SSC CGL Statistics – Moments, Skewness & Kurtosis – Exam-Level Drill

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SSC CGL Statistics – Moments, Skewness & Kurtosis – Exam-Level Drill – 10 questions, subject: Statistics (Tier II Paper II). Every question has the correct answer with a one-line explanation; tap "Show Answer". Free practice for TNPSC, UPSC, SSC, Bank and RRB exams – Shanmugam IAS Academy, Coimbatore.

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1.  

Directions (Q 36–38): Five values: 2, 4, 6, 8, 20.

What is the third central moment μ₃?

Correct Answer: C [288]

Explanation:

Mean 8; deviations −6, −4, −2, 0, 12; cubes −216 − 64 − 8 + 0 + 1728 = 1440 → 1440/5 = 288.

2.  
(Use the directions above – Q 36–38)
What is β₁ (to two decimals)?

Correct Answer: B [1.30]

Explanation:

β₁ = 288²/40³ = 82944/64000 = 1.296 ≈ 1.30.

3.  
(Use the directions above – Q 36–38)
What is β₂ (to two decimals), and the type of kurtosis?

Correct Answer: D [2.79, platykurtic]

Explanation:

μ₄ = (1296 + 256 + 16 + 0 + 20736)/5 = 4460.8; β₂ = 4460.8/1600 = 2.788 < 3.

4.  
About A = 20, the raw moments are μ′₁ = −1, μ′₂ = 5, μ′₃ = −7, μ′₄ = 41. Find the mean and μ₃.

Correct Answer: A [Mean 19, μ₃ = 6]

Explanation:

Mean = 20 − 1 = 19; μ₃ = −7 − 3(5)(−1) + 2(−1)³ = −7 + 15 − 2 = 6.

5.  
For the same raw moments (μ′₁ = −1, μ′₂ = 5, μ′₃ = −7, μ′₄ = 41), find μ₄ and β₂.

Correct Answer: C [μ₄ = 40, β₂ = 2.5]

Explanation:

μ₂ = 5 − 1 = 4; μ₄ = 41 − 4(−7)(−1) + 6(5)(1) − 3 = 41 − 28 + 30 − 3 = 40; β₂ = 40/16 = 2.5 (platykurtic; and β₂ ≥ β₁ + 1 = 1.5625, so the data are possible).

6.  
A grouped distribution with class width 10 has μ₂ = 120 and μ₄ = 30,000. After Sheppard's corrections, μ₂ and μ₄ are (approximately):

Correct Answer: B [111.67 and 24,291.67]

Explanation:

μ₂ − 100/12 = 111.67; μ₄ − 50 × 120 + 7 × 10⁴/240 = 30,000 − 6,000 + 291.67 = 24,291.67.

7.  
Mean = 25, median = 23 and Karl Pearson's coefficient of skewness (median form) is 0.5. What is the coefficient of variation?

Correct Answer: A [48%]

Explanation:

0.5 = 3(25 − 23)/σ → σ = 12 → CV = 12/25 × 100 = 48%.

8.  
Statement-basedFor y = 7 − 3x, which of the following is/are correct?
I. β₁(y) = β₁(x)
II. γ₁(y) = γ₁(x)
III. μ₂(y) = 9μ₂(x)

Correct Answer: C [Only I and III]

Explanation:

β₁ and μ₂ behave as stated; γ₁ changes sign because b = −3 < 0.

9.  
Two-quantity comparison
Quantity I: β₂ of 1, 3, 5, 7, 14
Quantity II: β₂ of a distribution with μ₂ = 9 and μ₄ = 162

Correct Answer: A [Quantity I > Quantity II]

Explanation:

QI = 960.8/400 = 2.402; QII = 162/81 = 2.

10.  
μ₂ = 25 and μ₃ = −125. What is the moment coefficient of skewness γ₁?

Correct Answer: D [−1]

Explanation:

γ₁ = μ₃/σ³ = −125/125 = −1 (β₁ = 1, sign taken from μ₃).

➤ Full notes, shortcuts and every answer explained – SSC CGL 2026 Statistics (Tier II Paper II) book (₹149): www.shanmugamiasacademy.in/study-materials/ssc-cgl-statistics

📘 SSC CGL 2026 – Statistics (Tier II Paper II)
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