Standard Deviation about mean

Q.

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

1. $\frac{7}{10}$

2. $\frac{9}{10}$

3. $\sqrt{\frac{3}{5}}$

4. $\frac{4}{5}$

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.

A scientist is weighing each of 30 fishes. Their mean weight worked out is 30 gm and a standard deviaiton of 2 gm. Later, it was found that the measuring scale was misaligned and always under-reported every fish weight by 2gm. The correct mean and standard deviation (in gm) of fishes are respectively:

1. $32,2$

2. $32,4$

3. $28,2$

4. $28,4$

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.

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.

For a frequency distribution consisting of 18 observations, the mean and the standard deviation were found to be 7 and 4 respectively. But on comparison with the original data, it was found that a figure 12 was miscopied as 21 in calculations. Find the correct mean and standard deviation.___

Q. The standard deviation of 4, 7, 10, 13, 16, 19, 22 is

1. 2
2. 6
3. 8
4. 16

Q.

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.

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

2. $\sqrt{2}$

3. 2

4. $\frac{1}{n}$

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.

The standard deviation of the first n natural numbers is

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

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

3. none of these

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

Q.

The standard deviation of a variate x is $\sigma$. Then the standard deviation of the variate $\frac{ax+b}{c}$ where a, b, c are constants, is

1. $\left(\frac{a}{c}\right)\sigma$
   

2. $\left|\frac{a}{c}\right|\sigma$
   

3. $\left(\frac{a^2}{c^2}\right)\sigma$
   

4. none of these