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The mean of the numbers \( a, b, 8, 5, 10 \) is \( 6 \) and their variance is \( 6.8 \). If \( M \) is the mean deviation of the numbers about the mean, then \( 25M \) is equal to:

The mean and variance of a set of \( 15 \) numbers are \( 12 \) and \( 14 \) respectively. The mean and variance of another set of \( 15 \) numbers are \( 14 \) and \( \sigma^{2} \) respectively. If the variance of all the \( 30 \) numbers in the two sets is \( 13 \), then \( \sigma^{2} \) is equal to:

The variance \( \sigma^2 \) of the data

is _______.

Let \( \mu \) be the mean and \( \sigma \) be the standard deviation of the distribution,

Where \( \sum f_{i} = 62 \). If \( [x] \) denotes the greatest integer \( \le x \), then \( \left[ \mu^{2} + \sigma^{2} \right] \) is equal to:

Let \( a_1, a_2, \ldots, a_{10} \) be \( 10 \) observations such that \( \displaystyle \sum_{K=1}^{10}a_k=50 \quad and \sum_{\forall\, k < j} a_k \cdot a_j = 1100 \). Then the standard deviation of \( a_1, a_2, \ldots, a_{10} \) is equal to:

Let sets \( A \) and \( B \) have \( 5 \) elements each. Let the mean of the elements in sets \( A \) and \( B \) be \( 5 \) and \( 8 \) respectively and the variance of the elements in sets \( A \) and \( B \) be \( 12 \) and \( 20 \) respectively. A new set \( C \) of \( 10 \) elements is formed by subtracting \( 3 \) from each element of \( A \) and adding \( 2 \) to each element of \( B \). Then the sum of the mean and variance of the elements of \( C \) is ________ .

Let \( 9 = x_1 < x_2 < \ldots < x_7 \) be in an A.P. with common difference \( d \). If the standard deviation of \( x_1, x_2, \ldots, x_7 \) is \( 4 \) and the mean is \( \bar{x} \), then \( \bar{x} + x_6 \) is equal to:

The sum and product of the mean and variance of a binomial distribution are \( 82\cdot 5 \) and \( 1350 \) respectively. Then the number of trials in the binomial distribution is __________.

From a lot of \( 10 \) items, which include \( 3 \) defective items, a sample of \( 5 \) items is drawn at random. Let the random variable \( X \) denote the number of defective items in the sample. If the variance of \( X \) is \( \sigma^{2} \), then \( 96\sigma^{2} \) is equal to \( \_\_\_\_\_\_\_\_\_\_\_ \).

10.

Let \( \alpha,\beta \in \mathbb{R} \). Let the mean and the variance of \( 6 \) observations \( -3,4,7,-6,\alpha,\beta \) be \( 2 \) and \( 23 \), respectively. The mean deviation about the mean of these \( 6 \) observations is:

11.

For \( 10 \) observations \( x_{1},x_{2},\ldots,x_{10} \), if and then their standard deviation is:

12.

Let the mean and standard deviation of marks of class \( A \) of \( 100 \) students be respectively \( 40 \) and \( \alpha \ (>\ 0) \), and the mean and standard deviation of marks of class \( B \) of \( n \) students be respectively \( 55 \) and \( 30-\alpha \). If the mean and variance of the marks of the combined class of \( 100+n \) students are respectively \( 50 \) and \( 350 \), then the sum of variances of classes \( A \) and \( B \) is:

13.

The mean and standard deviation of \( 15 \) observations were found to be \( 12 \) and \( 3 \) respectively. On rechecking it was found that an observation was read as \( 10 \) in place of \( 12 \). If \( \mu \) and \( \sigma^{2} \) denote the mean and variance of the correct observations respectively, then \( 15(\mu + \mu^{2} + \sigma^{2}) \) is equal to __________.

14.

The mean and variance of \( 7 \) observations are \( 8 \) and \( 16 \) respectively. If two observation are \( 6 \) and \( 8 \), then the variance of the remaining \( 5 \) observations is:

15.

If the mean and variance of the following data: \( 6, 10, 7, 13, a, 12, b, 12 \) are \( 9 \) and \( \dfrac{37}{4} \) respectively, then \( (a-b)^2 \) is equal to:

16.

A set of four observations has mean \( 1 \) and variance \( 13 \). Another set of six observations has mean \( 2 \) and variance \( 1 \). Then, the variance of all these \( 10 \) observations is equal to:

17.

From a lot of \( 12 \) items containing \( 3 \) defectives, a sample of \( 5 \) items is drawn at random. Let the random variable \( X \) denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of \( X \) is \( \dfrac{m}{n} \), where \( \gcd(m,n)=1 \), then \( n-m \) is equal to ______.

