Measures of Dispersion

Two classes can have the same average mark and be nothing alike: in one every student scored near 50, in the other half scored 20 and half 80. Dispersion measures that spread. This chapter covers the range, the quartile deviation, the mean deviation and the standard deviation with their coefficients, the coefficient of variation used to compare variability, the effect of changing origin and scale, and the combined standard deviation of two groups.

Absolute and relative measures

An absolute measure of dispersion is in the units of the data (rupees, kg). A relative measure — a coefficient — divides by an average or a total and is a pure number, so it can compare series in different units or with very different averages. Every absolute measure below has a matching coefficient.

The four measures and their coefficients
MeasureAbsolute formCoefficient
RangeR = L − S(L − S) ÷ (L + S)
Quartile deviationQ.D. = (Q₃ − Q₁) ÷ 2(Q₃ − Q₁) ÷ (Q₃ + Q₁)
Mean deviationM.D. = Σ|x − A| ÷ nM.D. ÷ A (A = mean or median)
Standard deviationσ = √(Σ(x − x̄)² ÷ n)σ ÷ x̄ (× 100 = C.V.)

Range, quartile deviation and mean deviation

The range is the largest value minus the smallest. It is quick and easily distorted by one extreme value. The quartile deviation (semi-interquartile range) uses only the middle half: Q.D. = (Q₃ − Q₁)/2, so it ignores the extreme quarters and suits open-end distributions. The mean deviation averages the absolute distances from a centre: M.D. = Σ|x − A|/n, and it is smallest when A is the median.

Worked: for 4, 8, 6, 10, 12 the mean is 8; the absolute deviations are 4, 0, 2, 2, 4, so M.D. about the mean = 12/5 = 2.4; the range is 12 − 4 = 8 and its coefficient is 8/16 = 0.5.

Standard deviation and variance

The standard deviation squares the deviations instead of dropping their signs, averages them and takes the square root: σ = √(Σ(x − x̄)²/n). Its square σ² is the variance. A convenient form is σ² = Σx²/n − x̄² ('mean of squares minus square of mean'), and for grouped data σ² = Σfx²/N − x̄², or with step deviations σ = h√(Σfu²/N − (Σfu/N)²).

Worked: 4, 8, 6, 10, 12
xx − x̄(x − x̄)²
4-416
800
6-24
1024
12416
Total040

σ² = 40/5 = 8 and σ = √8 ≈ 2.83. Check with the short form: Σx² = 360, 360/5 − 8² = 72 − 64 = 8.

Change of origin and scale. If y = a + bx, then σᵧ = |b|·σₓ and Var(y) = b²·Var(x). Adding a constant moves the data without spreading it, so the SD is unchanged; multiplying by b stretches it by |b|. Two standard results: the SD of the first n natural numbers is √((n² − 1)/12), and for two values a and b the SD is |a − b|/2.

⚠️ Exam trap
Variance responds to the square of the multiplier. Doubling every value doubles the SD but multiplies the variance by 4; multiplying by −2 still gives SD × 2, never a negative SD.

Coefficient of variation and combined SD

The coefficient of variation C.V. = (σ ÷ x̄) × 100 expresses the SD as a percentage of the mean. The series with the smaller C.V. is more consistent (more stable, more homogeneous). For 4, 8, 6, 10, 12, C.V. = 2.83/8 × 100 ≈ 35.4%.

For two groups of sizes n₁, n₂ with means x̄₁, x̄₂ and SDs σ₁, σ₂, the combined variance is σ² = [n₁(σ₁² + d₁²) + n₂(σ₂² + d₂²)] ÷ (n₁ + n₂), where d₁ = x̄₁ − x̄ and d₂ = x̄₂ − x̄ measure how far each group's mean sits from the combined mean. Forgetting the d² terms understates the spread whenever the group means differ.

ℹ️ Note
For a normal distribution the measures stand in fixed ratios: Q.D. ≈ (2/3)σ and M.D. ≈ (4/5)σ, so Q.D. : M.D. : S.D. ≈ 10 : 12 : 15.

Exam traps and a quick revision

  • Range and Q.D. use two values each; M.D. and S.D. use every value.
  • S.D. is unchanged by a change of origin and changes by |b| with a change of scale.
  • Smaller C.V. = more consistent, whatever the raw SDs say.
  • SD of 1, 2, …, n is √((n² − 1)/12); SD of two numbers is half their difference.
  • Variance can never be negative; it is zero only when all values are equal.

