Mean Deviation about Mean for Discrete Frequency Distributions

Q. An online exam is attempted by $50$ candidates out of which $20$ are boys. The average marks obtained by boys is $12$ with a variance $2.$ The variance of marks obtained by $30$ girls is also $2.$ The average marks of all $50$ candidates is $15.$ If $\mu$ is the average marks of girls and ${\sigma }^2$ is the variance of marks of 50 candidates, then $\mu +{\sigma }^2$ is equal to

Q.

Arjun and Karna had an archery competition where they had to shoot a target board with 7 concurrent circles dividing the board into 8 concurrent circular strips. Score for each outcome decreases by unity with centermost circle having score of 8. Dronacharya had difficulty in deciding who the winner was after the scores were recorded. Calculate the mean deviation about the score of innermost circle so as to decide who the winner is, assuming the winner is less deviated from the bull's eye.

$\begin{array}{ccccccccc}Points& 8& 7& 6& 5& 4& 3& 2& 1\\ Arjun\left(f_i\right)& 5& 7& 14& 23& 11& 10& 4& 3\\ Karna\left(f_i\right)& 5& 8& 11& 24& 14& 8& 5& 3\end{array}$

1. Arjun

2. Karna

3. Both of them

4. Insufficient data

Q. There are $60$ students in a class. The following is the frequency distribution of the marks obtained by the students in a test

$\begin{array}{cc}\text{Marks}& \text{Frequency}\\ 0& x-2\\ 1& x\\ 2& x^2\\ 3& (x+1{)}^2\\ 4& 2x\\ 5& x+1\end{array}$

Where $x$ is positive integer. Then the variance of the marks is

1. $1.26$
2. $1.28$
3. $1.34$
4. $1.24$

Q. 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 $0\ne p\in R$, 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. Let the observations $x_i$ $(1\le i\le 10)$ satisfy $\sum _{i=1}^{10}(x_i-5)=10$ and $\sum _{i=1}^{10}(x_i-5{)}^2=40$. For the observations $(x_1-3),(x_2-3),...,(x_{10}-3)$, which of the following is/are correct?

1. Mean $=3$
2. Mean $=5$
3. Variance $=3$
4. Variance $=5$

Q. Calculate the mean deviation about mean for the data given here:

Class interval	$5$	$15$	$25$	$35$	$45$
Frequency	$5$	$3$	$9$	$12$	$6$

1. $10.2$
2. $10.4$
3. $10.5$
4. $11.4$

Q. Calculate mean deviation about mean for the given data.

Score $\left(x\right)$	$6$	$20$	$8$	$18$	$16$	$12$	$14$	$10$
Frequency $\left(f\right)$	$2$	$7$	$11$	$27$	$18$	$13$	$17$	$5$

1. $3.117$
2. $3.217$
3. $4.212$
4. $6.21$

Q.

Given are the observations of runs scored by batsmen between India - Sri Lanka combined. Find the mean deviation about mean for the scores of batsmen

$\begin{array}{ccccccc}Score& 0-10& 10-20& 20-30& 30-40& 40-50& 50-60\\ numberofBatsmen& 12& 18& 27& 20& 17& 6\end{array}$

1. 15

2. 99

3. 10

4. 50

Q. Mean deviation for $n$ observations $x_1,x_2,.....x_n$ from their mean $\overline{X}$ is given by:

1. ${\sum }_{i=1}^n(x_i-\overline{X})$
2. $\frac{1}{n}{\sum }_{i=1}^n|x_i-\overline{X}|$
3. ${\sum }_{i=1}^n{{(x}_i-\overline{X})}^2$
4. $\frac{1}{n}{\sum }_{i=1}^n{{(x}_i-\overline{X})}^2$

Q. Coefficient of deviation is calculated by the formula:

1. $\frac{\overline{X}}{\sigma }\times 100$
2. $\frac{\overline{X}}{\sigma }$
3. $\frac{\sigma }{\overline{X}}\times 100$
4. $\frac{\sigma }{\overline{X}}$

Q. If $\sum _{i=1}^9(x_i-5)=9$ and $\sum _{i=1}^9(x_i-5{)}^2=45,$ then the standard deviation of the $9$ items $x_1,x_2,\cdots ,x_9$ is

1. $3$
2. $9$
3. $4$
4. $2$

Q. If the following frequency distribution
$\begin{array}{ccccccc}x& A& 2A& 3A& 4A& 5A& 6A\\ f& 2& 1& 1& 1& 1& 1\end{array}$,
where $A$ is a positive integer has a variance of $160$, then the value of $A$ is