18.

The variance of the numbers \( 8, 21, 34, 47, \ldots, 320 \) is __________.

19.

If the mean deviation about the median of the numbers \( k, 2k, 3k, \ldots, 1000k \) is \( 500 \), then \( k^{2} \) is equal to:

20.

The mean and standard deviation of \( 15 \) observations are found to be \( 8 \) and \( 3 \) respectively. On rechecking it was found that, in the observations, \( 20 \) was misread as \( 5 \). Then, the correct variance is equal to ____________.

21.

Let in a series of \( 2n \) observations, half of them are equal to \( a \) and remaining half are equal to \( -a \). Also by adding a constant \( b \) in each of these observations, the mean and standard deviation of new set becomes \( 5 \) and \( 20 \), respectively. Then the value of \( a^2 + b^2 \) is equal to:

22.

If the variance of \( 10 \) natural numbers \( 1, 1, 1, \ldots, 1, k \) is less than \( 10 \), then the maximum possible value of \( k \) is ________.

23.

Let \( n \) be an odd natural number such that the variance of \( 1, 2, 3, 4, \ldots, n \) is \( 14 \). Then \( n \) is equal to __________.

24.

Let the mean and the variance of \( 6 \) observations \( a, b, 68, 44, 48, 60 \) be \( 55 \) and \( 194 \), respectively. If \( a > b \), then \( a + 3b \) is:

25.

Let the median and the mean deviation about the median of \( 7 \) observation \( 170, 125, 230, 190, 210, a, b \) be \( 170 \) and \( \dfrac{205}{7} \) respectively. Then the mean deviation about the mean of these \( 7 \) observations is:

26.

Let \( X = \{ x \in N : 1 \le x \le 19 \} \) and for some \( a, b \in R \), \( Y = \{ ax + b : x \in X \} \). If the mean and variance of the elements of \( Y \) are \( 30 \) and \( 750 \), respectively, then the sum of all possible values of \( b \) is:

27.

A data consists of \( 20 \) observations \( x_{1}, x_{2}, \ldots, x_{20} \). If \( \sum_{i=1}^{20} (x_{i}+5)^{2} = 2500 \) and \( \sum_{i=1}^{20} (x_{i}-5)^{2} = 100 \), then the ratio of mean to standard deviation of this data is:

28.

If the mean and variance of the data \( 65, 68, 58, 44, 48, 45, 60, \alpha, \beta, 60 \) where \( \alpha>\beta \), are \( 56 \) and \( 66.2 \) respectively, then \( \alpha^{2}+\beta^{2} \) is equal to __________.

29.

If the mean and variance of the frequency distribution

are \( 9 \) and \( 15.08 \) respectively, then the value of \( \alpha^{2}+\beta^{2}-\alpha\beta \) is __________.

30.

The mean age of \( 25 \) teachers in a school is \( 40 \) years. A teacher retires at the age of \( 60 \) years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is \( 39 \) years, then the age (in years) of the newly appointed teacher is ____________ .

31.

The frequency distribution of the age of students in a class of \( 40 \) students is given below.

If the mean deviation about the median is \( 1.25 \), then \( 4x + 5y \) is equal to:

32.

The mean and variance of a data of \( 10 \) observations are \( 10 \) and \( 2 \), respectively. If an observations \( \alpha \) in this data is replaced by \( \beta \), then the mean and variance become \( 10.1 \) and \( 1.99 \), respectively. Then \( \alpha + \beta \) equals.

33.

Let the mean of the data

Be \( 5 \). If \( m \) and \( \sigma^2 \) are respectively the mean deviation about the mean and the variance of the data, then \( \dfrac{3\alpha}{m + \sigma^2} \) is equal to _____.

34.

If the variance of the frequency distribution

is \( 160 \), then the value of \( c \in \mathbb{N} \) is

35.

The mean deviation about the mean for the data

\( x_i \)	\( 5 \)	\( 7 \)	\( 9 \)	\( 10 \)	\( 12 \)	\( 15 \)
\( f_i \)	\( 8 \)	\( 6 \)	\( 2 \)	\( 2 \)	\( 2 \)	\( 6 \)

is equal to:

36.

The mean and variance of \( 10 \) observations were calculated as \( 15 \) and \( 15 \) respectively by a student who took by mistake \( 25 \) instead of \( 15 \) for one observation. Then, the correct standard deviation is ______.

37.

