Standard Deviation about Mean

Q. 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 option(s) is/are

1. new mean $=10.2$
2. new mean $=10.1$
3. new variance $=3.99$
4. new variance $=4.01$

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

Q. Mean of five observations is $4.4$ and variance is $8.24$. If three of the observations are $1,2$ and $6$, then the other two observations are

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

Q.

The standard deviation of the first n natural numbers is

1. $\frac{n+1}{2}$
   

2. $\sqrt{\frac{n(n+1)}{2}}$
   

3. $\sqrt{\frac{n^2-1}{12}}$
   

4. none of these

Q. Let $x_1,x_2,\dots ,x_{100}$ be $100$ observations such that $\sum _{i=1}^{100}{x}_i=0,$ $\sum _{1\le i<j\le 100}|x_i{x}_j|=80000$ and mean deviation from their mean be $5.$ Then their standard deviation is

1. $10$
2. $30$
3. $40$
4. $50$

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. $7$
3. $90$
4. $81$

Q. Mean of five observations is $4.4$ and variance is $8.24$. If three of the observations are $1,2$ and $6$, then the other two observations are

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

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.

Standard deviation for $x_1,x_2,.....x_n$ about mean can be expressed as $\sqrt{\frac{1}{n}{\sum }_{i=1}^n(x_i-\overline{x}{)}^2}$.

1. True

2. False

Q. Consider the first $10$ positive integers, if we multiply each number by $-1$ and then add $1$ to each number, then standard deviation of the numbers, so obtained is approximately equal to

1. $6.5$
2. $3.87$
3. $2.87$
4. $8.25$

Q. Mean and variance of $20$ observations are $10$ and $4$, respectively. It was found, that in place of $11,9$ was taken by mistake, then correct variance is

1. $3.99$
2. $3.98$
3. $4.01$
4. $4.02$

Q.

Which of the following are true statements.

1. Variance gives a better picture of dispersion or scatter compared to mean deviation

2. For given n observation x₁, x₂, ... x_n, with mean variance will be ${\sum }_{i=1}^n$ ${(x_i-\overline{x})}^2$

3. Variance can be zero even if all observations are not equal.

4. Variance can be a negative quantity if all the observations are negative

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

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

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

Q.

If variance of first $n$ natural number is $10$ and variance of first $m$ even natural number is $16$, then the value of $m+n$ is