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CBSE Class 11 Economics★ 6-8 marks · CBSE Class 11 Economics (Statistics for Economics) Chapter 6

Measures of Dispersion (Range, Quartile Deviation, Standard Deviation)

The 'average' tells you the CENTRE of the data. But how SPREAD OUT is it? Two datasets can have the same mean but wildly different spreads. This chapter covers range, quartile deviation, mean deviation, standard deviation, and the Lorenz Curve. CBSE Class 11 Economics (Statistics for Economics) Chapter 6.

⏱ 20 min readHard difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Calculate range and quartile deviation and explain what each measures
2Compute mean deviation from mean and from median for individual and discrete series
3Calculate variance and standard deviation using the direct method for individual and continuous series
4Compute the coefficient of variation (CV) and use it to compare consistency between two datasets
5Explain the Lorenz Curve, what it represents, and how it shows inequality

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Why this chapter matters

The average alone is incomplete — two countries can have the same per capita income but wildly different inequality. Dispersion measures (especially standard deviation and CV) are essential for understanding income inequality, investment risk, and the consistency of economic data.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Measures of Central Tendency

Mean and median are needed to calculate mean deviation and standard deviation

Measures of Dispersion

"The mean tells you where the centre is. The standard deviation tells you how far from centre the data wanders."

1. Chapter Overview

CENTRAL TENDENCY tells you WHERE the data clusters. DISPERSION tells you how SPREAD OUT it is. Both are essential. This chapter covers: absolute and relative measures of dispersion — Range, Quartile Deviation, Mean Deviation, Standard Deviation (the most important), Coefficient of Variation, and the Lorenz Curve (for inequality).

2. Why Dispersion Matters

Two datasets with IDENTICAL means:

Dataset A: [50, 50, 50, 50, 50] → Mean = 50. Variation = ZERO.
Dataset B: [0, 25, 50, 75, 100] → Mean = 50. Variation = LARGE.

The mean is the SAME. The story is COMPLETELY DIFFERENT.

Absolute vs Relative Dispersion

Absolute: Expressed in the SAME UNITS as the data (dollars, kg, cm). Example: Range, Standard Deviation.
Relative: Expressed as a RATIO or PERCENTAGE. UNIT-FREE. Allows COMPARISON across datasets with different units. Example: Coefficient of Variation.

3. Measures of Dispersion

Range

Range = Maximum value — Minimum value
Simplest measure. ONLY uses two extreme values. IGNORES everything in between. Highly affected by outliers.

Quartile Deviation (Semi-Interquartile Range)

Q.D. = (Q₃ — Q₁) / 2
Q₁ (First quartile): 25% of observations below it. Q₃ (Third quartile): 75% below it.
Covers the MIDDLE 50% of data. Not affected by extreme values at either end.

Mean Deviation (M.D.)

Average of the ABSOLUTE deviations from a measure of central tendency (usually mean or median)
M.D. = Σ|X — A| / N (where A is mean or median)
Uses ALL values. But: using absolute values makes it mathematically less tractable.

Standard Deviation (S.D., σ) — THE MOST IMPORTANT

σ = √[Σ(X — X̄)² / N] (for population)
The SQUARE ROOT of the average of SQUARED deviations from the mean
Why squared? (1) Eliminates sign (negative deviations become positive when squared). (2) Gives MORE WEIGHT to larger deviations. (3) Mathematically TRACTABLE.

Variance

Variance = σ² = Σ(X — X̄)² / N
The standard deviation SQUARED

Coefficient of Variation (C.V.)

C.V. = (σ / X̄) × 100
Relative measure. Allows COMPARISON of dispersion across datasets with DIFFERENT MEANS or DIFFERENT UNITS.
LOWER C.V. = more consistent/less variability.

4. Lorenz Curve — Measuring Inequality

Graphical representation of INEQUALITY (income, wealth, land distribution)
Horizontal axis: CUMULATIVE % of population (from poorest to richest)
Vertical axis: CUMULATIVE % of income (or whatever is being measured)
Line of perfect equality: 45° diagonal. Bottom 20% of population have 20% of income; bottom 50% have 50%; etc.
Lorenz Curve: the ACTUAL distribution. BOWS BELOW the line of equality.
The FARTHER the Lorenz Curve is from the line of equality → the GREATER the inequality
Gini Coefficient (not in detail in Class 11, but related): numerical measure derived from Lorenz Curve

5. Which Measure to Use?

Measure	Pros	Cons
Range	Simplest	Uses only 2 values. Unstable.
Quartile Deviation	Ignores extremes	Ignores most data
Mean Deviation	Uses all data	Absolute values — mathematically less useful
Standard Deviation	Mathematically BEST. Foundation of statistics.	Sensitive to outliers
Coefficient of Variation	Allows cross-dataset comparison	Derived from mean and SD — inherits their assumptions

Standard Deviation is the default choice. Complement with C.V. for relative comparison.

