE Mains 2019 MCQ

A data consists of n observations : x₁, x₂, . . . . . . ., x_n.    

If     $\sum\limits_{i = 1}^n {{{\left( {{x_i} + 1} \right)}^2}} = 9n$    and

$\sum\limits_{i = 1}^n {{{\left( {{x_i} - 1} \right)}^2}} = 5n,$

then the standard deviation of this data is :

A

2

B

$\sqrt 5 $

C

5

D

$\sqrt 7 $

Question 85: Palette 85/112

JEE Mains 2019 MCQ

5 students of a class have an average height 150 cm and variance 18 cm². A new student, whose height is 156 cm, joined them. The variance (in cm²) of the height of these six students is :

A

16

B

22

C

20

D

18

Question 86: Palette 86/112

JEE Mains 2018 MCQ

The mean and the standard deviation(s.d.) of five observations are9 and 0, respectively. If one of the observations is changed such that the mean of the new set of five observations becomes 10, then their s.d. is :

A

0

B

1

C

2

D

4

Question 87: Palette 87/112

JEE Mains 2018 MCQ

If $\sum\limits_{i = 1}^9 {\left( {{x_i} - 5} \right)} = 9$ and

$\sum\limits_{i = 1}^9 {{{\left( {{x_i} - 5} \right)}^2}} = 45$, then the standard deviation of the 9 items
${x_1},{x_2},.......,{x_9}$ is

A

3

B

9

C

4

D

2

Question 88: Palette 88/112

JEE Mains 2018 MCQ

If the mean of the data : 7, 8, 9, 7, 8, 7, $\lambda $, 8 is 8, then the variance of this data is :

A

${7 \over 8}$

B

1

C

${9 \over 8}$

D

2

Question 89: Palette 89/112

JEE Mains 2018 MCQ

The mean of set of 30 observations is 75. If each observation is multiplied by a non-zero number $\lambda $ and then each of them is decreased by 25, their mean remains the same. Then $\lambda $ is equal to :

A

${1 \over 3}$

B

${2 \over 3}$

C

${4 \over 3}$

D

${10 \over 3}$

Question 90: Palette 90/112

JEE Mains 2017 MCQ

The sum of 100 observations and the sum of their squares are 400 and 2475, respectively. Later on, three observations, 3, 4 and 5, were found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations is :

A

8.25

B

8.50

C

8.00

D

9.00

Question 91: Palette 91/112

JEE Mains 2017 MCQ

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If now the mean age of the teachers in this school is 39 years, then the age (in years) of the newly appointed teacher is :

A

25

B

30

C

35

D

40

Question 92: Palette 92/112

JEE Mains 2016 MCQ

The mean of 5 observations is 5 and their variance is 124. If three of the observations are 1, 2 and 6 ; then the mean deviation from the mean of the data is :

A

2.4

B

2.8

C

2.5

D

2.6

Question 93: Palette 93/112

JEE Mains 2016 MCQ

If the mean deviation of the numbers 1, 1 + d, ..., 1 +100d from their mean is 255, then a value of d is :

A

10.1

B

20.2

C

10

D

5.05

Question 94: Palette 94/112

JEE Mains 2016 MCQ

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?

A

3$a$² - 26$a$ + 55 = 0

B

3$a$² - 32$a$ + 84 = 0

C

3$a$² - 34$a$ + 91 = 0

D

3$a$² - 23$a$ + 44 = 0

Question 95: Palette 95/112

JEE Mains 2015 MCQ

The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is :

A

15.8

B

14.0

C

16.8

D

16.0

Question 96: Palette 96/112

JEE Mains 2014 MCQ

The variance of first 50 even natural numbers is

A

833

B

437

C

${{437} \over 4}$

D

${{833} \over 4}$

Question 97: Palette 97/112

JEE Mains 2013 MCQ

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given?

A

median

B

mode

C

variance

D

mean

Question 98: Palette 98/112

JEE Mains 2012 MCQ

Let x₁, x₂,........., x_n be n observations, and let $\overline x $ be their arithematic mean and ${\sigma ^2}$ be their variance.