Let the mean and variance of \( 8 \) numbers \( -10, -7, -1, x, y, 9, 2, 16 \) be \( \frac{7}{2} \) and \( \frac{293}{4} \), respectively. Then the mean of \( 4 \) numbers \( x, y, x + y + 1, |x - y| \) is:

38.

Let \( S \) be the set of all values of \( a_{1} \) for which the mean deviation about the mean of \( 100 \) consecutive positive integers \( a_{1}, a_{2}, a_{3}, \ldots, a_{100} \) is \( 25 \). Then \( S \) is:

39.

If the mean deviation about median for the numbers \( 3, 5, 7, 2k, 12, 16, 21, 24 \), arranged in the ascending order, is \( 6 \) then the median is:

40.

The mean of \( 6 \) distinct observations is \( 6.5 \) and their variance is \( 10.25 \). If \( 4 \) out of \( 6 \) observations are \( 2,4,5 \) and \( 7 \), then the remaining two observations are:

41.

If the mean of the data

Class	\( 5\text{-}10 \)	\( 10\text{-}15 \)	\( 15\text{-}20 \)	\( 20\text{-}25 \)	\( 25\text{-}30 \)	\( 30\text{-}35 \)
Frequency	\( 2 \)	\( k \)	\( 28 \)	\( 54 \)	\( k+1 \)	\( 5 \)

is \( 21 \), then \( 1 \) is one of the roots of the equation:

42.

For a statistical data \( x_1, x_2, \ldots, x_{10} \) of \( 10 \) values, a student obtained the mean as \( 5.5 \) and \( \sum_{i=1}^{10} x_i^2 = 371 \). He later found that he had noted two values in the data incorrectly as \( 4 \) and \( 5 \), instead of the correct values \( 6 \) and \( 8 \), respectively. The variance of the corrected data is:

43.

Let the mean and variance of \( 7 \) observations \( 2, 4, 10, x, 12, 14, y \), \( x > y \), be \( 8 \) and \( 16 \) respectively. Two numbers are chosen from \( \{1, 2, 3, x - 4, y, 5\} \) one after another without replacement, then the probability, that the smaller number among the two chosen numbers is less than \( 4 \), is:

44.

The mean and variance of \( n \) observations are \( 8 \) and \( 16 \), respectively. If the sum of the first \( (n-1) \) observations is \( 48 \) and the sum of squares of the first \( (n-1) \) observations is \( 496 \), then the value of \( n \) is:

45.

The first of the two samples in a group has \( 100 \) items with mean \( 15 \) and standard deviation \( 3 \). If the whole group has \( 250 \) items with mean \( 15.6 \) and standard deviation \( \sqrt{13.44} \), then the standard deviation of the second sample is:

46.

Let \( X = \{11, 12, 13, \ldots, 40, 41\} \) and \( Y = \{61, 62, 63, \ldots, 90, 91\} \) be the two sets of observations. If \( \bar{x} \) and \( \bar{y} \) are their respective means and \( \sigma^2 \) is the variance of all the observations in \( X \cup Y \), then \( \left| \bar{x} + \bar{y} - \sigma^2 \right| \) is equal to ______.

47.

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

48.

The mean and variance of \( 7 \) observations are \( 8 \) and \( 16 \) respectively. If one observation \( 14 \) is omitted and \( a \) and \( b \) are respectively mean and variance of remaining \( 6 \) observation, then \( a+3b-5 \) is equal to \( \_\_\_\_\_ \).

49.

Let the mean and the variance of seven observations \( 2,\, 4,\, \alpha,\, 8,\, \beta,\, 12,\, 14 \), \( \alpha < \beta \), be \( 8 \) and \( 16 \) respectively. Then the quadratic equation whose roots are \( 3\alpha + 2 \) and \( 2\beta + 1 \) is:

50.

Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \displaystyle \sum_{i=1}^{10} (x_i-2)=30 \), \( \displaystyle \sum_{i=1}^{10} (x_i-\beta)^2=98 \), \( \beta>2 \), and their variance is \( \frac{4}{5} \). If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1-1)+4\beta \), \( 2(x_2-1)+4\beta \), \( \ldots \), \( 2(x_{10}-1)+4\beta \), then \( \frac{\beta \mu}{\sigma^2} \) is equal to:

51.

The mean and standard deviation of \( 10 \) observations are \( 20 \) and \( 8 \) respectively. Later on, it was observed that one observation was recorded as \( 50 \) instead of \( 40 \). Then the correct variance is:

52.

Let the mean and the standard deviation of the observation \( 2, 3, 3, 4, 5, 7, a, b \) be \( 4 \) and \( \sqrt{2} \) respectively. Then the mean deviation about the mode of these observations is:

53.