6. Exam Focus

Range, Quartile Deviation — definitions and formulas
Standard Deviation — formula, calculation, why squared deviations
Variance — definition
Coefficient of Variation — formula, when used
Lorenz Curve — what it shows, how to read it, line of equality

7. Conclusion

Knowing the AVERAGE is not enough. You must know the SPREAD:

RANGE: Max — Min. Crude but instant.
STANDARD DEVIATION: The gold standard. Uses all data. Mathematically powerful. The foundation of inferential statistics.
C.V.: For comparing apples and oranges (data with different means/units).
LORENZ CURVE: For SEEING inequality.

'The mean is a seductive liar. The standard deviation is the honest friend who tells you the truth about how messy the data really is.'

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Range

Range = L − S (Largest value − Smallest value)

Simplest measure; only uses two extreme values; highly sensitive to outliers

Quartile Deviation (Semi-Interquartile Range)

Q.D. = (Q₃ − Q₁) / 2

Q₁ = value below which 25% of data falls; Q₃ = value below which 75% falls; covers middle 50% of data; ignores extremes

Mean Deviation from Mean

M.D.(X̄) = Σ|X − X̄| / N (individual); Σf|X − X̄| / Σf (discrete)

Average of absolute deviations from mean; uses all values; absolute value removes negative signs

Mean Deviation from Median

M.D.(M) = Σ|X − M| / N (individual); Σf|X − M| / Σf (discrete)

Mean deviation is minimum when calculated from the median — but mean deviation from mean is more commonly asked

Variance

σ² = Σ(X − X̄)² / N (individual); Σf(X − X̄)² / Σf (discrete/continuous using mid-points)

Average of squared deviations from mean; squaring eliminates negative signs and gives MORE weight to larger deviations

Standard Deviation

σ = √[Σ(X − X̄)² / N]

Square root of variance; in original units; the most important measure of dispersion in statistics

Coefficient of Variation

C.V. = (σ / X̄) × 100

Relative measure; unit-free; lower CV = more consistent; used to compare dispersion across datasets with different means or units

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Not taking the absolute value when calculating mean deviation

✓ Mean deviation uses |X − X̄| or |X − M| — the ABSOLUTE value (ignore the negative sign). Without absolute values, deviations above and below the mean cancel out and sum to zero, making the result meaningless.

WATCH OUT

✗ Confusing variance with standard deviation

✓ Variance = σ² (units are squared, e.g., ₹²). Standard deviation = σ = √variance (same units as original data, e.g., ₹). Variance is the intermediate step; standard deviation is the final, interpretable result. Do not stop at variance unless specifically asked.

WATCH OUT

✗ Using CV to compare datasets without computing it correctly

✓ C.V. = (σ / X̄) × 100. Both σ and X̄ must be in the same units. Lower CV means LESS variability (more consistent), not more. When comparing two investments or datasets, the one with lower CV is more stable.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 6

Exercise 6

Calculation of range, quartile deviation, mean deviation, standard deviation, and coefficient of variation for individual, discrete, and continuous series; Lorenz Curve interpretation

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· range-QD

For the following data: 15, 22, 8, 35, 12, 28, 45, 20. Calculate (a) Range (b) If Q₁ = 13 and Q₃ = 31.5, find Quartile Deviation.

▶ Show solution

(a) Range = L − S = 45 − 8 = 37. This means the entire spread of data is 37 units. (b) Quartile Deviation = (Q₃ − Q₁) / 2 = (31.5 − 13) / 2 = 18.5 / 2 = 9.25. Interpretation: The middle 50% of data is spread over a range of 18.5 units, and the typical deviation from the median is about 9.25 units.

Q2MEDIUM· standard-deviation

Calculate the Standard Deviation and Variance for the following data: 4, 8, 12, 16, 20.

▶ Show solution

Step 1: Mean X̄ = (4 + 8 + 12 + 16 + 20) / 5 = 60 / 5 = 12. Step 2: Calculate (X − X̄) and (X − X̄)²: X=4: deviation = 4−12 = −8, (−8)² = 64. X=8: deviation = 8−12 = −4, (−4)² = 16. X=12: deviation = 12−12 = 0, 0² = 0. X=16: deviation = 16−12 = 4, 4² = 16. X=20: deviation = 20−12 = 8, 8² = 64. Step 3: Σ(X − X̄)² = 64 + 16 + 0 + 16 + 64 = 160. Step 4: Variance σ² = 160 / 5 = 32. Step 5: Standard Deviation σ = √32 = 5.66 (approx).

Q3HARD· coefficient-of-variation

Two batsmen's scores across 6 innings are: Batsman A: 48, 52, 55, 60, 42, 55 (X̄ = 52, σ = 5.77). Batsman B: 30, 75, 20, 85, 15, 90 (X̄ = 52.5, σ = 30.4). (a) Calculate the Coefficient of Variation for each. (b) Which batsman is more consistent and why? (c) Why is CV a better measure of comparison than SD here?

▶ Show solution

(a) C.V. for Batsman A = (σ/X̄) × 100 = (5.77/52) × 100 = 11.1%. C.V. for Batsman B = (30.4/52.5) × 100 = 57.9%. (b) Batsman A is more consistent because his CV (11.1%) is far lower than Batsman B's (57.9%). A lower CV means the scores are clustered close to the mean — A scores consistently around 52 runs. B's high CV means his performance is highly variable — sometimes very high, sometimes very low. (c) Why CV is better than SD here: Both batsmen have nearly the same mean (~52 runs). If we only compared SD (5.77 vs 30.4), we'd also conclude A is more consistent. But what if one batsman had a mean of 30 and another had a mean of 80? Comparing SD directly when means differ is misleading — a high SD on a mean of 80 might be proportionally smaller than a lower SD on a mean of 20. CV expresses dispersion as a PERCENTAGE of the mean, making it unit-free and comparable regardless of the magnitude of mean scores.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Range = Largest − Smallest; simple but ignores all values between extremes
•Quartile Deviation = (Q₃ − Q₁) / 2; covers middle 50% of data; not affected by extreme outliers
•Mean Deviation = Σ|X − A| / N where A is mean or median; absolute value is MANDATORY
•Variance σ² = Σ(X − X̄)² / N; Standard Deviation σ = √variance; SD is in same units as original data
•Coefficient of Variation = (σ / X̄) × 100; lower CV = more consistent (less variable)
•Absolute dispersion: in original units (Range, SD, MD); Relative dispersion: unit-free ratio (CV)
•Lorenz Curve: shows income/wealth inequality; the farther from the 45° line of equality, the greater the inequality
•Property of variance: Σ(X − X̄)² is MINIMUM when deviations are from the mean (not any other value)

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 6-8 marks

Question type	Marks each	Typical count	What it tests
Short Answer	3	1	Range, quartile deviation, or mean deviation calculation; or conceptual difference between absolute and relative dispersion
Long Answer	6	1	Standard deviation calculation for individual or continuous series, followed by coefficient of variation comparison

Prep strategy

Standard deviation must be practised thoroughly — the step-by-step table (X, X−X̄, (X−X̄)², f(X−X̄)²) must be presented neatly; each step earns marks
Practise CV comparison problems: compute CV for both datasets and explicitly state 'lower CV = more consistent' in your conclusion for full credit
Lorenz Curve: be able to explain it in words (what it shows, what the line of equality represents, and what greater distance from the line means) — a guaranteed 3-mark theory question

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Income Inequality — Lorenz Curve for India

India's Lorenz Curve has shifted further from the line of equality over the past 30 years, reflecting rising income inequality. The top 10% hold a disproportionate share of national income — visible as the bow in India's Lorenz Curve.

Coefficient of Variation in Investment

Mutual fund analysts use CV to compare two investment options with different expected returns. A fund with CV = 15% is more consistently managed than one with CV = 40% — even if both have the same average return.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Always present SD calculations as a neat table with columns: X, (X − X̄), (X − X̄)², and frequency if needed — messy calculations lose follow-through marks
For CV comparison problems: compute CV for both series, state which is lower, and write one sentence of interpretation ('Batsman A is more consistent because...') for full marks
Lorenz Curve theory questions: draw a clear sketch with axes labelled, 45° line of equality drawn, and actual Lorenz Curve bowing below — a good diagram earns 2-3 marks
Mean deviation question: show absolute values explicitly as |X − X̄| in the working — forgetting the absolute value sign will cost marks

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Explore the coefficient of skewness using Pearson's formula: SK = (Mean − Mode) / SD — connects central tendency and dispersion to describe the shape of a distribution
Study the properties of variance: Var(aX + b) = a²·Var(X) — important for understanding how transformations affect dispersion, used in financial modelling

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 11 BoardHigh

CUETHigh

Class 12 Statistics / MathsMedium

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

SD uses ALL values in the dataset (unlike range or quartile deviation). Squaring the deviations makes it mathematically tractable — it connects directly to the normal distribution, hypothesis testing, and regression analysis in higher statistics. It is also the building block for the Coefficient of Variation.

The Lorenz Curve plots cumulative % of population (X-axis, poorest to richest) against cumulative % of income (Y-axis). The 45° diagonal is perfect equality. The actual Lorenz Curve bows below it — the greater the bow, the greater the inequality. The Gini Coefficient (not required in detail for Class 11) is the ratio of the area between the curve and the line of equality to the total area below the line — ranges from 0 (perfect equality) to 1 (perfect inequality).

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