If L = 18 and S = 3, the coefficient of range is:

5/7. (18 − 3)/(18 + 3) = 15/21 = 5/7.

If Q₁ = 20 and Q₃ = 40, the coefficient of quartile deviation is:

1/3. (40 − 20)/(40 + 20) = 20/60 = 1/3.

Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 2 Chapter 2: Measures of Dispersion

Short notes, key concepts, quick revision and practice MCQs with answers

Averages alone do not show how spread out the data are. Dispersion measures the scatter of observations around a central value. The chapter covers range, quartile deviation, variance, standard deviation (including the step-deviation method) and the coefficient of variation used to compare consistency.

Key concepts: Measures of Dispersion

Dispersion: The extent to which observations are scattered about a central value. Absolute measures have the units of the data; relative measures (coefficients) are unit free and used for comparison.
Range: Range = L − S, the largest minus the smallest value. Coefficient of range = (L − S)/(L + S).
Quartile deviation: Inter-quartile range = Q₃ − Q₁. Quartile deviation (semi inter-quartile range) = (Q₃ − Q₁)/2. Coefficient of Q.D. = (Q₃ − Q₁)/(Q₃ + Q₁).
Variance of raw data: σ² = (1/n) Σ(xᵢ − x̄)² = (1/n) Σxᵢ² − x̄².
Variance of a frequency distribution: σ² = (1/N) Σfᵢ(xᵢ − x̄)² = (1/N) Σfᵢxᵢ² − x̄², where N = Σfᵢ and x̄ = Σfᵢxᵢ/N.
Standard deviation: S.D. σ = √variance. It is the most widely used measure of dispersion.
Step-deviation method: Let u = (x − A)/h, where A is the assumed mean and h the class width. Then Var(X) = h² Var(U) and σₓ = h σᵤ. S.D. is independent of change of origin but not of scale.
Coefficient of variation: C.V. = (σ/x̄) × 100. The series with the smaller C.V. is more consistent (more stable).

Quick revision: important points

Variance of a constant is 0.
Adding a constant to every observation does not change the variance or S.D.
Multiplying every observation by k multiplies S.D. by |k| and variance by k².
S.D. of aX + b = |a| × S.D. of X.
Lower C.V. means greater consistency or uniformity.

Practice MCQs with answers

The range of 5, 12, 3, 18, 9 is:
1. 13
2. 15
3. 18
4. 21
Show answer
Answer: B. 15
18 − 3 = 15.
If L = 18 and S = 3, the coefficient of range is:
1. 5/7
2. 7/5
3. 15
4. 1/7
Show answer
Answer: A. 5/7
(18 − 3)/(18 + 3) = 15/21 = 5/7.
If Q₁ = 20 and Q₃ = 40, the quartile deviation is:
1. 20
2. 10
3. 30
4. 60
Show answer
Answer: B. 10
(40 − 20)/2 = 10.
If Q₁ = 20 and Q₃ = 40, the coefficient of quartile deviation is:
1. 1/2
2. 1/3
3. 2/3
4. 3
Show answer
Answer: B. 1/3
(40 − 20)/(40 + 20) = 20/60 = 1/3.
Standard deviation is the positive square root of:
1. Mean
2. Range
3. Variance
4. Q.D.
Show answer
Answer: C. Variance
σ = √variance.

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