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Mean and variance of two groups combined (or shifted and combined)

This type of question, asked 8 times: 2021 Paper 6, 2021 Paper 8, 2021 Paper 21, 29 Jan 2023, 31 Jan 2023, 6 Apr 2023, 11 Apr 2023, 8 Apr 2026 S2. All 8 questions below, oldest first.

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  1. Question 1

    2021 Paper 6

    82
    Consider the statistics of two sets of observations as follows:
    SetSizeMeanVariance
    Observation I1022
    Observation IInn31
    If the variance of the combined set of these two observations is 179\frac{17}{9}, then the value of nn is equal to

  2. Question 2

    2021 Paper 8

    85
    Consider a set of 3n3n numbers having variance 4 In this set, the mean of first 2n2n numbers is 6 and the mean of the remaining numbers is 3. A new set is constructed by adding 1 into each of first 2 n numbers, and subtracting 1 from each of the remaining nn numbers. If the variance of the new set is kk, then 9k9k is equal to

  3. Question 3

    2021 Paper 21

    89
    An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2 . The variance of marks obtained by 30 girls is also 2 . The average marks of all 50 candidates is 15. If μ\mu is the average marks of girls and σ2\sigma^{2} is the variance of marks of 50 candidates, then μ+σ2\mu+\sigma^{2} is equal to

  4. Question 4

    29 Jan 2023

    175
    Let X={11,12,13,…,40,41}X = \{ 11,12,13,\ldots,40,41\} and Y={61,62,63,…,90,91}Y = \{ 61,62,63,\ldots,90,91\} be the two sets of observations. If xˉ\bar{x} and yˉ\bar{y} are their respective means and σ2\sigma^{2} is the variance of all the observations in X∪YX \cup Y, then ∣xˉ+yˉ−σ2∣\left| \bar{x}+\bar{y}-\sigma^{2} \right| is equal to

  5. Question 5

    31 Jan 2023

    159
    Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and α\alpha ( >> 0)0), and the mean and standard deviation of marks of class B of nn students be respectively 55 and 30 −α- \alpha. If the mean and variance of the marks of the combined class of 100+n100 + n students are respectively 50 and 350, then the sum of variances of classes AA and BB is :

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  6. Question 6

    6 Apr 2023

    69
    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 σ2\sigma^{2} respectively. If the variance of all the 30 numbers in the two sets is 13, then σ2\sigma^{2} is equal to

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  7. Question 7

    11 Apr 2023

    64
    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 AA and adding 2 to each element of B. Then the sum of the mean and variance of the elements of CC is

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  8. Question 8

    8 Apr 2026 S2

    57
    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 :

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