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

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:

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:

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:

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:

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:

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:

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:

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:

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JEE Main 2025 4th April Evening Shift

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: