Class 11 Economics Chapter 6 — Measures of Dispersion (ASSEB)
Welcome to HSLC Guru! This English-medium guide for ASSEB Class 11 Economics (Statistics for Economics) Chapter 6 — Measures of Dispersion — covers complete summary, textbook question answers, additional MCQs, fill-in-the-blanks, true/false, glossary and a ready-reckoner formula table. Use it to master Range, Quartile Deviation, Mean Deviation, Standard Deviation, Variance, Coefficient of Variation and the Lorenz Curve for your half-yearly and final exams.
Summary
Meaning of Dispersion. Measures of central tendency (mean, median, mode) tell us about the centre of a distribution but they do not reveal how the individual observations are spread around that centre. Dispersion is the extent to which values in a distribution differ from their average. Two series can have the same mean yet be very different in spread. For example, the daily incomes of two groups may both average Rs 100, but in one group every worker earns near Rs 100 while in the other some earn Rs 20 and some Rs 200. Studying dispersion is therefore essential to judge the reliability of an average, to compare variability across series, to facilitate further statistical analysis (correlation, regression, quality control) and to support policy decisions on inequality.
Absolute and Relative Measures. An absolute measure of dispersion is expressed in the same unit as the original data (rupees, kilograms, marks). It cannot be used to compare two series measured in different units or having very different averages. A relative measure (also called a coefficient) is a pure number obtained by dividing the absolute measure by a suitable average; it is independent of units and is used for comparison. The four absolute measures discussed in the chapter are Range, Quartile Deviation, Mean Deviation and Standard Deviation, with their corresponding coefficients.
Range and Quartile Deviation. Range (R) is the difference between the largest (L) and smallest (S) observation: R = L − S. Coefficient of Range = (L − S)/(L + S). It is simple but uses only two extreme values. The Inter-Quartile Range = Q3 − Q1 covers the middle 50% of data. Quartile Deviation (Q.D.) or semi-inter-quartile range = (Q3 − Q1)/2. Coefficient of Q.D. = (Q3 − Q1)/(Q3 + Q1). Q.D. is not affected by extreme values and is useful for open-end distributions, but it ignores 50% of the data.
Mean Deviation, Standard Deviation, Variance, CV and Lorenz Curve. Mean Deviation (M.D.) is the arithmetic mean of the absolute deviations of items from a chosen average (mean or median). M.D. about median is least. Standard Deviation (σ), introduced by Karl Pearson (1893), is the positive square root of the arithmetic mean of squared deviations from the mean. It is the most widely used and best measure because it is based on every observation, is rigidly defined, suitable for further algebraic treatment, and least affected by sampling fluctuations. Variance = σ². Coefficient of Variation (C.V.) = (σ/Mean) × 100; the series with lower C.V. is more consistent/uniform/stable. The Lorenz Curve is a graphical device (cumulative percentage of variable on Y-axis vs cumulative percentage of frequency on X-axis) used to study inequality of income, wealth, profits etc.; the farther the curve from the line of equal distribution, the greater the inequality. Important properties of S.D.: (i) independent of change of origin but not of scale, (ii) it is the minimum root-mean-square deviation, (iii) for combined series a special formula combines means and S.D.s.
Textbook Questions and Answers
Very Short Answer Type (1 mark)
Q1. Define dispersion.
Answer: Dispersion is the extent to which the individual values in a distribution scatter around the average value.
Q2. Name any two absolute measures of dispersion.
Answer: Range and Standard Deviation (also Quartile Deviation, Mean Deviation).
Q3. Write the formula for Range.
Answer: Range (R) = L − S, where L = largest value and S = smallest value.
Q4. What is the coefficient of Range?
Answer: Coefficient of Range = (L − S) / (L + S).
Q5. Define Quartile Deviation.
Answer: Q.D. = (Q3 − Q1)/2; it is half of the inter-quartile range.
Q6. Mean Deviation is least when calculated from which average?
Answer: From the median.
Q7. Who introduced the concept of Standard Deviation?
Answer: Karl Pearson, in 1893.
Q8. What is Variance?
Answer: Variance is the square of the Standard Deviation, i.e. Variance = σ².
Q9. Write the formula for Coefficient of Variation.
Answer: C.V. = (σ / Mean) × 100.
Q10. What does the Lorenz Curve show?
Answer: It is a graphical measure of dispersion showing the degree of inequality in the distribution of income, wealth, profits, etc.
