Moments, Skewness, and Kurtosis for SSC CGL Tier 2 Paper 2

In SSC CGL Tier 2 Quantitative Aptitude, questions on Moments, Skewness, and Kurtosis are part of the Statistics section. These concepts help in understanding the shape, spread, and symmetry of a data distribution. Let’s understand each of them step-by-step with simple explanations and formulas.

1. Moments

Definition

Moments are the statistical measures used to describe the shape and characteristics of a frequency distribution. They provide information about the mean, variance, skewness, and kurtosis of data.

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Types of Moments

Moments can also be taken about an arbitrary origin (raw moments) or about the mean (central moments). The central moments are more useful in describing the shape of the distribution.

Type of Moment	About	Formula (for ungrouped data)	Purpose
1st Moment (Mean)	Measures central tendency	μ₁ = Σ(x − 𝑥̄)/N = 0	Represents mean position
2nd Moment (Variance)	Measures dispersion	μ₂ = Σ(x − 𝑥̄)² / N	Represents spread of data
3rd Moment	Measures skewness	μ₃ = Σ(x − 𝑥̄)³ / N	Tells about asymmetry
4th Moment	Measures kurtosis	μ₄ = Σ(x − 𝑥̄)⁴ / N	Describes peakness or flatness

Also check out Most Repeated Quantitative Aptitude Questions for SSC CGL Tier 2

2. Skewness

Definition

Skewness measures the asymmetry of a data distribution about its mean. A perfectly symmetrical distribution has zero skewness.

Types of Skewness

Type	Shape	Characteristics
Symmetrical (Zero Skewness)	Mean = Median = Mode	Both sides are mirror images.
Positively Skewed (Right Skewed)	Mean > Median > Mode	Tail on the right side is longer.
Negatively Skewed (Left Skewed)	Mean < Median < Mode	Tail on the left side is longer.

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Formulas for Skewness

Shortcut Tip:

• Sk > 0 → Positively skewed
• Sk < 0 → Negatively skewed
• Sk = 0 → Symmetrical

Formula Type	Formula	Meaning
Karl Pearson’s Coefficient	Sk = (Mean − Mode) / S.D.	Commonly used in SSC exams
Alternative Formula	Sk = 3(Mean − Median) / S.D.	Used when mode is not known
Moment Coefficient	Sk = μ₃ / (μ₂)^(3/2)	Based on moments

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3. Kurtosis

Definition

Kurtosis measures the degree of peakedness or flatness of a data distribution compared to a normal distribution.

Types of Kurtosis

Type	Shape of Curve	Characteristic
Mesokurtic	Normal curve	Moderate peak (β₂ = 3)
Leptokurtic	More peaked	Higher peak, heavy tails (β₂ > 3)
Platykurtic	Flatter curve	Lower peak, light tails (β₂ < 3)

Formula for Kurtosis

β₂=μ4(μ2)2\text{β₂} = \frac{μ₄}{(μ₂)²}β₂=(μ2​)2μ4​​

or Excess Kurtosis=β2−3\text{Excess Kurtosis} = β₂ – 3Excess Kurtosis=β2​−3

If β₂ = 3 → Normal curve
If β₂ > 3 → Leptokurtic
If β₂ < 3 → Platykurtic

4. Quick Comparison Table

Aspect	Moments	Skewness	Kurtosis
Purpose	Describe shape and features of data	Measure of asymmetry	Measure of peakedness
Uses Moments	Yes (μ₁, μ₂, μ₃, μ₄)	Yes (μ₃)	Yes (μ₄)
Value Interpretation	–	Sk = 0 → Symmetrical Sk > 0 → Right tail Sk < 0 → Left tail	β₂ = 3 → Normal β₂ > 3 → Peaked β₂ < 3 → Flat
Based on	Mean deviations	Third moment	Fourth moment

Key Takeaways for SSC CGL Tier 2

• Remember:
  • Skewness → Asymmetry,
  • Kurtosis → Peakness,
  • Moments → Foundation for both.
• Practice numerical questions on:
  • Karl Pearson’s Skewness
  • Coefficient of Variation
  • Moments about mean

Check out SSC CGL Exam Pattern

FAQs

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• Sampling Theory for SSC CGL Tier 2 Paper 2
• Correlation and Regression SSC CGL Tier 2 Paper 2
• Moments, Skewness, and Kurtosis for SSC CGL Tier 2 Paper 2
• Measures of Dispersion for SSC CGL Tier 2 (JSO Post)
• Measures of Central Tendency for SSC CGL Tier 2, Paper 2
• Important Statistics Topics for SSC CGL Tier 2, Check Details Here

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