APPSC Paper I · Assistant Statistical Officer Paper II · Topic 8

8 APPSC Paper I Dispersion & Skewness questions with answers

26 questions from APPSC ASO 2025 Paper II question paper with answers and APPSC ASO Paper II Mock 1 with answers. Try each one, then tap Show answer.

Previous-paper questions are as in the original paper. Medha AI wrote the answers and explanations.

8.1Range, quartile deviation, mean deviation, SD & CVNotes

  1. Q1 ASO 2025 Paper II, Q61

    For the set of observations 7.4, 12.2, 8.6, 7.7, 9.3, 5.8, 11.6, 8.8, 9.0, the range is equal to:

    1. A7.8
    2. B6.4
    3. C5.9
    4. D2.3
    Show answerHide answer

    Answer B. 6.4

    Range = largest − smallest = 12.2 − 5.8 = 6.4.

    Source: APPSC Assistant Statistical Officer 2025, Paper II (Notification 12/2024), Q63, final key

  2. Q2 ASO 2025 Paper II, Q67

    Match the following:

    Column IColumn II
    1. Standard DeviationI. Variability in a data set
    2. Coefficient of VariationII. obtained by subtracting lowest observation from highest observation
    3. DispersionIII. an absolute measure of dispersion
    4. RangeIV. checks consistency in performance of cricket players

    Choose the correct option.

    1. A1-I; 2-IV; 3- III; 4-II
    2. B1-IV; 2-III; 3- II; 4-I
    3. C1-IV; 2-III; 3- I; 4-II
    4. DI- III; 2-IV; 3- I; 4-II
    Show answerHide answer

    Answer D. I- III; 2-IV; 3- I; 4-II

    Standard deviation is an absolute measure of dispersion (1-III), the coefficient of variation compares consistency, e.g. of batsmen (2-IV), dispersion means variability in data (3-I), and range is highest minus lowest (4-II). The keyed option prints '1' as 'I' by a typo.

    Source: APPSC Assistant Statistical Officer 2025, Paper II (Notification 12/2024), Q69, final key

  3. Q3 ASO 2025 Paper II, Q71

    If n =10, mean = 12 and standard deviation = 3, then coefficient of variation (in percentage) is equal to:

    1. A25
    2. B12
    3. C4
    4. D10
    Show answerHide answer

    Answer A. 25

    CV = (SD / mean) × 100 = (3 / 12) × 100 = 25%.

    Source: APPSC Assistant Statistical Officer 2025, Paper II (Notification 12/2024), Q74, final key

  4. Q4 ASO 2025 Paper II, Q80

    Given below are two statements labelled as Assertion (A) and Reasoning (R).

    Assertion (A): For the observations 2, 3, 5, 9 and 11, mean deviation about mean is 3.2 and the mean deviation about median is 3.

    Reasoning (R): Mean deviation is the least when measured about the median.

    In the light of above, choose the correct option.

    1. ABoth A and B are true and R is correct explanation of A.
    2. BBoth A and B are true and R is not correct explanation of A.
    3. CA is true but R is not true.
    4. DA is not true but R is true.
    Show answerHide answer

    Answer A. Both A and B are true and R is correct explanation of A.

    Mean = 6: deviations 4, 3, 1, 3, 5, MD = 16/5 = 3.2. Median = 5: deviations 3, 2, 0, 4, 6, MD = 15/5 = 3. A is true, and the median value is smaller because mean deviation is least about the median, so R explains A.

    Source: APPSC Assistant Statistical Officer 2025, Paper II (Notification 12/2024), Q84, final key

  5. Q5 ASO 2025 Paper II, Q107

    If the mean deviation of X from its mean is 5, then the mean deviation of Y = 2X+3 from its mean is:

    1. A10
    2. B5
    3. C20
    4. D15
    Show answerHide answer

    Answer A. 10

    Mean deviation is unaffected by change of origin but scales by |b|; MD(Y) = |2| x 5 = 10.

