8.0 Review: Measures of Central Tendency & Dispersion

Topic Description: This section reviews basic data analysis using marks of two students (A and B) in 8 subjects. It introduces central tendency (average performance) and dispersion (spread/variability). Graph representation helps visualize consistency. Key questions compare totals, averages, scatter, and best methods for measuring variability/consistency.

Data Example (Marks in 8 subjects):

• Student A: English 40, Nepali 50, C.Maths 65, Science 60, Social 58, Population 62, Opt.Maths 55, Computer 46
• Student B: English 50, Nepali 65, C.Maths 80, Science 35, Social 55, Population 70, Opt.Maths 85, Computer 25

Questions & Answers: i. Total & Average:

• Total A: 436, Average A: 54.5
• Total B: 465, Average B: 58.125

ii. More Scattered: B (range 25-85 vs A's 46-65; wider extremes).

iii. Better Achievement: B (higher average).

iv. Methods for Consistency/Variability: Range, Quartile Deviation (QD), Mean Deviation (MD), Standard Deviation (SD). Best: SD (uses all data, algebraic).

v. Comparison: Coefficient of Variation (CV = SD/Mean × 100%) for relative consistency.

Concepts:

• Central Tendency: Mean (arithmetic average), Median (middle value), Mode (most frequent) → concentration around center.
• Dispersion: Scatter/variability from average → homogeneity (low) vs heterogeneity (high).
  • Absolute: Same units as data (e.g., marks).
  • Relative: Unit-free coefficients for comparison.

8.1 Quartile Deviation (Semi-Interquartile Range)

Topic Description: Quartile deviation measures spread using middle 50% of data (interquartile range). Ignores extremes (unlike range), focuses on central spread. Good for skewed data.

Definition: QD = (Q3 - Q1)/2 (half interquartile range).

• Coefficient of QD = (Q3 - Q1)/(Q3 + Q1)

Quartiles (Continuous Series):

• Q1 (Lower): 25th percentile (N/4 position).
• Q2 (Median): 50th percentile (N/2).
• Q3 (Upper): 75th percentile (3N/4).
• Formula: Q = l + [(position - Cf)/f] × i
  • l: lower limit of quartile class
  • Cf: cumulative frequency before class
  • f: frequency of class
  • i: class width

Meaning:

• Q1: 25% data below.
• Q3: 75% data below (25% above).

Example: Given marks → Q1=43.5, Q3=64.375 → QD=10.437, Coeff≈0.193

8.2 Mean Deviation (Average Deviation)

Topic Description: Measures average absolute deviation from mean/median. Better than QD (uses more data, not just quartiles). Less influenced by extremes than range.

Definition: Average of |deviations| from central value (mean preferred).

• From Mean: MD = Σf|m - Mean| / N
• From Median: Similar with Median.
• Coefficient: MD / Mean (or Median)

Steps:

1. Compute Mean/Median.
2. Absolute deviations |m - central|.
3. Weighted average.

Example: Marks → MD from mean=10, Coeff=0.4

8.3 Standard Deviation (SD)

Topic Description: Most reliable dispersion measure (Karl Pearson). Square root of variance (average squared deviations from mean). Uses all data, sensitive to extremes, algebraic properties (e.g., for normal distribution).

Definition: σ = √(variance) = positive √[mean of (deviations)² from mean].

Methods (Continuous Series):

1. Actual Mean: √[Σf(m-Mean)² / N]
2. Direct: √[Σfm²/N - (Σfm/N)²]
3. Assumed Mean: √[Σfd²/N - (Σfd/N)²] (d=m-A)
4. Step Deviation: √[Σfd'²/N - (Σfd'/N)²] × i (d'=d/i)

Coefficient of SD: σ / Mean Coefficient of Variation (CV): (σ / Mean) × 100%

• Interpretation: Lower CV → more consistent/homogeneous/uniform. Higher CV → more variable/scattered.

Example: Marks → σ≈16.39, CV=21%

Comparison Example: Firm A (CV=1.53%) more uniform than B (1.73%).

Key Concepts Clarified

• Dispersion: Scatter from average → low: homogeneous/consistent; high: heterogeneous/variable.
• Absolute Measures: Units same as data (Range, QD, MD, SD).
• Relative Measures: Unit-free (coefficients) → compare different scales/units.
• Best Measure: SD → all data used, stable, mathematical (e.g., normal curve).
• Consistency: Low dispersion/CV → reliable/uniform performance.
• Variability: High dispersion/CV → scattered/inconsistent.

5 Most Common Questions

1. Define Quartile Deviation & Coefficient.
   • QD = (Q3-Q1)/2; Coeff = (Q3-Q1)/(Q3+Q1)
2. Compute QD from frequency table.
   • Find Q1/Q3 positions, use quartile formula → QD & Coeff.
3. Define Mean Deviation & Coefficient.
   • MD = Σf|dev| / N from mean/median; Coeff = MD/Mean
4. Compute SD (Assumed Mean/Step Deviation).
   • Table: m, f, d/d', fd/fd', d'², fd'² → apply formula.
5. Compare series using CV.
   • Lower CV → more consistent (e.g., better uniformity).

10 Important Solved Examples – Measures of Central Tendency & Dispersion

Example 1: Total and Average Marks
Marks of Student A: 40, 50, 65, 60, 58, 62, 55, 46
Marks of Student B: 50, 65, 80, 35, 55, 70, 85, 25

Total A = 436, Average A = 54.5

Total B = 465, Average B = 58.125

Example 2: Identify More Scattered

Student A: 46–65 → Range = 19

Student B: 25–85 → Range = 60

Conclusion: B is more scattered.

Example 3: Better Achievement

Compare averages: A = 54.5, B = 58.125

Conclusion: B has better overall performance.

Example 4: Compute Quartile Deviation (QD)
Given marks (continuous series): Q1 = 43.5, Q3 = 64.375

QD = (Q3 − Q1)/2 = (64.375 − 43.5)/2 = 10.437

Coefficient of QD = (Q3 − Q1)/(Q3 + Q1) = 20.875/107.875 ≈ 0.193

Example 5: Find Quartiles from Frequency Table

Steps:

Arrange cumulative frequency

Find Q1 position = N/4, Q3 = 3N/4

Apply formula: Q = l + [(position − Cf)/f] × i

Example 6: Mean Deviation from Mean
Marks: 40, 50, 65, 60, 58, 62, 55, 46

Mean = 54.5

Absolute deviations: |40−54.5|=14.5, |50−54.5|=4.5, …

MD = Σ|deviations|/N = 10

Coefficient MD = 10/54.5 ≈ 0.183

Example 7: Standard Deviation (Direct Method)
Marks: 40, 50, 65, 60, 58, 62, 55, 46

Mean = 54.5

Σm² = 40²+50²+65²+…+46² = 25384

σ = √[Σm²/N − (Mean)²] = √[25384/8 − 54.5²] ≈ 16.39

CV = σ / Mean × 100% = 16.39/54.5 ×100 ≈ 30%

Example 8: Comparison Using Coefficient of Variation

Firm A: CV = 1.53%

Firm B: CV = 1.73%

Conclusion: A is more consistent/uniform than B

Example 9: Compute Mean Deviation from Median

Median = Q2 = 56.5

Absolute deviations: |40−56.5|, |50−56.5|, …

MD = Σ|m−Median| / N = 9.75

Coefficient = 9.75 / 56.5 ≈ 0.172

Example 10: Absolute vs Relative Measures

Range = 60 marks (absolute, unit = marks)

SD = 16.39 (absolute)

CV = 30% (relative, unit-free, compare different series)

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