Measures of Dispersion: Range, Quartile & Mean Deviation

1. What Is Dispersion? (Why Averages Are Not Enough)

A Measure of Central Tendency (Mean, Median) gives us the center of a distribution, but it reveals nothing about how widely the observations are scattered around that center.

Real-Life Business Example:

Consider the monthly profits of two startup branches in Indore over 3 months:

• Branch A: ₹48,000, ₹50,000, ₹52,000 → Average Profit = ₹50,000
• Branch B: ₹10,000, ₹50,000, ₹90,000 → Average Profit = ₹50,000

Both branches have the exact same Mean profit (₹50,000). But Branch A is stable, reliable, and low-risk, while Branch B has huge volatility and high risk.

Definition:
Dispersion is the measure of the extent to which individual observations are scattered or spread out around their central average.

2. Absolute vs. Relative Measures of Dispersion

When analyzing variation, statisticians calculate two distinct metrics:

3. Range & Coefficient of Range

The Range is the simplest measure of dispersion. It is the absolute difference between the largest and smallest values in a dataset.

Formulas:

• Absolute Range: Range = L - S
  (where L = Largest value, S = Smallest value)
• Coefficient of Range:
  Coefficient of Range = (L - S) ÷ (L + S)

Example:

Stock price of Tata Motors during a week: High = ₹980, Low = ₹920.

• Range = 980 - 920 = ₹60
• Coefficient of Range = (980 - 920) ÷ (980 + 920) = 60 ÷ 1900 = 0.0316

4. Quartile Deviation (Semi-Interquartile Range)

The simple Range is heavily distorted by freak extreme values. The Quartile Deviation (QD) solves this by focusing strictly on the middle 50% of the distribution, completely ignoring the bottom 25% and top 25% extremes!

5. Mean Deviation (Average Deviation)

The Mean Deviation (denoted by MD) is defined as the arithmetic average of the deviations of observations from a central measure (Mean or Median), ignoring algebraic plus (+) and minus (-) signs (taking absolute values |d|).

Master Mathematical Property:

Mean Deviation is MINIMUM when deviations are taken from the MEDIAN!
Σ|X - Median| < Σ|X - Mean|

Calculation Formulas:

(where |d| indicates absolute deviation without negative signs, m is the mid-point, and N = Σf)

Relative Coefficient Formulas:

• Coefficient of MD from Mean: MD_x̄ ÷ x̄
• Coefficient of MD from Median: MD_M ÷ M

Solved Example: Mean Deviation from Median (Individual Series)

Problem: Find Mean Deviation from Median for the marks: 10, 15, 20, 25, 30.

1. Calculate Median (M):
   Data is ordered (N = 5). Middle item = 3rd item = 20.
2. Compute Absolute Deviations |X - 20|:
   • |10 - 20| = 10
   • |15 - 20| = 5
   • |20 - 20| = 0
   • |25 - 20| = 5
   • |30 - 20| = 10
   • Sum (Σ|X - M|): 10 + 5 + 0 + 5 + 10 = 30
3. Apply Formula:
   MD_M = 30 ÷ 5 = 6 marks
Coefficient of MD = 6 ÷ 20 = 0.30

6. Summary Comparison Table