141- Referring to the frequency histogram below, what is the total variance of the data?
\begin{minipage}{0.45\textwidth} [Figure: Frequency histogram with x-axis values 3, 5, 7, 9, 11 and y-axis (frequency) values up to 8, showing a roughly triangular distribution with peak around x=7] \end{minipage} \begin{minipage}{0.45\textwidth} (1) $4.5$
(2) $4.8$
(3) $4.92$
(4) $5.12$ \end{minipage}
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142. The data $x_i = 1, 2, 3, 4, 5$ are given. Suppose the data are transformed by $u_i = 12x_i + 6$. What is $\bar{u}_i$?
(1) $0/4$ (2) $0/48$ (3) $0/52$ (4) $0/6$

141. We represent the statistical data with stem-and-leaf plot as shown below. We show the variance of the data inside the box. Which is it?

Stem	\multicolumn{4}{l}{Leaf}
2	5	6	7	9
3	1	3	4	5 6
4	$\circ$	1	2	4

(1) $9/25$ (2) $9/75$ (3) $10/15$ (4) $10/85$

142. A population with mean $12$ and variance $12/6$, and another population with mean $24$ and variance $7/2$, form a new combined population. If the two populations have equal size, what is the standard deviation of the new population?
(1) $7/9$ (2) $3$ (3) $3/1$ (4) $3/2$

142-- In the following frequency table, what is the standard deviation using the quick method?

$x$	27	29	31	33	35
$f$	7	15	13	11	9

(1) $2.6$ (2) $2.7$ (3) $2.8$ (4) $2.9$

140. The mathematics scores of 40 students of a class are given in the following table. What is the weighted mean of the scores?

$x$	10	12	14	15	17	18
$f$	5	8	7	10	6	4

(1) $14.2$ (2) $14.25$ (3) $14.4$ (4) $14.75$

141. The unemployment rate of a country over 10 years is given below. What is the value of $\dfrac{Q_1 + Q_3 - 2Q_2}{Q_3 - Q_1}$?
\fbox{$11.5,\ 12.8,\ 13.5,\ 11.2,\ 12.3,\ 12.6,\ 11.9,\ 10.6,\ 10.2,\ 30,\ 12.7$}
(1) $-0.225$ (2) $-0.125$ (3) $0.175$ (4) $0.275$

147 -- Referring to the relative frequency histogram of grouped quantitative data, what is the mean?
[Figure: A relative frequency histogram with data values 7, 12, 13, 17, 19 on the horizontal axis and relative frequencies approximately 12, 35, 18, 25, 10 on the vertical axis]

• [(1)] $13$
• [(2)] $13.8$
• [(3)] $14$
• [(4)] $14.2$

128- In the frequency table below, the median is $13.5$ and the first quartile minus the third quartile is $17$. If we add $4$ units to each data value in the table, what is the new variance?

Data	11	12	13	14	28	31	$a$
Frequency	3	2	6	3	2	5	1

(1) $71$ (2) $71.5$ (3) $72$ (4) $72.5$

128. To estimate the mean income of individuals in a community, we randomly select two samples. We use the standard deviation of the second sample as an estimate for the mean of the first sample, which equals $\frac{2}{\overline{x}}$ times the calculated value for the first sample. The size of the second sample is how many times the size of the first sample?
(1) $1/5$ (2) $2/25$ (3) $2/75$ (4) $3/5$

129. The mean of six statistical data is a natural number, and the variance of these data is $1$, $9$, $b^2$, $5$, $\pi^2$, $9$. If the variance of these data equals $4$, what is the value of $ab$? $(a, b \in \mathbb{Z})$
(1) $-4$ (2) $4$ (3) $2$ (4) $-2$
*130. In isosceles triangle $ABC$, point $M$ is the midpoint of $AB$, and the perpendicular bisector of $AB$ cuts side $AC$ at point $N$. If $\widehat{NBC} = 54°$, what is the measure of angle $\widehat{MNB}$?
(1) $48$ (2) $56$ (3) $66$ (4) $78$

