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Frequency table with unknown frequencies or values: use the mean or variance to find them

This type of question, asked 8 times: 2021 Paper 17, 31 Jan 2023, 6 Apr 2023, 10 Apr 2023, 13 Apr 2023, 9 Apr 2024, 23 Jan 2026 S2, 2 Apr 2026 S1. All 8 questions below, oldest first.

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  1. Question 1

    2021 Paper 17

    68
    Let the mean and variance of the frequency distribution
    xxx1=2x_{1}= 2x2=6x_{2}= 6x3=8x_{3}= 8x4=9x_{4}= 9
    ff44α\alphaβ\beta
    be 6 and 6.8 respectively. If x3x_{3} is changed from 8 to 7, then the mean for the new data will be:

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  2. Question 2

    31 Jan 2023

    84
    If the variance of the frequency distribution
    xix_{i}2345678
    Frequency fif_{i}3616α\alpha956
    is 3, then α\alpha is equal to

  3. Question 3

    6 Apr 2023

    180
    If the mean and variance of the frequency distribution
    xix_{i}246810121416
    fif_{i}44α\alpha158β\beta45
    are 9 and 15.08 respectively, then the value of α2+β2−αβ\alpha^{2}+\beta^{2}-\alpha\beta is

  4. Question 4

    10 Apr 2023

    170
    Let μ\mu be the mean and σ\sigma be the standard deviation of the distribution
    xix_{i}012345
    fif_{i}k+2k + 22k2kk2−1k^{2}- 1k2−1k^{2}- 1k2+1k^{2}+ 1k−3k - 3
    where ∑fi=62\sum f_{i}= 62. If [x]\lbrack x\rbrack denotes the greatest integer ≤x\leq x, then [μ2+σ2]\left\lbrack \mu^{2}+\sigma^{2} \right\rbrack is equal to

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  5. Question 5

    13 Apr 2023

    90
    Let the mean of the data
    xx13579
    Frequency (f)(f)42428α\alpha8
    be 5. If mm and σ2\sigma^{2} are respectively the mean deviation about the mean and the variance of the data, then 3αm+σ2\frac{3\alpha}{m +\sigma^{2}} is equal to

  6. Question 6

    9 Apr 2024

    157
    If the variance of the frequency distribution
    xxcc2c2c3c3c4c4c5c5c6c6c
    ff211111
    is 160, then the value of c∈Nc \in N is

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  7. Question 7

    23 Jan 2026 S2

    68
    If the mean and the variance of the data
    Class4-88-1212-1616-20
    Frequency3λ\lambda47
    are μ\mu and 19 respectively, then the value of λ+μ\lambda + \mu is

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  8. Question 8

    2 Apr 2026 S1

    58
    If the mean of the data
    ClassFrequency
    5-102
    10-15kk
    15-2028
    20-2554
    25-30k+1k + 1
    30-355
    is 21, then kk is one of the roots of the equation :

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