Statistics · JEE Main previous year questions

Let the mean and variance of four numbers $3,7,x$ and $y$ ($x>y$) be $5$ and $10$ respectively. Then the mean of four numbers $3+2x$, $7+2y$, $x+y$ and $x-y$ is ________.

Let the mean and variance of the frequency distribution $$\begin{array}{lcccc}x:& x_1=2& x_2=6& x_3=8& x_4=9\\ f:& 4& 4& \alpha & \beta \end{array}$$ be 6 and 6.8 respectively. If $x_3$ is changed from 8 to 7, then the mean for the new data will be :

1. A.4
2. B.5
3. C.$\frac{17}{3}$
4. D.$\frac{16}{3}$

Consider the following frequency distribution : $$\begin{array}{lccccc}\text{Class :}& 10\text{-}20& 20\text{-}30& 30\text{-}40& 40\text{-}50& 50\text{-}60\\ \text{Frequency :}& \alpha & 110& 54& 30& \beta \end{array}$$ If the sum of all frequencies is 584 and median is 45, then $|\alpha -\beta |$ is equal to ____.

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 :

1. A.8
2. B.6
3. C.4
4. D.5

Consider the following frequency distribution : | Class : | 0-6 | 6-12 | 12-18 | 18-24 | 24-30 | |---|---|---|---|---|---| | Frequency : | a | b | 12 | 9 | 5 | If $\text{mean}=\frac{309}{22}$ and $\text{median}=14$, then the value $(a-b{)}^2$ is equal to ________.

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 :

1. A.10, 11
2. B.3, 18
3. C.8, 13
4. D.1, 20

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 the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is _________.

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 $a^2+b^2$ is equal to :

1. A.425
2. B.250
3. C.925
4. D.650

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?

1. A.$b^2=3(a^2+c^2)+9d^2$
2. B.$b^2=3(a^2+c^2)-9d^2$
3. C.$b^2=3(a^2+c^2+d^2)$
4. D.$b^2=a^2+c^2+3d^2$

Let $X_1,X_2,\dots \dots ,X_{18}$ be eighteen observations such that ${\sum }_{i=1}^{18}(X_i-\alpha )=36$ and ${\sum }_{i=1}^{18}(X_i-\beta {)}^2=90$, where $\alpha$ and $\beta$ are distinct real numbers. If the standard deviation of these observations is 1, then the value of $|\alpha -\beta |$ is _________.

If ${\sum }_{i=1}^n(x_i-a)=n$ and ${\sum }_{i=1}^n(x_i-a{)}^2=na$ $(n,a>1)$ then the standard deviation of $n$ observations $x_1,x_2,\dots ,x_n$ is :

1. A.$a-1$
2. B.$n\sqrt{a-1}$
3. C.$\sqrt{n(a-1)}$
4. D.$\sqrt{a-1}$

Consider the data on x taking the values $0,2,4,8,\dots ,2^n$ with frequencies ${}^n{C}_0,{}^n{C}_1,{}^n{C}_2,\dots ,{}^n{C}_n$ respectively. If the mean of this data is $\frac{728}{2^n}$, then n is equal to ______ .

For the frequency distribution : Variate ($x$) : $x_1{x}_2{x}_3\dots x_{15}$ Frequency ($f$) : $f_1{f}_2{f}_3\dots f_{15}$ where $0<x_1<x_2<x_3<\cdots <x_{15}=10$ and ${\sum }_{i=1}^{15}{f}_i>0$, the standard deviation cannot be :

1. A.6
2. B.1
3. C.4
4. D.2

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

1. A.$\frac{7}{10}$
2. B.$\frac{9}{10}$
3. C.$\frac{4}{5}$
4. D.$\sqrt{\frac{3}{5}}$

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

1. A.7
2. B.9
3. C.-7
4. D.-27

If the variance of the terms in an increasing A.P., $b_1,b_2,b_3,\dots ,b_{11}$ is 90, then the common difference of this A.P. is _______.

Let the observations $x_i(1\le i\le 10)$ satisfy the equations, ${\sum }_{i=1}^{10}(x_i-5)=10$ and ${\sum }_{i=1}^{10}(x_i-5{)}^2=40$. If $\mu$ and $\lambda$ are the mean and the variance of the observations, $x_1-3,x_2-3,\dots ,x_{10}-3$, then the ordered pair $(\mu ,\lambda )$ is equal to :

1. A.$(6,6)$
2. B.$(3,3)$
3. C.$(3,6)$
4. D.$(6,3)$

The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. 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

1. A.10
2. B.-20
3. C.-10
4. D.-5

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

1. A.3.98
2. B.3.99
3. C.4.01
4. D.4.02

If the variance of the first n natural numbers is 10 and the variance of the first m even natural numbers is 16, then m + n is equal to_____.

If the mean and variance of eight numbers 3, 7, 9, 12, 13, 20, x and y be 10 and 25 respectively, then x.y is equal to _______.

If the data $x_1,x_2,\dots ,x_{10}$ 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 :

1. A.$\sqrt{2}$
2. B.2
3. C.$2\sqrt{2}$
4. D.4

Let a random variable X have a binomial distribution with mean 8 and variance 4. If $P(X\le 2)=\frac{k}{2^{16}}$, then k is equal to :

1. A.17
2. B.1
3. C.137
4. D.121

If for some $x\in \mathrm{R}$, the frequency distribution of the marks obtained by 20 students in a test is : | Marks | 2 | 3 | 5 | 7 | | Frequency | $(x+1{)}^2$ | $2x-5$ | $x^2-3x$ | $x$ | then the mean of the marks is

1. A.3.0
2. B.2.8
3. C.2.5
4. D.3.2

If both the mean and the standard deviation of 50 observations $x_1,x_2,\dots ,x_{50}$ are equal to 16, then the mean of $(x_1-4{)}^2,(x_2-4{)}^2,\dots ,(x_{50}-4{)}^2$ is :

1. A.400
2. B.480
3. C.380
4. D.525