Let the mean and variance of \( 8 \) numbers \( x, y, 10, 12, 6, 12, 4, 8 \) be \( 9 \) and \( 9.25 \) respectively. If \( x>y \), then \( 3x-2y \) is equal to ________.

54.

The mean and variance of a binomial distribution are \( \alpha \) and \( \dfrac{\alpha}{3} \) respectively. If \( P(X=1)=\dfrac{4}{243} \), then \( P(X=4 \text{ or } 5) \) is equal to:

55.

If the mean and the variance of \( 6,4,a,8,b,12,10,13 \) are \( 9 \) and \( 9.25 \) respectively, then \( a+b+ab \) is equal to:

56.

If the mean of the following probability distribution of a random variable \( X \):

\( X \)	\( 0 \)	\( 2 \)	\( 4 \)	\( 6 \)	\( 8 \)
\( P(X) \)	\( a \)	\( 2a \)	\( a + b \)	\( 2b \)	\( 3b \)

is \( \dfrac{46}{9} \), then the variance of the distribution is:

57.

Let \( M \) denote the median of the following frequency distribution

Then \( 20\,M \) is equal to:

58.

Marks obtains by all the students of class \( 12 \) are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be \( 14 \) with median class interval \( 12\text{–}18 \) and median class frequency \( 12 \). If the number of students whose marks are less than \( 12 \) is \( 18 \), then the total number of students is:

59.

The mean and variance of the marks obtained by the students in a test are \( 10 \) and \( 4 \) respectively. Later, the marks of one of the students is increased from \( 8 \) to \( 12 \). If the new mean of the marks is \( 10.2 \), then their new variance is equal to:

60.

Let the six numbers \( a_1, a_2, a_3, a_4, a_5, a_6 \), be in A. P. and \( a_1 + a_3 = 10 \). If the mean of these six numbers is \( \frac{19}{2} \) and their variance is \( \sigma^2 \), then \( 8\sigma^2 \) is equal to:

61.

The mean and variance of \( 10 \) observations are \( 9 \) and \( 34.2 \), respectively. If \( 8 \) of these observations are \( 2, 3, 5, 10, 11, 13, 15, 21 \), then the mean deviation about the median of all the \( 10 \) observations is:

62.

Consider \( 10 \) observations \( x_1, x_2, \ldots, x_{10} \) such that \( \sum\limits_{i=1}^{10} (x_i-\alpha)=2 \) and \( \sum\limits_{i=1}^{10} (x_i-\beta)^2=40 \), where \( \alpha, \beta \) are positive integers. Let the mean and the variance of the observations be \( \dfrac{6}{5} \) and \( \dfrac{84}{25} \) respectively. Then \( \dfrac{\beta}{\alpha} \) is equal to:

63.

Let the mean and variance of four numbers \( 3, 7, x \) and \( y \, (x>y) \) be \( 5 \) and \( 10 \) respectively. Then the mean of four numbers \( 3+2x, 7+2y, x+y \) and \( x-y \) is ________.

64.

If the mean deviation about the mean of the numbers \( 1, 2, 3, \ldots, n \), where \( n \) is odd, is \( \frac{5(n+1)}{n} \), then \( n \) is equal to __________.

65.

Suppose a class has \( 7 \) students. The average marks of these students in the mathematics examination is \( 62 \), and their variance is \( 20 \). A student fails in the examination if he/she gets less than \( 50 \) marks, then in worst case, the number of students can fail is \( \_ \_ \_ \_ \_ \_ \).

66.

Let the mean and the variance of \( 20 \) observations \( x_1,x_2,\ldots,x_{20} \) be \( 15 \) and \( 9 \), respectively. For \( \alpha \in R \), if the mean of \( (x_1+\alpha)^2,(x_2+\alpha)^2,\ldots,(x_{20}+\alpha)^2 \) is \( 178 \), then the square of maximum value of \( \alpha \) is equal to _________.

67.

If the variance of the frequency distribution

is \( 3 \), then \( \alpha \) is equal to _____.

68.

The mean and variance of \( 5 \) observations are \( 5 \) and \( 8 \) respectively. If \( 3 \) observations are \( 1,3,5 \), then the sum of the cubes of the remaining two observations is:

69.

Two dice \( A \) and \( B \) are rolled. Let the numbers obtained on \( A \) and \( B \) be \( \alpha \) and \( \beta \) respectively. If the variance of \( \alpha - \beta \) is \( \dfrac{p}{q} \), where \( p \) and \( q \) are co-prime, then the sum of the positive divisors of \( p \) is equal to:

70.

