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Measures of Dispersion: Range, Quartile Deviation, Mean Deviation, Standard Deviation, Variance & CV

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UPSC EPFO APFC • EPFO EO/AO • RBI Grade B • SEBI Grade A
⚡ Quick Answer & AI Exam Snapshot (TL;DR)

While Measures of Central Tendency give a single central value, Measures of Dispersion (‘Averages of the Second Order’ — Bowley) measure the degree of scatter, spread, or variation of items around that central value. Dispersion measures are classified into Absolute Measures (expressed in original units: Range, Quartile Deviation, Mean Deviation, Standard Deviation) and Relative Measures (unit-free pure ratios/percentages used to compare two series: Coefficient of Range, Coefficient of QD, Coefficient of MD, and Karl Pearson’s Coefficient of Variation CV = [σ / X̄] × 100).

1. Absolute vs Relative Measures of Dispersion (Master Formula Table)

Measure Absolute Measure Formula (In Original Units) Relative Measure (Unit-Free Coefficient) Key Exam Property
1. Range Range = L − S
(Largest Value − Smallest Value)		 Coeff. of Range = (L − S) / (L + S) Simplest measure; used in Statistical Quality Control (SQC) Charts & daily stock/weather ranges. Most affected by extreme outliers.
2. Quartile Deviation (QD)
(Semi-Interquartile Range)		 QD = (Q₃ − Q₁) / 2
(Where Q₃ − Q₁ is Interquartile Range)		 Coeff. of QD = (Q₃ − Q₁) / (Q₃ + Q₁) Covers the central 50% of observations. Best measure of dispersion for Open-End Class Distributions.
3. Mean Deviation (MD)
(Average Deviation)		 MD = ∑|Xᵢ − A| / N
(Where A = Mean, Median, or Mode)		 Coeff. of MD = MD / A
(Divided by the average from which it was taken)		 Ignores algebraic signs (±); therefore unsuitable for further algebraic treatment. Minimum when taken from Median.
4. Standard Deviation (σ)
(Root-Mean-Square Deviation)		 σ = √[ ∑(Xᵢ − X̄)² / N ]
Variance = σ²		 Coefficient of Variation (CV):
CV = (σ / X̄) × 100		 Coined by Karl Pearson (1893). Always calculated from Arithmetic Mean; never negative. Best & most widely used measure.
5. Lorenz Curve Graphical Measure of Dispersion Related to Gini Coefficient (0 to 1) Devised by Max O. Lorenz (1905) to measure income and wealth inequality (Line of Equal Distribution at 45°).

2. High-Yield Mathematical Properties of Standard Deviation & Variance

UPSC EPFO APFC tests these five properties of Standard Deviation (σ) and Variance (σ²) repeatedly:

  1. Independent of Change of Origin, But Affected by Change of Scale: If every observation Xᵢ is transformed to Yᵢ = a + bXᵢ (adding/subtracting a constant a and multiplying by b):
    New Standard Deviation (σᵧ) = |b| × σₓ (Adding/subtracting ‘a’ has ZERO effect on SD!)
    New Variance (σᵧ²) = b² × σₓ²

    Example: If the SD of 10 numbers is 6, and every number is increased by 15 and then multiplied by −2, the new SD is |−2| × 6 = 12, and the new Variance is (−2)² × 6² = 4 × 36 = 144.

  2. SD of First ‘n’ Natural Numbers (1, 2, 3, …, n):
    σ = √[ (n² − 1) / 12 ]   and   Variance (σ²) = (n² − 1) / 12
  3. SD of Any Two Numbers (a and b): The Standard Deviation of just two numbers is simply half of their range: σ = |a − b| / 2.
  4. Empirical Relationship Between QD, MD, and SD (Normal Distribution):
    QD : MD : SD = 10 : 12 : 15   (or approximately 4 : 5 : 6)
    6 QD = 5 MD = 4 SD ⇒ QD = (2/3) SD ≈ 0.6745 σ   |   MD = (4/5) SD ≈ 0.7979 σ
  5. Karl Pearson’s Coefficient of Variation (CV) — Testing Consistency vs Variability:
    • A series with a LOWER Coefficient of Variation (CV) is MORE CONSISTENT, MORE UNIFORM, MORE STABLE, and MORE HOMOGENEOUS.
    • A series with a HIGHER Coefficient of Variation (CV) is MORE VARIABLE and LESS CONSISTENT.