23 -- The integers from 9 to 19 are chosen at random. Two numbers are removed from these numbers and replaced by their difference. This process continues until all numbers are even, non-repeating, and the mean is as large as possible. What is the standard deviation of the new data?
(1) $\sqrt{10}$ (2) $\sqrt{11}$ (3) $\sqrt{21}$ (4) $\sqrt{28}$

Nine students in a class gave a test for 50 marks. Let $S _ { 1 } \leq S _ { 2 } \leq \cdots \leq S _ { 5 } \leq \cdots \leq S _ { 8 } \leq S _ { 9 }$ denote their ordered scores. Given that $S _ { 1 } = 20$ and $\sum _ { i = 1 } ^ { 9 } S _ { i } = 250$, let $m$ be the smallest value that $S _ { 5 }$ can take and $M$ be the largest value that $S _ { 5 }$ can take. Then the pair $( m , M )$ is given by
(A) $( 20,35 )$
(B) $( 20,34 )$
(C) $( 25,34 )$
(D) $( 25,50 )$.

If the $n$ terms $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$ are in arithmetic progression with increment $r$, then the difference between the mean of their squares and the square of their mean is
(A) $\frac { r ^ { 2 } \left( ( n - 1 ) ^ { 2 } - 1 \right) } { 12 }$
(B) $\frac { r ^ { 2 } } { 12 }$
(C) $\frac { r ^ { 2 } \left( n ^ { 2 } - 1 \right) } { 12 }$
(D) $\frac { n ^ { 2 } - 1 } { 12 }$

Consider the given data with frequency distribution

$x _ { i }$	3	8	11	10	5	4
$f _ { i }$	5	2	3	2	4	4

Match each entry in List-I to the correct entries in List-II.
List-I
(P) The mean of the above data is
(Q) The median of the above data is
(R) The mean deviation about the mean of the above data is
(S) The mean deviation about the median of the above data is
List-II
(1) 2.5
(2) 5
(3) 6
(4) 2.7
(5) 2.4
The correct option is:
(A) $( P ) \rightarrow ( 3 )$ $( Q ) \rightarrow ( 2 )$ $( R ) \rightarrow ( 4 )$ $( S ) \rightarrow ( 5 )$
(B) $( P ) \rightarrow ( 3 )$ $( Q ) \rightarrow ( 2 )$ $( R ) \rightarrow ( 1 )$ $( S ) \rightarrow ( 5 )$
(C) $( P ) \rightarrow ( 2 )$ $( Q ) \rightarrow ( 3 )$ $( R ) \rightarrow ( 4 )$ $( S ) \rightarrow ( 1 )$
(D) $( P ) \rightarrow ( 3 )$ $( Q ) \rightarrow ( 3 )$ $( R ) \rightarrow ( 5 )$ $( S ) \rightarrow ( 5 )$

Let $X$ be a random variable, and let $P ( X = x )$ denote the probability that $X$ takes the value $x$. Suppose that the points $( x , P ( X = x ) ) , x = 0,1,2,3,4$, lie on a fixed straight line in the $x y$-plane, and $P ( X = x ) = 0$ for all $x \in \mathbb { R } - \{ 0,1,2,3,4 \}$. If the mean of $X$ is $\frac { 5 } { 2 }$, and the variance of $X$ is $\alpha$, then the value of $24 \alpha$ is $\_\_\_\_$ .

