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

is _______.

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:

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 \( \_\_\_\_\_\_\_\_\_\_\_ \).

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:

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 __________.

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 ______.

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:

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:

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 __________.

10.

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:

11.

If the variance of the frequency distribution

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

12.

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:

13.

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

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

14.

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:

15.

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:

16.

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 _________.

17.

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:

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