3. Normal Distribution Area Property & Moments / Skewness / Kurtosis

  • Normal Distribution Coverage around Mean (X̄):
    • X̄ ± 1σ covers 68.27% of total observations.
    • X̄ ± 2σ covers 95.45% of total observations.
    • X̄ ± 3σ covers 99.73% of total observations.
  • Skewness (Lack of Symmetry) vs Kurtosis (Peakedness of Curve):
    • Karl Pearson’s Coefficient of Skewness (Sₖ): Sₖ = (Mean − Mode) / σ = 3(Mean − Median) / σ.
    • Bowley’s Quartile Coefficient of Skewness: Sₖ = [ (Q₃ − Median) − (Median − Q₁) ] / (Q₃ − Q₁) (lies between −1 and +1).
    • Kurtosis (β₂ = μ₄ / μ₂²): For a Normal Curve (Mesokurtic), β₂ = 3 (and excess kurtosis γ₂ = β₂ − 3 = 0). A high-peaked curve is Leptokurtic (β₂ > 3); a flat-topped curve is Platykurtic (β₂ < 3).

🎯 High-Yield Exam One-Liners (Last-Minute Revision Box)

  1. Measures of Dispersion are called ‘Averages of the Second Order’ (A.L. Bowley).
  2. Range (L − S) is based on only the two extreme values; its relative measure is (L − S) / (L + S).
  3. Interquartile Range is (Q₃ − Q₁); Quartile Deviation (Semi-Interquartile Range) is (Q₃ − Q₁) / 2.
  4. Quartile Deviation is the most appropriate measure of dispersion for Open-End Class Distributions.
  5. Mean Deviation (MD) is always minimum when calculated about the MEDIAN.
  6. Root-Mean-Square Deviation is minimum when calculated about the Arithmetic Mean, where it is called Standard Deviation (σ).
  7. The term ‘Standard Deviation’ was first introduced by Karl Pearson in 1893.
  8. Variance is the square of Standard Deviation (σ²); Standard Deviation is the positive square root of Variance.
  9. Standard Deviation is independent of Change of Origin (adding/subtracting a constant) but directly affected by Change of Scale (multiplying/dividing by a constant).
  10. If each observation in a dataset is increased by 20, the Standard Deviation and Variance remain completely UNCHANGED.
  11. If each observation in a dataset is multiplied by 3, the Standard Deviation becomes 3 times and the Variance becomes 9 (3²) times.
  12. Variance of the first ‘n’ natural numbers is (n² − 1) / 12.
  13. In a symmetrical normal distribution, the ratio of QD : MD : SD is 10 : 12 : 15 (or 4 : 5 : 6), meaning 6 QD = 5 MD = 4 SD.
  14. Always holds true for any distribution: Quartile Deviation (QD) < Mean Deviation (MD) < Standard Deviation (SD).
  15. Coefficient of Variation (CV) = (Standard Deviation / Mean) × 100; it is a pure unit-free percentage.
  16. Lower Coefficient of Variation (CV) indicates HIGHER consistency and stability (e.g., deciding the more consistent batsman).
  17. Lorenz Curve is a graphical method of studying dispersion (inequality of income/wealth); the 45° diagonal is the Line of Equal Distribution.
  18. In a Normal Distribution, Mean ± 1σ covers 68.27%, Mean ± 2σ covers 95.45%, and Mean ± 3σ covers 99.73% of total area.
  19. For a Mesokurtic (Normal) distribution, Pearson’s Kurtosis measure β₂ = 3; for Leptokurtic (peaked) β₂ > 3; for Platykurtic (flat) β₂ < 3.
  20. Standard Deviation of two values ‘a’ and ‘b’ is simply |a − b| / 2.