Suppose that the mean and median of the non-negative numbers \( 21, 8, 17, a, 51, 103, b, 13, 67 \), \( (a > b) \), are \( 40 \) and \( 21 \), respectively. If the mean deviation about the median is \( 26 \), then \( 2a \) is equal to:

71.

Let the mean and the variance of \( 5 \) observations \( x_1, x_2, x_3, x_4, x_5 \) be \( \frac{24}{5} \) and \( \frac{194}{25} \) respectively. If the mean and variance of the first \( 4 \) observation are \( \frac{7}{2} \) and \( a \) respectively, then \( (4a + x_5) \) is equal to:

72.

The mean and standard deviation of \( 20 \) observation are found to be \( 10 \) and \( 2 \), respectively. On rechecking, it was found that an observation by mistake was taken \( 8 \) instead of \( 12 \). The correct standard deviation is:

73.

The mean and standard deviation of the marks of \( 10 \) students were found to be \( 50 \) and \( 12 \) respectively. Later, it was observed that two marks \( 20 \) and \( 25 \) were wrongly read as \( 45 \) and \( 50 \) respectively. Then the correct variance is __________.

74.

The random variable \( X \) follows binomial distribution \( B(n,p) \), for which the difference of the mean and the variance is \( 1 \). If \( 2P(X=2)=3P(X=1) \), then \( n^{2}P(X>1) \) is equal to:

75.

A coin is tossed three times. Let \( X \) denote the number of times a tail follows a head. If \( \mu \) and \( \sigma^2 \) denote the mean and variance of \( X \), then the value of \( 64\left(\mu + \sigma^2\right) \) is:

76.

The mean and standard deviation of \( 50 \) observations are \( 15 \) and \( 2 \) respectively. It was found that one incorrect observation was taken such that the sum of correct and incorrect observations is \( 70 \). If the correct mean is \( 16 \), then the correct variance is equal to:

77.

The number of values of \( a \in \mathbb{N} \) such that the variance of \( 3,7,12,a,43-a \) is a natural number is:

78.

Let the Mean and Variance of five observations \( x_1 = 1, x_2 = 3, x_3 = a, x_4 = 7 \) and \( x_5 = b, \; a > b \), be \( 5 \) and \( 10 \) respectively. Then the Variance of the observations \( n + x_n, \; n = 1, 2, \ldots, 5 \) is:

79.

Let the mean and variance of \( 12 \) observations be \( \dfrac{9}{2} \) and \( 4 \) respectively. Later on, it was observed that two observations were considered as \( 9 \) and \( 10 \) instead of \( 7 \) and \( 14 \) respectively. If the correct variance is \( \dfrac{m}{n} \), where \( m \) and \( n \) are coprime, then \( m+n \) is equal to:

80.

Let \( a,b,c \in \mathbb{N} \) and \( a <b <c\). Let the mean , the mean deviation about the mean and the variance of the 5 observation 9, 25 \(a,b,c\) be 18, 4 and \(\dfrac{136}{5},\) respectively. Then \(2a+b-c\) is equal to _________.

81.

If the mean and variance of five observations are \(\displaystyle \frac{24}{5} \) and \(\displaystyle \frac{194}{25} \) respectively and the mean of the first four observations is \(\displaystyle \frac{7}{2} \), then the variance of the first four observations is equal to:

82.

The mean and standard deviation of \( 20 \) observations were calculated as \( 10 \) and \( 2.5 \) respectively. It was found that by mistake one data value was taken as \( 25 \) instead of \( 35 \). If \( \alpha \) and \( \sqrt{\beta} \) are the mean and standard deviation respectively for correct data, then \( (\alpha, \beta) \) is:

83.

The mean and standard deviation of \( 100 \) observations are \( 40 \) and \( 5.1 \), respectively. By mistake one observation is taken as \( 50 \) instead of \( 40 \). If the correct mean and the correct standard deviation are \( \mu \) and \( \sigma \) respectively, then \( 10(\mu + \sigma) \) is equal to:

84.

If the mean of the frequency distribution

is \( 28 \), then its variance is __________.

85.

Let the mean of \( 6 \) observations \( 1, 2, 4, 5, x \) and \( y \) be \( 5 \) and their variance be \( 10 \). Then their mean deviation about the mean is equal to:

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JEE Main 2024 4th April Evening Shift

If the mean of the following probability distribution of a random variable \( X \):

\( X \)	\( 0 \)	\( 2 \)	\( 4 \)	\( 6 \)	\( 8 \)
\( P(X) \)	\( a \)	\( 2a \)	\( a + b \)	\( 2b \)	\( 3b \)

is \( \dfrac{46}{9} \), then the variance of the distribution is:

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