Consider the following frequency distribution:

Value	4	5	8	9	6	12	11
Frequency	5	$f _ { 1 }$	$f _ { 2 }$	2	1	1	3

Suppose that the sum of the frequencies is 19 and the median of this frequency distribution is 6. For the given frequency distribution, let $\alpha$ denote the mean deviation about the mean, $\beta$ denote the mean deviation about the median, and $\sigma ^ { 2 }$ denote the variance.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I

(P) $7 f _ { 1 } + 9 f _ { 2 }$ is equal to (Q) $19 \alpha$ is equal to (R) $19 \beta$ is equal to (S) $19 \sigma ^ { 2 }$ is equal to

List-II

(1) 146
(2) 47
(3) 48
(4) 145
(5) 55

(A)	$( \mathrm { P } ) \rightarrow ( 5 )$	$( \mathrm { Q } ) \rightarrow ( 3 )$	$( \mathrm { R } ) \rightarrow ( 2 )$	$( \mathrm { S } ) \rightarrow ( 4 )$
(B)	$( \mathrm { P } ) \rightarrow ( 5 )$	$( \mathrm { Q } ) \rightarrow ( 2 )$	$( \mathrm { R } ) \rightarrow ( 3 )$	$( \mathrm { S } ) \rightarrow ( 1 )$
(C)	$( \mathrm { P } ) \rightarrow ( 5 )$	$( \mathrm { Q } ) \rightarrow ( 3 )$	$( \mathrm { R } ) \rightarrow ( 2 )$	$( \mathrm { S } ) \rightarrow ( 1 )$
(D)	$( \mathrm { P } ) \rightarrow ( 3 )$	$( \mathrm { Q } ) \rightarrow ( 2 )$	$( \mathrm { R } ) \rightarrow ( 5 )$	$( \mathrm { S } ) \rightarrow ( 4 )$

The average marks of boys in a class is 52 and that of girls is 42 . The average marks of boys and girls combined is 50 . The percentage of boys in the class is
(1) 40
(2) 20
(3) 80
(4) 60

If the mean deviation about the median of the numbers $\mathrm{a},2\mathrm{a},\ldots,50\mathrm{a}$ is 50, then $|\mathrm{a}|$ equals
(1) 3
(2) 4
(3) 5
(4) 2

If the mean of $4, 7, 2, 8, 6$ and $a$ is 7, then the mean deviation from the median of these observations is
(1) 8
(2) 5
(3) 1
(4) 3

The frequency distribution of daily working expenditure of families in a locality is as follows:

\begin{tabular}{ c } Expenditure

in ₹. $( x )$ :

& $0 - 50$ & $50 - 100$ & $100 - 150$ & $150 - 200$ & $200 - 250$ \hline

No. of

families $( f )$ :

& 24 & 33 & 37 & $b$ & 25 \hline \end{tabular}
If the mode of the distribution is Rs. 140, then the value of $b$ is
(1) 34
(2) 31
(3) 26
(4) 36

Statement 1: The variance of first $n$ odd natural numbers is $\frac { n ^ { 2 } - 1 } { 3 }$ Statement 2: The sum of first $n$ odd natural numbers is $n ^ { 2 }$ and the sum of squares of first $n$ odd natural numbers is $\frac { n \left( 4 n ^ { 2 } + 1 \right) } { 3 }$.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.
(3) Statement 1 is false, Statement 2 is true.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.

Let $x_{1}, x_{2}, \ldots, x_{n}$ be $n$ observations, and let $\bar{x}$ be their arithmetic mean and $\sigma^{2}$ be their variance. Statement 1: Variance of $2x_{1}, 2x_{2}, \ldots, 2x_{n}$ is $4\sigma^{2}$. Statement 2: Arithmetic mean of $2x_{1}, 2x_{2}, \ldots, 2x_{n}$ is $4\bar{x}$.
(1) Statement 1 is false, Statement 2 is true
(2) Statement 1 is true, Statement 2 is false
(3) Statement 1 is true, Statement 2 is the correct explanation for Statement 1
(4) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1

If the median and the range of four numbers $\{ x , y , 2 x + y , x - y \}$, where $0 < y < x < 2 y$, are 10 and 28 respectively, then the mean of the numbers is :
(1) 18
(2) 10
(3) 5
(4) 14

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) mode
(2) variance
(3) mean
(4) median