Statement 1 : Variance of 2x₁, 2x₂,......., 2x_n is 4${\sigma ^2}$.
Statement 2 : : Arithmetic mean of 2x₁, 2x₂,......, 2x_n is 4$\overline x $.

A

Statement 1 is false, statement 2 is true

B

Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

C

Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

D

Statement 1 is true, statement 2 is false

Question 99: Palette 99/112

JEE Mains 2011 MCQ

If the mean deviation about the median of the numbers a, 2a,........., 50a is 50, then |a| equals

A

4

B

5

C

2

D

3

Question 100: Palette 100/112

JEE Mains 2010 MCQ

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is

A

${5 \over 2}$

B

${11 \over 2}$

C

6

D

${13 \over 2}$

Question 101: Palette 101/112

JEE Mains 2009 MCQ

If the mean deviation of number 1, 1 + d, 1 + 2d,........, 1 + 100d from their mean is 255, then the d is equal to

A

20.0

B

10.1

C

20.2

D

10.0

Question 102: Palette 102/112

JEE Mains 2009 MCQ

Statement - 1 : The variance of first n even natural numbers is ${{{n^2} - 1} \over 4}$

Statement - 2 : The sum of first n natural numbers is ${{n\left( {n + 1} \right)} \over 2}$ and the sum of squares of first n natural numbers is ${{n\left( {n + 1} \right)\left( {2n + 1} \right)} \over 6}$

A

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

B

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

C

Statement-1 is true, Statement-2 is false

D

Statement-1 is false, Statement-2 is true

Question 103: Palette 103/112

JEE Mains 2008 MCQ

The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?

A

a = 0, b = 7

B

a = 5, b = 2

C

a = 1, b = 6

D

a = 3, b = 4

Question 104: Palette 104/112

JEE Mains 2007 MCQ

The average marks of boys in a class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is

A

80

B

60

C

40

D

20

Question 105: Palette 105/112

JEE Mains 2006 MCQ

Suppose a population A has 100 observations 101, 102,........, 200, and another population B has 100 observations 151, 152,......., 250. If V_A and V_B represent the variances of the two populations, respectively, then ${{{V_A}} \over {{V_B}}}$ is

A

1

B

${9 \over 4}$

C

${4 \over 9}$

D

${2 \over 3}$

Question 106: Palette 106/112

JEE Mains 2005 MCQ

If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately :

A

20.5

B

22.0

C

24.0

D

25.5

Question 107: Palette 107/112

JEE Mains 2005 MCQ

Let x₁, x₂,...........,x_n be n observations such that

$\sum {x_i^2} = 400$ and $\sum {{x_i}} = 80$. Then a possible value of n among the following is

A

18

B

15

C

12

D

9

Question 108: Palette 108/112

JEE Mains 2004 MCQ

Consider the following statements:
(a) Mode can be computed from histogram
(b) Median is not independent of change of scale
(c) Variance is independent of change of origin and scale.
Which of these is/are correct?

A

only (a)

B

only (b)

C

only (a) and (b)

D

(a), (b) and (c)

Question 109: Palette 109/112

JEE Mains 2004 MCQ

In a series of 2n observations, half of them equal $a$ and remaining half equal $–a$. If the standard deviation of the observations is 2, then $|a|$ equals

A

2

B

$\sqrt 2 $

C

${1 \over n}$

D

${{\sqrt 2 } \over n}$

Question 110: Palette 110/112

JEE Mains 2003 MCQ

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set :

A

is increased by 2

B

is decreased by 2

C

is two times the original median

D

remains the same as that of the original set

Question 111: Palette 111/112

JEE Mains 2003 MCQ

In an experiment with 15 observations on $x$, then following results were available:
$\sum {{x^2}} = 2830$, $\sum x = 170$
One observation that was 20 was found to be wrong and was replaced by the correct value 30. Then the corrected variance is :

A

188.66

B

177.33

C

8.33

D

78.00

Question 112: Palette 112/112

JEE Mains 2002 MCQ

In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?

A

73

B

65

C

68

D

74