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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:

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:

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:

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 ________.

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 __________.

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 ____________ .

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:

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:

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 ________.

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 ________.

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:

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JEE Main 2021 27th July Morning Shift

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: