Standard Deviation

Q. The variance of first $50$ even natural numbers is:

1. $\frac{833}{4}$
2. $833$
3. $437$
4. $\frac{437}{4}$

Q. If the arithmetic mean and harmonic mean between two numbers are $27$ and $12$ respectively, then the geometric mean of those two numbers is

1. $14$
2. $18$
3. $24$
4. $9$

Q. Let $\alpha$ and $\beta$ be two numbers where $\alpha <\beta .$ The geometric mean of these numbers exceeds the smaller number $\alpha$ by $12$ and the arithmetic mean of the same numbers is smaller by $24$ than the larger number $\beta .$ Then the value of $|\beta -\alpha |$ is

1. $36$
2. $48$
3. $6$
4. $30$

Q. For the given data
$36,38,33,34,32,30,34,35,65$
Which among the following has highest value

1. mode
2. mean
3. median
4. range

Q. Let ${\sigma }_1^2$ be the variance of the dataset $\{1,4,7,10,\dots ,301\}$ and ${\sigma }_2^2$ be the variance of the dataset $\{7,13,19,25,\dots ,607\}$. Then the value of $\frac{{\sigma }_2^2-{\sigma }_1^2}{{\sigma }_1^2}$ is

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

1. $\frac{9}{8}$
2. $2$
3. $\frac{7}{8}$
4. $1$

Q. If $S_1,S_2,S_3\dots \dots ,S_n,\dots$ are the sums of infinite geometric series whose first terms are $1,2,3,\dots \dots n,\dots \dots$ and whose common ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots \dots \frac{1}{n+1}\dots \dots$ respectively, then the value of $\sum _{r=1}^{2n-1}{S}_r^2$ is

1. $\frac{n(n+1)(2n+1)}{2}-1$
2. $\frac{n(n+1)(n+2)}{2}-1$
3. $\frac{n(2n+1)(4n+1)}{3}-1$
4. $\frac{n(2n+1)(4n+1)}{3(n+1)}-1$

Q.

Let $S$ be the standard deviation of $n$ observations. Each of the $n$ observations is multiplied by a constant $C$. Then the standard deviation of the resulting number is

1. S

2. CS

3. $S\sqrt{C}$

4. none of these

Q.

The mean square deviations of a set of observations $x_1,x_2,\cdots ,x_n$about a point c is defined to be $\frac{1}{n}{\sum }_{i=1}^n(x_i-c{)}^2$
The mean square deviations about -1 and +1 of a set of observations are 7 and 3, respectively. Find the standard deviation of this set of observations.

1. $\sqrt{3}$

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

Q. What is the variance of the following data ?
$$\begin{array}{cc}\text{Value}& \text{Frequency}\\ 4& 1\\ 2& 2\\ 0& 2\\ -2& 2\\ -4& 1\end{array}$$

1. 6
2. 3
3. 5
4. \N

Q.

The variance of first 50 even natural numbers is

1. $\frac{833}{4}$

2. 833

3. 437

4. $\frac{437}{4}$

Q. The mean and 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 standard deviation become half of their original values, then $q$ is equal to:

1. $-20$
2. $-5$
3. $10$
4. $-10$

Q.

The standard deviation of 25 numbers is 40. If each of the numbers is increased by 5, then the new standard deviation will be

1. 40

2. 45

3. $40+\frac{21}{25}$

4. none of these

Q. In a data set consisting of 10 values
$x_1,x_2,....,x_{10}$ such that $-10\le x_i\le 10$ with mean 0, what can be the maximum standard deviation?

1. 10
2. 100
3. 81
4. Insufficient data

Q.

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 $\frac{V_A}{V_B}$ is

1. 1

2. $\frac{9}{4}$

3. $\frac{4}{9}$

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

Q. If the standard deviation of $0,1,2,\mathrm{3...9}$ is $K$, then the standard deviation of $10,11,12,\mathrm{13...19}$ is

1. $K+10$
2. $K$
3. $\sqrt{10}+K$
4. $10K$

Q. Which has more standard deviation?
Set 1 : {10, 20, 30, 40, 50} or
Set 2 : {1, 2, 3, 4, 5}

1. Set 1
2. Set 2
3. Equal

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

Q.

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, respectivley. The variance of the combined data set is

1. $\frac{13}{2}$

2. $\frac{5}{2}$

3. $\frac{11}{2}$

4. 6

Q.

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?

1. Mean

2. Median

3. Mode

4. Variance

Q.

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

1. $3a^2–26a+55=0$

2. $3a^2–32a+84=0$

3. $3a^2–34a+91=0$

4. $3a^2–23a+44=0$

Q.

Which of the following gives a description about standard and mean deviation.

1. measures of dispersion which depends upon the deviations of observations from central tendency

2. measures of central tendency of the given observations

3. maximum and minimum values taken

4. all of the above

Q. If the standard deviation of $X$ is $\sigma$, then s.d. of the variable $U=\frac{aX+b}{c}$ where $a,b,c$ are constants is

1. $\left|\frac{c}{a}\right|\sigma$
2. $\left|\frac{a}{c}\right|\sigma$
3. $\left|\frac{b}{c}\right|\sigma$
4. $\frac{c^2}{a^2}\sigma$

Q. 3 out of 5 observations of a data set are known to be 10, 10, 10. The mean and the variance of this data are both 10. Find the remaining values.

1. 10 , 10
2. 5, 15
3. 2, 18
4. 12, 8

Q.

Let $x_1,x_2,\cdots ,x_n$ be n observations, and let $\overline{x}$ be their arithmetic mean and ${\sigma }^2$ be the variance.
Statement – 1: Variance of $2x_1,2x_2,\cdots ,2x_n\text{is}4{\sigma }^2$
Statement – 2: Arithmetic mean $2x_1,2x_2,\cdots ,2x_n\text{is}4\overline{x}$

1. Statement – 1 is false, Statement – 2 is true

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

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

4. Statement – 1 is true, statement – 2 is false.

Q. If the median and mode of 4 observations is 4, 6 and the sum of squares of observations is 48. Then
standard deviation is

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