Lecture on Measures of Dispersion

Introduction

• The lecture covers the chapter on measures of dispersion.
• Focus on summarizing formulas and understanding application on questions.
• Important to learn the formulas and their applications.

Topics Covered

1. Range

   • Formula: ( R = L - S )
     • ( L ) is the largest value, ( S ) is the smallest value.
   • Coefficient of Range: ( \frac{L - S}{L + S} )
   • Formula applies to individual, discrete, and continuous series.
   • Practice: Find the range and coefficient using examples.

2. Interquartile Range and Quartile Deviation

   • Interquartile Range (IQR): ( Q3 - Q1 )
   • Quartile Deviation (QD): ( \frac{Q3 - Q1}{2} )
   • Coefficient of Quartile Deviation: ( \frac{Q3 - Q1}{Q3 + Q1} )
   • Requires calculation of ( Q1 ) and ( Q3 ) first.
   • Different formulas for individual, discrete, and continuous series.

3. Mean Deviation

   • Mean Deviation from Mean and Median.
   • For Individual Series:
     • From Median: ( \frac{\Sigma |x - M|}{n} )
     • From Mean: ( \frac{\Sigma |x - \overline{x}|}{n} )
   • Discrete Series requires frequency in calculations.
   • Continuous Series uses mid-values.
   • Coefficient involves dividing deviation by the respective mean or median.

4. Standard Deviation

   • Multiple methods: Actual Mean, Assumed Mean, and Step Deviation.
   • Formula varies for individual, discrete, and continuous series.
   • Important for calculating variance: Variance is the square of standard deviation.
   • Coefficient of Standard Deviation: ( \frac{\sigma}{\overline{x}} )

5. Combined Standard Deviation

   • Formula involves calculating ( \sigma ) for combined data sets.
   • Use ( n1, n2, \sigma1, \sigma2 ) and means ( \bar{x1}, \bar{x2} ).

6. Variance and Coefficient of Variation

   • Variance: Square the standard deviation result.
   • Coefficient of Variation: ( \frac{\sigma}{\overline{x}} \times 100 )
   • Useful for comparing variability between different data sets.

Special Topics

• Lorenz Curve: Graphical representation of distribution.
  • Plot cumulative percentage of income against cumulative percentage of population.
  • Line of equal distribution is a benchmark.

Conclusion

• Mastery of formulas and understanding their application is crucial.
• Practice with examples to ensure comprehension.
• Use the provided notes and examples to reinforce learning.