Key takeaways

  • Absolute measures carry the data's units; coefficients are pure numbers that compare series.
  • Q.D. = (Q₃ − Q₁)/2; M.D. is least about the median; σ² = Σx²/n − x̄².
  • For y = a + bx: σᵧ = |b|σₓ and Var(y) = b²Var(x).
  • C.V. = σ/x̄ × 100; the lower C.V. marks the more consistent series.
  • Combined variance adds each group's d² = (x̄ᵢ − x̄)² to its own variance before weighting.

Practice questions (10)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. Which measure of dispersion is based only on the middle 50% of the observations?

    1. range
    2. quartile deviation
    3. standard deviation
    4. mean deviation
    Show answer

    Answer: B — quartile deviation

    Q.D. = (Q₃ − Q₁)/2 uses only the stretch between the first and third quartiles, which holds the middle half of the data. The range uses only the two extremes.

  2. If 10 is added to every observation, the standard deviation:

    1. remains unchanged
    2. becomes 10 times
    3. increases by 100
    4. increases by 10
    Show answer

    Answer: A — remains unchanged

    Adding a constant shifts every value and the mean by the same amount, so every deviation x − x̄ is unchanged and so is σ.

  3. The variance of a variable X is 9. The variance of Y = 5 − 2X is:

    1. 18
    2. −18
    3. 13
    4. 36
    Show answer

    Answer: D — 36

    Var(a + bX) = b²·Var(X) = (−2)² × 9 = 36. The constant 5 has no effect, and the negative sign disappears on squaring.

  4. The standard deviation of the observations 4, 8, 6, 10, 12 is:

    1. 2√2
    2. 4
    3. 2.4
    4. 8
    Show answer

    Answer: A — 2√2

    Mean = 8; squared deviations 16, 0, 4, 4, 16 sum to 40; variance = 40/5 = 8 and σ = √8 = 2√2 ≈ 2.83. 8 is the variance and 2.4 the mean deviation.

  5. Batsman A averages 50 runs with SD 10; batsman B averages 40 runs with SD 9. Which is more consistent, and why?

    1. Both are equally consistent
    2. A, because his average is higher
    3. B, because his SD is lower
    4. A, because his C.V. is lower
    Show answer

    Answer: D — A, because his C.V. is lower

    Consistency is judged by the coefficient of variation: A = 10/50 × 100 = 20%, B = 9/40 × 100 = 22.5%. A's lower C.V. makes him more consistent even though his SD is larger.

  6. For a distribution Q₁ = 30 and Q₃ = 50. The coefficient of quartile deviation is:

    1. 0.25
    2. 0.2
    3. 0.5
    4. 10
    Show answer

    Answer: A — 0.25

    Coefficient of Q.D. = (Q₃ − Q₁)/(Q₃ + Q₁) = 20/80 = 0.25. The quartile deviation itself is 20/2 = 10.

  7. The standard deviation of the first 9 natural numbers 1, 2, …, 9 is:

    1. √(20/3)
    2. 20/3
    3. 3
    4. √10
    Show answer

    Answer: A — √(20/3)

    SD of 1 to n = √((n² − 1)/12) = √(80/12) = √(20/3) ≈ 2.58. 20/3 is the variance.

  8. The coefficient of variation is defined as:

    1. (range ÷ mean) × 100
    2. (standard deviation ÷ mean) × 100
    3. (mean ÷ standard deviation) × 100
    4. (variance ÷ mean) × 100
    Show answer

    Answer: B — (standard deviation ÷ mean) × 100

    C.V. = σ/x̄ × 100, the SD as a percentage of the mean. It is a pure number, which is what lets it compare series in different units.

  9. Group A has 10 items with mean 20 and variance 4; group B has 10 items with mean 30 and variance 4. The variance of all 20 items together is:

    1. 4
    2. 8
    3. 29
    4. 25
    Show answer

    Answer: C — 29

    Combined mean = 25, so d₁ = −5 and d₂ = 5. σ² = [10(4 + 25) + 10(4 + 25)]/20 = 29. Averaging only the two variances gives 4 and ignores the gap between the group means.

  10. For a normal distribution with standard deviation 15, the quartile deviation is approximately:

    1. 7.5
    2. 12
    3. 15
    4. 10
    Show answer

    Answer: D — 10

    For a normal distribution Q.D. ≈ (2/3)σ = 10 and M.D. ≈ (4/5)σ = 12. The ratio Q.D. : M.D. : S.D. ≈ 10 : 12 : 15 is exactly this case.