    Source: APPSC Assistant Statistical Officer 2025, Paper II (Notification 12/2024), Q113, final key

  6. Q6 ASO 2025 Paper II, Q123

    For the set of observations 1216, 1167, 1232, 1202, 1278, 1141 and 1221, the mean deviation about mean is:

    1. A80.5
    2. B78.9
    3. C67.82
    4. D32.69
    Show answerHide answer

    Answer D. 32.69

    Mean = 8457/7 = 1208.14; absolute deviations 7.86, 41.14, 23.86, 6.14, 69.86, 67.14, 12.86 sum to 228.86, so MD = 228.86/7 = 32.69.

    Source: APPSC Assistant Statistical Officer 2025, Paper II (Notification 12/2024), Q130, final key

  7. Q7 ASO Paper II Mock 1, Q108

    Of 10 observations, five are equal to 4 and five are equal to −4. What is their standard deviation?

    1. A0
    2. B4
    3. C16
    4. D8
    Show answerHide answer

    Answer B. 4

    Mean = 0; every squared deviation is 16; σ² = 16, σ = 4.

    Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)

  8. Q8 ASO Paper II Mock 1, Q110

    What is the variance of the first ten even numbers 2, 4, …, 20?

    1. A16.5
    2. B8.25
    3. C5.74
    4. D33
    Show answerHide answer

    Answer D. 33

    They are 2 × (1, 2, …, 10). Variance of 1 … 10 = (100 − 1)/12 = 8.25; multiply by 2² = 4: 33.

    Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)

  9. Q9 ASO Paper II Mock 1, Q112

    The coefficient of variation of a distribution is 25% and its mean is 48. What is its variance?

    1. A12
    2. B1.92
    3. C3.69
    4. D144
    Show answerHide answer

    Answer D. 144

    σ = CV × mean/100 = 0.25 × 48 = 12; variance = 12² = 144.

    Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)

  10. Q10 ASO Paper II Mock 1, Q114

    The coefficient of standard deviation of a series is 0.15 and its standard deviation is 6. What is the mean?

    1. A0.9
    2. B90
    3. C240
    4. D40
    Show answerHide answer

    Answer D. 40

    Coefficient of SD = σ/x̄ ⇒ x̄ = 6/0.15 = 40.

    Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)

  11. Q11 ASO Paper II Mock 1, Q116

    What is the standard deviation of 6, 8, 10, 12 and 14?

    1. A2.83
    2. B8
    3. C3.16
    4. D2.4
    Show answerHide answer

    Answer A. 2.83

    Mean = 10; squared deviations 16, 4, 0, 4, 16, sum 40; variance = 40/5 = 8; SD = √8 = 2.83.

    Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)

  12. Q12 ASO Paper II Mock 1, Q118

    A series has mean 40 and SD 6. What is the root mean square deviation of the observations about the value 48?

    1. A6
    2. B10
    3. C14
    4. D2
    Show answerHide answer

    Answer B. 10

    s² = σ² + (x̄ − A)² = 36 + 64 = 100, so s = 10. It is least (= σ) when A is the mean.

    Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)

  13. Q13 ASO Paper II Mock 1, Q120

    If y = a + bx, where a and b are constants (b ≠ 0), consider the following:

    1. ȳ = a + b x̄

    2. σᵧ = |b| σₓ

    3. The coefficient of variation of y is always equal to that of x

    4. Range of y = |b| × range of x

    Which of the above are correct?

    1. A1 and 2 only
    2. B1, 2 and 4 only
    3. C2, 3 and 4 only
    4. D1, 2, 3 and 4
    Show answerHide answer

    Answer B. 1, 2 and 4 only

    The mean follows both origin and scale (ȳ = a + bx̄); SD and range change only with scale (|b|). CVᵧ = |b|σₓ/|a + bx̄| × 100, which differs from CVₓ whenever a ≠ 0 (or b < 0 with a positive mean), so statement 3 is not always true.

    Source: Gupta, S.C. & Kapoor, V.K., Fundamentals of Mathematical Statistics, Measures of Dispersion

  14. Q14 ASO Paper II Mock 1, Q122

    The standard deviation of a set of observations is zero if and only if:

    1. Aall the observations are equal
    2. Bthe mean is zero
    3. Cthe sum of deviations from the mean is zero
    4. Dthe mean equals the median
    Show answerHide answer

    Answer A. all the observations are equal

    σ² = Σ(x − x̄)²/n is a sum of squares; it is zero only when every x equals x̄, i.e. all observations are equal.

    Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)

8.1Skewness, kurtosis & the normal distributionNotes

  1. Q15 ASO 2025 Paper II, Q66

    Match the following:

    Column IColumn II
    1. Normal Distribution isI. Mode
    2. Mean of Normal Distribution is same asII. unimodal
    3. Quartile deviation for normal distributionIII. symmetric
    4. Normal distributionIV. two third of standard deviation

    Choose the correct option.

    1. A1-I; 2-IV; 3- III; 4-II
    2. B1-IV; 2-I ; 3- II; 4-III
    3. C1-IV; 2-III; 3- I; 4-II
    4. DI- III; 2-I ; 3- IV; 4-II
    Show answerHide answer

    Answer D. I- III; 2-I ; 3- IV; 4-II

    The normal distribution is symmetric (1-III), its mean equals its mode (2-I), its quartile deviation is about two-thirds of the standard deviation (3-IV), and it is unimodal (4-II). The keyed option prints '1' as 'I' by a typo.

    Source: APPSC Assistant Statistical Officer 2025, Paper II (Notification 12/2024), Q68, final key

  2. Q16 ASO 2025 Paper II, Q95

    For Normal distribution, the value of coefficient of skewness is:

    1. AOne
    2. BZero
    3. CThree
    4. DMinus Three
    Show answerHide answer

    Answer B. Zero

    The normal distribution is perfectly symmetric (mean = median = mode), so its coefficient of skewness is zero.

    Source: APPSC Assistant Statistical Officer 2025, Paper II (Notification 12/2024), Q100, final key

  3. Q17 ASO 2025 Paper II, Q96

    A negative value of coefficient of skewness implies that:

    1. AMode is more than mean
    2. BMean=median
    3. CMean=mode
    4. DMode is less than mean
    Show answerHide answer

    Answer A. Mode is more than mean

    Pearson's coefficient of skewness is (Mean - Mode)/SD, so a negative value means the mean is below the mode; in a negatively skewed distribution Mean < Median < Mode.

    Source: APPSC Assistant Statistical Officer 2025, Paper II (Notification 12/2024), Q101, final key

  4. Q18 ASO 2025 Paper II, Q97

    Suppose that for a frequency distribution, Mean = 164.73, First Quartile = 160.95, Second Quartile = 164.76 and Third Quartile= 168.45, then on the basis of Bowley's coefficient of skewness, the distribution is:

    1. ANegatively Skewed
    2. BPositively Skewed
    3. CSymmetric
    4. DMesokurtic
    Show answerHide answer

    Answer A. Negatively Skewed

    Bowley's Sk = (Q3 + Q1 - 2Q2)/(Q3 - Q1) = (168.45 + 160.95 - 329.52)/7.50 = -0.12/7.50 = -0.016, a small negative value, so the distribution is negatively skewed.

    Source: APPSC Assistant Statistical Officer 2025, Paper II (Notification 12/2024), Q102, final key

  5. Q19 ASO 2025 Paper II, Q122

    For normal distribution, consider the following statements

    1. Mean = Mode =Median;

    2. Odd order central moments are identically equal to zero;

    3. Coefficient of skewness is zero.

    Which of the following holds?

    1. A2 and 3 are true
    2. B1 and 3 are true
    3. C1 and 2 are true
    4. DAll three are true
    Show answerHide answer

    Answer D. All three are true

    The normal curve is symmetric about its mean, so mean, median and mode coincide, all odd central moments vanish and the skewness coefficient is zero.

    Source: APPSC Assistant Statistical Officer 2025, Paper II (Notification 12/2024), Q128, final key

  6. Q20 ASO Paper II Mock 1, Q109

    The first four moments of a distribution about the value 10 are −1, 10, −20 and 300. The value of β₂ and the nature of the curve are

    1. A3.70, leptokurtic
    2. B3.42, leptokurtic
    3. C2.77, platykurtic
    4. D3.42, platykurtic
    Show answerHide answer

    Answer B. 3.42, leptokurtic

    μ₂ = 10 − (−1)² = 9; μ₄ = 300 − 4(−20)(−1) + 6(10)(−1)² − 3(−1)⁴ = 300 − 80 + 60 − 3 = 277; β₂ = 277/81 = 3.42 > 3, leptokurtic.

    Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)

  7. Q21 ASO Paper II Mock 1, Q111

    Match List I with List II:

    List I:

    A. Mesokurtic

    B. Leptokurtic

    C. Platykurtic

    D. γ₂ (excess kurtosis)

    List II:

    1. β₂ < 3

    2. β₂ − 3

    3. β₂ = 3

    4. β₂ > 3

    1. AA-3, B-1, C-4, D-2
    2. BA-3, B-4, C-1, D-2
    3. CA-2, B-4, C-1, D-3
    4. DA-3, B-2, C-1, D-4
    Show answerHide answer

    Answer B. A-3, B-4, C-1, D-2

    Mesokurtic (normal) β₂ = 3; leptokurtic, more peaked, β₂ > 3; platykurtic, flatter, β₂ < 3; γ₂ = β₂ − 3.

    Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)

  8. Q22 ASO Paper II Mock 1, Q113

    Consider the following statements about normal variables:

    1. If X is normal, aX + b (a ≠ 0) is also normal.

    2. The sum of two independent normal variables is normal.

    3. For independent normal X and Y, the variance of X − Y is σ₁² − σ₂².

    4. The normal curve is unimodal.

    Select the correct answer:

    1. A1, 2 and 4 only
    2. B1, 2, 3 and 4
    3. C1 and 2 only
    4. D2, 3 and 4 only
    Show answerHide answer

    Answer A. 1, 2 and 4 only

    A linear function of a normal variable and the sum or difference of independent normal variables are normal; variances add for both X + Y and X − Y, so Var(X − Y) = σ₁² + σ₂². The curve has a single peak at μ.

    Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)

  9. Q23 ASO Paper II Mock 1, Q115

    The second and fourth central moments of a distribution are μ2 = 2 and μ4 = 18. What is β2 and how is the distribution described?

    1. A4.5; leptokurtic
    2. B4.5; platykurtic
    3. C9.0; leptokurtic
    4. D2.25; platykurtic
    Show answerHide answer

    Answer A. 4.5; leptokurtic

    β2 = μ4 / μ2² = 18 / 2² = 4.5. Since β2 > 3 (the normal value), the distribution is leptokurtic (more peaked than normal).

    Source: S.C. Gupta & V.K. Kapoor, Fundamentals of Mathematical Statistics, Moments, Skewness and Kurtosis

  10. Q24 ASO Paper II Mock 1, Q117

    Marks are normally distributed with mean 70 and standard deviation 8. The standard normal variate Z for a score of 82 is

    1. A12
    2. B1.5
    3. C0.1875
    4. D10.25
    Show answerHide answer

    Answer B. 1.5

    Z = (X − μ)/σ = (82 − 70)/8 = 1.5.

    Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)

  11. Q25 ASO Paper II Mock 1, Q119

    The first four moments of a distribution about the value 5 are 1, 4, 10 and 46. What are β2 and the nature of kurtosis?

    1. A5.11; leptokurtic
    2. B2.875; platykurtic
    3. C3.0; mesokurtic
    4. D1.69; platykurtic
    Show answerHide answer

    Answer C. 3.0; mesokurtic

    Convert to central moments: μ2 = μ2′ − μ1′² = 4 − 1 = 3; μ4 = μ4′ − 4μ3′μ1′ + 6μ2′μ1′² − 3μ1′⁴ = 46 − 40 + 24 − 3 = 27. β2 = μ4 / μ2² = 27 / 9 = 3, so the distribution is mesokurtic.

    Source: S.C. Gupta & V.K. Kapoor, Fundamentals of Mathematical Statistics, Moments, Skewness and Kurtosis

  12. Q26 ASO Paper II Mock 1, Q121

    In a positively skewed distribution, the usual relationship among the three averages is

    1. AMean < Median < Mode
    2. BMean > Median > Mode
    3. CMean = Median = Mode
    4. DMedian > Mean > Mode
    Show answerHide answer

    Answer B. Mean > Median > Mode

    With positive skew the tail extends to the right; extreme high values raise the mean most, so Mean > Median > Mode.

    Source: S.P. Gupta, Statistical Methods, Skewness, Moments and Kurtosis

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