Short Answer Type (2-3 marks) — with short numericals
Q11. Distinguish between absolute and relative measures of dispersion.
Answer: Absolute measures are expressed in the same units as the data (e.g., rupees, kg) and cannot compare two series in different units. Relative measures are pure numbers (coefficients) obtained by dividing the absolute measure by a suitable average and are used to compare variability of two or more distributions.
Q12. Find the Range and Coefficient of Range for the data: 25, 30, 18, 42, 50, 12, 38.
Answer: L = 50, S = 12. Range = 50 − 12 = 38. Coefficient of Range = (50 − 12)/(50 + 12) = 38/62 = 0.613.
Q13. Calculate Quartile Deviation from the data: 4, 6, 9, 12, 15, 18, 22, 24, 28.
Answer: n = 9. Q1 = size of (n+1)/4 = 2.5th item = (6 + 9)/2 = 7.5. Q3 = size of 3(n+1)/4 = 7.5th item = (22 + 24)/2 = 23. Q.D. = (Q3 − Q1)/2 = (23 − 7.5)/2 = 7.75. Coefficient of Q.D. = (23 − 7.5)/(23 + 7.5) = 15.5/30.5 = 0.508.
Q14. Compute Mean Deviation from mean for: 10, 20, 30, 40, 50.
Answer: Mean = 150/5 = 30. Deviations |x − 30|: 20, 10, 0, 10, 20; Σ|d| = 60. M.D. = 60/5 = 12. Coefficient of M.D. = 12/30 = 0.40.
Q15. Why is Standard Deviation considered the best measure of dispersion?
Answer: Because it (i) is rigidly defined, (ii) is based on all observations, (iii) is capable of further algebraic treatment, (iv) is least affected by sampling fluctuations, and (v) is the minimum root-mean-square deviation from any chosen value.
Q16. Mean and S.D. of a series are 50 and 5 respectively; of another series 100 and 8. Which series is more variable?
Answer: C.V. of first = (5/50)×100 = 10%. C.V. of second = (8/100)×100 = 8%. The first series is more variable; the second is more consistent.
Long Answer Type (5-6 marks) — Step-by-step S.D. problems
Q17. Calculate the Standard Deviation of the following individual series by direct method: 6, 7, 10, 12, 13, 4, 8, 12.
Answer:
Step 1 — Mean: ΣX = 6+7+10+12+13+4+8+12 = 72; n = 8; Mean (X̄) = 72/8 = 9.
| X | (X − X̄) | (X − X̄)² |
|---|---|---|
| 6 | −3 | 9 |
| 7 | −2 | 4 |
| 10 | 1 | 1 |
| 12 | 3 | 9 |
| 13 | 4 | 16 |
| 4 | −5 | 25 |
| 8 | −1 | 1 |
| 12 | 3 | 9 |
| Σ | 0 | 74 |
Step 2 — Variance = Σ(X − X̄)²/n = 74/8 = 9.25.
Step 3 — S.D. (σ) = √9.25 = 3.04 (approx.).
Step 4 — Coefficient of Variation = (3.04/9) × 100 = 33.78%.
Q18. Calculate Standard Deviation of the discrete series given below using the assumed-mean (short-cut) method.
| X | 5 | 10 | 15 | 20 | 25 |
|---|---|---|---|---|---|
| f | 2 | 4 | 8 | 4 | 2 |
Answer: Take A = 15.
| X | f | d = X − 15 | fd | fd² |
|---|---|---|---|---|
| 5 | 2 | −10 | −20 | 200 |
| 10 | 4 | −5 | −20 | 100 |
| 15 | 8 | 0 | 0 | 0 |
| 20 | 4 | 5 | 20 | 100 |
| 25 | 2 | 10 | 20 | 200 |
| Σ | 20 | — | 0 | 600 |
σ = √[ Σfd²/N − (Σfd/N)² ] = √[600/20 − 0²] = √30 = 5.477.
Mean = A + Σfd/N = 15 + 0 = 15. C.V. = (5.477/15) × 100 = 36.51%.