Q. If the mean deviation about mean $1,1+d,1+2d,....1+100d$ from their mean is $255$, then the $d$ is equal to

1. $20$
2. $10$
3. $10.1$
4. $20.2$

Q. The mean deviation about the mean for the following data :
$\begin{array}{cccccccc}{x}_i& 2& 3& 4& 5& 6& 7& 8\\ f_i& 5& 2& 3& 4& 5& 4& 2\end{array}$
is

1. $2.3$
2. $3.4$
3. $4$
4. $1.66$

Q.

Given are the observations of runs scored by batsmen between India - Sri Lanka combined. Find the mean deviation about mean for the scores of batsmen

$\begin{array}{ccccccc}Score& 0-10& 10-20& 20-30& 30-40& 40-50& 50-60\\ numberofBatsmen& 12& 18& 27& 20& 17& 6\end{array}$

1. 15

2. 99

3. 10

4. 50

Q.

Arjun and Karna had an archery competition where they had to shoot a target board with 7 concurrent circles dividing the board into 8 concurrent circular strips. Score for each outcome decreases by unity with centermost circle having score of 8. Dronacharya had difficulty in deciding who the winner was after the scores were recorded. Calculate the mean deviation about the score of innermost circle so as to decide who the winner is, assuming the winner is less deviated from the bull's eye.

$\begin{array}{ccccccccc}Points& 8& 7& 6& 5& 4& 3& 2& 1\\ Arjun\left(f_i\right)& 5& 7& 14& 23& 11& 10& 4& 3\\ Karna\left(f_i\right)& 5& 8& 11& 24& 14& 8& 5& 3\end{array}$

1. Arjun

2. Karna

3. Both of them

4. Insufficient data

Q. Given a number $n$ and an array containing $1$ to $(2n+1)$ consecutive numbers. Three elements are chosen at random. Find the probability that the elements chosen are in A.P.

Q.

Arjun and Karna had an archery competition where they had to shoot a target board with 7 concurrent circles dividing the board into 8 concurrent circular strips. Score for each outcome decreases by unity with centermost circle having score of 8. Dronacharya had difficulty in deciding who the winner was after the scores were recorded. Calculate the mean deviation about the score of innermost circle so as to decide who the winner is, assuming the winner is less deviated from the bull's eye.

$\begin{array}{ccccccccc}Points& 8& 7& 6& 5& 4& 3& 2& 1\\ Arjun\left(f_i\right)& 5& 7& 14& 23& 11& 10& 4& 3\\ Karna\left(f_i\right)& 5& 8& 11& 24& 14& 8& 5& 3\end{array}$

1. Arjun

2. Karna

3. Both of them

4. Insufficient data

Q.

Steven Gerard, arguably one of the most consistent and skilled player in English premier league was given monthly ratings by a Football website. The ratings are given on monthly basis. The following are the ratings for the past 40 months.

$\begin{array}{ccccccc}Rating& 6& 8& 8.5& 9& 7& 10\\ Numberofmonthswiththerating& 2& 8& 10& 7& 8& 5\end{array}$

Calculate how his rating is deviating from his average performance in terms of mean deviation about mean.

1. 0.825

2. 0.85

3. 0.9

4. 1

Q. The mean deviation about the mean for the following data :
$\begin{array}{cccccccc}{x}_i& 2& 3& 4& 5& 6& 7& 8\\ f_i& 5& 2& 3& 4& 5& 4& 2\end{array}$
is

1. $2.3$
2. $3.4$
3. $4$
4. $1.66$

Q. There are $60$ students in a class. The following is the frequency distribution of the marks obtained by the students in a test

$\begin{array}{cc}\text{Marks}& \text{Frequency}\\ 0& x-2\\ 1& x\\ 2& x^2\\ 3& (x+1{)}^2\\ 4& 2x\\ 5& x+1\end{array}$

Where $x$ is positive integer. Then the variance of the marks is

1. $1.26$
2. $1.28$
3. $1.24$
4. $1.34$

Q. If the following frequency distribution
$\begin{array}{ccccccc}x& A& 2A& 3A& 4A& 5A& 6A\\ f& 2& 1& 1& 1& 1& 1\end{array}$
, where $A$ is a positive integer has a variance of $160$, then the value of $A$ is