📝 UPSC & Regulatory Level Practice MCQs (With Detailed Solutions)

Q1. The Variance of a set of 15 observations is found to be 25. If every observation in the dataset is first increased by 10 and then multiplied by 2, what will be the new Standard Deviation and the new Variance respectively?
  • A. New SD = 20; New Variance = 110
  • B. New SD = 10; New Variance = 100
  • C. New SD = 15; New Variance = 50
  • D. New SD = 5; New Variance = 25
Correct Answer: (B) — Original Variance (σ²) = 25, so original SD (σ) = 5. Adding 10 (change of origin) has zero effect on SD or Variance. Multiplying by 2 (change of scale) makes New SD = 2 × 5 = 10, and New Variance = 2² × 25 = 4 × 25 = 100.

Q2. In a symmetrical (normal) distribution, if the Standard Deviation (SD) is 30, what are the approximate values of Quartile Deviation (QD) and Mean Deviation (MD) respectively?
  • A. QD = 20 and MD = 24
  • B. QD = 24 and MD = 20
  • C. QD = 15 and MD = 25
  • D. QD = 25 and MD = 28
Correct Answer: (A) — Using the empirical ratio QD : MD : SD = 4 : 5 : 6 (or QD = 2/3 SD and MD = 4/5 SD): QD = (2/3) × 30 = 20; MD = (4/5) × 30 = 24.

Q3. Batsman A has a mean score of 50 with a standard deviation of 10, whereas Batsman B has a mean score of 40 with a standard deviation of 6. Which batsman is more consistent?
  • A. Batsman A (because his mean score is higher)
  • B. Batsman B (because his Coefficient of Variation is 15% compared to A’s 20%)
  • C. Both are equally consistent
  • D. Consistency cannot be compared when means are different
Correct Answer: (B) — CV of A = (10 / 50) × 100 = 20%. CV of B = (6 / 40) × 100 = 15%. Since Batsman B has a lower Coefficient of Variation (15% < 20%), Batsman B is more consistent.

Q4. What is the Variance of the first 11 natural numbers (1, 2, 3, …, 11)?
  • A. 10
  • B. 11
  • C. 12
  • D. √10
Correct Answer: (A) — Variance of first n natural numbers = (n² − 1) / 12 = (11² − 1) / 12 = (121 − 1) / 12 = 120 / 12 = 10.

Q5. Which of the following is a graphical method used to measure the dispersion of income or wealth inequality in an economy?
  • A. Ogive Curve
  • B. Histogram
  • C. Lorenz Curve
  • D. Scatter Diagram
Correct Answer: (C) — The Lorenz Curve plots cumulative percentage of population against cumulative percentage of income/wealth; the farther the curve bows away from the 45° Line of Equal Distribution, the greater the dispersion/inequality.

❓ Frequently Asked Questions (Exam FAQs)

Q: Why is Coefficient of Variation (CV) preferred over Standard Deviation when comparing two datasets?

Standard Deviation is an absolute measure expressed in the units of the data (e.g., Rupees vs Kilograms) and depends on the magnitude of the mean. Coefficient of Variation (CV = [SD / Mean] × 100) is a unit-free relative percentage, allowing fair comparison of consistency across datasets with different units or vastly different means.

Q: How does adding or multiplying a constant affect Standard Deviation?

Standard Deviation is invariant to Change of Origin (adding or subtracting a constant leaves SD unchanged) but scales linearly with Change of Scale (multiplying every observation by k multiplies SD by |k| and Variance by k²).

Q: What is the empirical relationship between QD, MD, and SD?

In a normal distribution, 6 QD = 5 MD = 4 SD, meaning QD = (2/3) SD, MD = (4/5) SD, and the ratio QD : MD : SD is 4 : 5 : 6 (or more precisely 10 : 12 : 15).

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