Q19. Compute Mean, Standard Deviation and Coefficient of Variation of the following continuous series.
| Class | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
| f | 5 | 10 | 20 | 10 | 5 |
Answer: Take A = 25, h = 10, d’ = (m − 25)/10.
| Class | m (mid) | f | d’ | fd’ | fd’² |
|---|---|---|---|---|---|
| 0–10 | 5 | 5 | −2 | −10 | 20 |
| 10–20 | 15 | 10 | −1 | −10 | 10 |
| 20–30 | 25 | 20 | 0 | 0 | 0 |
| 30–40 | 35 | 10 | 1 | 10 | 10 |
| 40–50 | 45 | 5 | 2 | 10 | 20 |
| Σ | — | 50 | — | 0 | 60 |
Mean = A + (Σfd’/N) × h = 25 + (0/50) × 10 = 25.
σ = h × √[ Σfd’²/N − (Σfd’/N)² ] = 10 × √[60/50 − 0] = 10 × √1.2 = 10 × 1.0954 = 10.95.
C.V. = (10.95/25) × 100 = 43.82%.
Q20. The mean and S.D. of the marks of two sections are: Section A — Mean 60, σ = 6; Section B — Mean 50, σ = 8. Which section is more consistent? Also find the variance of each section.
Answer: Variance of A = σ² = 36; Variance of B = σ² = 64. C.V. (A) = (6/60) × 100 = 10%. C.V. (B) = (8/50) × 100 = 16%. Since C.V. of A < C.V. of B, Section A is more consistent (less variable). Section B has greater dispersion in marks.
Q21. From the following data, calculate Mean Deviation about Median and its coefficient: 8, 12, 15, 18, 20, 22, 25, 30, 33.
Answer: Arranged data (already sorted), n = 9. Median = (n+1)/2 = 5th item = 20.
| X | |X − 20| |
|---|---|
| 8 | 12 |
| 12 | 8 |
| 15 | 5 |
| 18 | 2 |
| 20 | 0 |
| 22 | 2 |
| 25 | 5 |
| 30 | 10 |
| 33 | 13 |
| Σ | 57 |
M.D. (Median) = Σ|X − Median|/n = 57/9 = 6.33. Coefficient of M.D. = 6.33/20 = 0.317.
Additional Practice — Multiple Choice Questions
Q1. The simplest measure of dispersion is —
(a) Mean Deviation (b) Range (c) Standard Deviation (d) Variance
Answer: (b) Range.
Q2. Quartile Deviation is equal to —
(a) Q3 − Q1 (b) (Q3 + Q1)/2 (c) (Q3 − Q1)/2 (d) Q3/Q1
Answer: (c) (Q3 − Q1)/2.
Q3. Mean Deviation is minimum when measured from —
(a) Mean (b) Mode (c) Median (d) Geometric Mean
Answer: (c) Median.
Q4. Variance is —
(a) Square root of S.D. (b) Square of S.D. (c) Mean of deviations (d) None
Answer: (b) Square of S.D.
Q5. Coefficient of Variation is expressed as —
(a) Pure number (b) Rupees (c) Percentage (d) Ratio
Answer: (c) Percentage.
Q6. The Lorenz Curve is used to study —
(a) Central tendency (b) Inequality (c) Correlation (d) Time series
Answer: (b) Inequality.
Q7. Standard Deviation is independent of change of —
(a) Origin only (b) Scale only (c) Both origin and scale (d) Neither
Answer: (a) Origin only.
Q8. If C.V. of A = 12% and of B = 18%, the more consistent series is —
(a) A (b) B (c) Both equal (d) Cannot say
Answer: (a) A.
Q9. S.D. was introduced by —
(a) Bowley (b) Karl Pearson (c) Fisher (d) Spearman
Answer: (b) Karl Pearson.
Q10. Coefficient of Range = —
(a) (L − S)/(L + S) (b) L − S (c) L/S (d) (L + S)/(L − S)
Answer: (a) (L − S)/(L + S).
Q11. The middle 50% of observations lie between —
(a) Q1 and Median (b) Q1 and Q3 (c) Mean and Median (d) S and L
Answer: (b) Q1 and Q3.
Q12. Which of the following is a relative measure?
(a) Range (b) Q.D. (c) S.D. (d) Coefficient of Variation
Answer: (d) Coefficient of Variation.
Fill in the Blanks
Q1. The difference between the largest and smallest value of a series is called __________.
Answer: Range.
Q2. Standard Deviation is the positive square root of __________.
Answer: Variance (mean of squared deviations from mean).
Q3. Coefficient of Variation = (σ/Mean) × __________.
Answer: 100.
