Standard Deviation about Mean

Q. If the mean and variance of the following data :
$6,10,7,13,a,12,b,12$ are $9$ and $\frac{37}{4}$ respectively, then $(a-b{)}^2$ is equal to

1. $32$
2. $12$
3. $24$
4. $16$

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

1. $7$
2. $1$
3. $11$
4. $9$

Q.

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 $b$?

1. $a=0,b=7$

2. $a=5,b=2$

3. $a=1,b=6$

4. $a=3,b=4$

Q. Let the mean and variance of the frequency distribution
$\begin{array}{ccccc}x:& x_1=2& x_2=6& x_3=8& x_4=9\\ f:& 4& 4& \alpha & \beta \end{array}$
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. $5$
2. $4$
3. $\frac{17}{3}$
4. $\frac{16}{3}$

Q. If the data $x_1,x_2,....,x_{10}$ 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 $2000,$ then the standard deviation of the data is

1. $4$
2. $2$
3. $2\sqrt{2}$
4. $\sqrt{2}$

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

1. $380$
2. $400$
3. $480$
4. $525$

Q. 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

1. $(11,25)$
2. $(11,26)$
3. $(10.5,25)$
4. $(10.5,26)$

Q. 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

1. $\frac{536}{25}$
2. $\frac{134}{5}$
3. $\frac{112}{5}$
4. $\frac{92}{5}$

Q. If the variance of $10$ natural numbers $1,1,1,\dots ,1,k$ is less than $10,$ then the maximum possible value of $k$ is

Q. The means of two samples of size $40$ and $50$ were found to be $54$ and $63$ respectively. Their standard deviations were $6$ and $9$ respectively. The variance of the combined sample of size $90$ is

1. $9$
2. $81$
3. $7$
4. $90$

Q. The mean of $6$ distinct observations is $6.5$ and their variance is $\mathrm{10.25.}$ If $4$ out of $6$ observations are $2,4,5$ and $7,$ then the remaining two observations are

1. $1,20$
2. $10,11$
3. $3,18$
4. $8,13$

Q.

A student is allowed to select at most n books from a collection of $(2n+1)$ books. If the total number of ways in which he can select at least one book is $63$, then the value of n is equal to

1. $6$

2. $5$

3. $4$

4. $3$

Q. 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. $5$
2. $6$
3. $4$
4. $8$

Q. 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 :

1. $4:9$
2. $5:8$
3. $10:3$
4. $6:7$

Q. 5 students of a class have an average height $150\text{cm}$ and variance $18{\text{cm}}^2$. A new student, whose height is $156\text{cm}$, joined them. The variance $\left(\text{in}{\text{cm}}^2\right)$ of the height of these six students is:

1. $20$
2. $16$
3. $18$
4. $22$

Q. 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

Q. The variance of data $1001,1003,1006,1007,1009,1010$ is

1. $20$
2. $50$
3. $10$
4. $15$

Q.

The mean and standard deviation of 6 observations are 8 and 4 respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.

Q. If the standard deviation of the numbers $-1,0,1,k$ is $\sqrt{5}$, then $k$ is equal to:

1. $\sqrt{6}$
2. 2$\sqrt{\frac{10}{3}}$
3. 2$\sqrt{6}$
4. 4$\sqrt{\frac{5}{3}}$

Q. The average marks of $10$ students in a class was $60$ with a standard deviation of $4,$ while the average marks of other ten students was $40$ with a standard deviation of $6.$ If all the $20$ students are taken together and $\sigma$ is the combined standard deviation, then the value of $\left[\sigma \right]$ is
($[\cdot ]$ represents the greatest integer function)

Q. The variance of $20$ observations is $5$. If each observation is multiplied by $2,$ then the new variance of the resulting observations, is

Q. If the arithmetic mean of two positive numbers a & b (a > b) is twice their G.M., then a:b is

1. 
2. 
3. 
4. 

Q.

If $x_1,x_2,...,x_{18}$ are observations such that $\sum _{j=1}^{18}\left(x_j-8\right)=9$ and $\sum _{j=1}^{18}{\left(x_j-8\right)}^2=45$, then the standard deviation of these observations is

1. $\sqrt{\frac{81}{34}}$

2. $5$

3. $\sqrt{5}$

4. $\frac{3}{2}$

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

1. $3a^2-32a+84=0$
2. $3a^2-34a+91=0$
3. $3a^2-23a+44=0$
4. $3a^2-26a+55=0$

Q. 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 diffrence of the remaining two observations is

1. $3$
2. $9$
3. $7$
4. $5$

Q. If the data $x_1,x_2,....,x_{10}$ 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. $4$
2. $2$
3. $2\sqrt{2}$
4. $\sqrt{2}$

Q. 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 :

1. $1$
2. $3$
3. $5$
4. $7$

Q. The variance of first $20$ natural number is

1. $\frac{279}{12}$
2. $\frac{133}{2}$
3. $\frac{399}{4}$
4. $\frac{133}{4}$

Q.

The standard deviation of 4 consecutive numbers which are in A.P is $\sqrt{5}$. The common difference (d) of this A.P is

1. $\pm \sqrt{5}$

2. $\pm 2\sqrt{5}$

3. $\pm 2$
   

4. $\pm \sqrt{2}$

Q. Standard Deviation of $n$ observation $a_1,a_2,a_3,........,a_n$ is $\sigma .$ Then the standard deviation of the observation $\lambda a_1,\lambda a_2,........,\lambda a_n$ is

1. $\lambda \sigma$
2. $\lambda \sigma$
3. $|\lambda |\sigma$
4. ${\lambda }^n\sigma$