Q4. The Lorenz Curve was developed by __________.
Answer: Max O. Lorenz.
Q5. Mean Deviation is least when measured from __________.
Answer: the median.
True or False
Q1. Range is based on all observations of the series.
Answer: False (only on the largest and smallest).
Q2. Standard Deviation cannot be negative.
Answer: True.
Q3. Variance and S.D. are measured in the same unit as data.
Answer: False (variance is in squared units).
Q4. Lower C.V. indicates higher consistency.
Answer: True.
Q5. Quartile Deviation is influenced by extreme values.
Answer: False.
Glossary
| Term | Meaning |
|---|---|
| Dispersion | Scatter of values around an average. |
| Range | L − S, the simplest absolute measure. |
| Inter-Quartile Range | Q3 − Q1; covers middle 50% of data. |
| Quartile Deviation | (Q3 − Q1)/2; semi-inter-quartile range. |
| Mean Deviation | Mean of absolute deviations from a chosen average. |
| Standard Deviation (σ) | Positive square root of arithmetic mean of squared deviations from mean. |
| Variance | σ² — square of S.D. |
| Coefficient of Variation | (σ/Mean) × 100; relative measure of variability. |
| Lorenz Curve | Graph showing inequality of distribution. |
| Line of Equal Distribution | 45° diagonal in Lorenz diagram representing perfect equality. |
Formula Table (Quick Revision)
| Measure | Absolute Formula | Relative Formula (Coefficient) |
|---|---|---|
| Range | R = L − S | (L − S)/(L + S) |
| Quartile Deviation | (Q3 − Q1)/2 | (Q3 − Q1)/(Q3 + Q1) |
| Mean Deviation (Mean) | Σ|X − X̄|/n or Σf|X − X̄|/N | M.D./Mean |
| Mean Deviation (Median) | Σ|X − M|/n or Σf|X − M|/N | M.D./Median |
| Standard Deviation (Direct) | σ = √[Σ(X − X̄)²/n] | — |
| S.D. (Short-cut, individual) | σ = √[Σd²/n − (Σd/n)²], d = X − A | — |
| S.D. (Discrete, short-cut) | σ = √[Σfd²/N − (Σfd/N)²] | — |
| S.D. (Continuous, step-deviation) | σ = h × √[Σfd’²/N − (Σfd’/N)²], d’ = (m − A)/h | — |
| Variance | σ² | — |
| Coefficient of Variation | — | C.V. = (σ/Mean) × 100 |
| Combined S.D. | σ₁₂ = √[(n₁σ₁² + n₂σ₂² + n₁d₁² + n₂d₂²)/(n₁ + n₂)] | — |
Tip: In examinations, always state the formula first, then substitute carefully, show every column of the working table, and finish with the unit / percentage. For comparison questions remember — lower C.V. → more consistent / uniform / stable / homogeneous; higher C.V. → more variable / heterogeneous.
Properties of Standard Deviation (Quick Notes)
- Standard Deviation is always non-negative; σ = 0 means all observations are equal.
- S.D. is independent of change of origin: adding/subtracting a constant from every value does not affect σ.
- S.D. is dependent on change of scale: multiplying every value by k multiplies σ by |k|.
- For a combined series of two groups with sizes n₁, n₂, means X̄₁, X̄₂ and standard deviations σ₁, σ₂, the combined S.D. is given by σ₁₂ = √[(n₁σ₁² + n₂σ₂² + n₁d₁² + n₂d₂²)/(n₁ + n₂)] where d₁ = X̄₁ − X̄, d₂ = X̄₂ − X̄, and X̄ is the combined mean.
- S.D. is the minimum of the root-mean-square deviations taken from any value; it is minimum when measured from the arithmetic mean.
- S.D. is suitable for further mathematical and statistical treatment such as correlation, regression, skewness and kurtosis.
Lorenz Curve — Construction Steps
- Arrange the data of the variable (income/wealth) in ascending order along with the corresponding number of persons.
- Compute cumulative totals for both variable and frequency.
- Convert the cumulative totals into cumulative percentages of the respective grand totals.
- Plot the cumulative percentage of frequency on the X-axis (0 to 100) and cumulative percentage of variable on the Y-axis (0 to 100).
- Join the points with a smooth curve. Draw the diagonal from (0,0) to (100,100) — this is the line of equal distribution.
- The greater the gap between the curve and the diagonal, the greater the inequality.
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