Standard Deviation, Variance & Coefficient of Variation
1. What Is Standard Deviation?
Introduced by Karl Pearson in 1893, the Standard Deviation (denoted by the Greek letter σ, read as “Sigma”) is universally regarded as the most important and scientifically rigorous measure of dispersion.
Definition:
The Standard Deviation is defined as the positive square root of the arithmetic mean of the squared deviations of observations taken from their Arithmetic Mean.
Why Is Standard Deviation Superior to Mean Deviation?
In Mean Deviation, we artificially dropped the minus signs by writing |d| — which violates mathematical algebraic laws. Standard Deviation squares each deviation (d²), naturally converting negative signs into positive values while preserving strict algebraic properties.
2. Calculation of Standard Deviation Across Series
where d = X - A (Assumed Mean)
where d = X - A and N = Σf
where d' = (m - A) ÷ i and i is class width
Solved Example: Step-Deviation Method for Continuous Series
Problem: Calculate the Standard Deviation from the following frequency distribution:
| Class Interval | 0 - 10 | 10 - 20 | 20 - 30 | 30 - 40 |
|---|---|---|---|---|
| Frequency (f) | 3 | 5 | 9 | 3 |
Solution Table:
- Let
A = 15(mid-point of class10 - 20). - Class width
i = 10.
| Class | Mid-point (m) | f | d = m - 15 | d’ = d ÷ 10 | fd’ | fd’² (= fd’ × d’) |
|---|---|---|---|---|---|---|
| 0 - 10 | 5 | 3 | -10 | -1 | -3 | +3 |
| 10 - 20 | 15 (A) | 5 | 0 | 0 | 0 | 0 |
| 20 - 30 | 25 | 9 | +10 | +1 | +9 | +9 |
| 30 - 40 | 35 | 3 | +20 | +2 | +6 | +12 |
| Total | — | N = 20 | — | — | Σfd’ = +12 | Σfd’² = 24 |
Calculation:
(Σfd'² ÷ N) = 24 ÷ 20 = 1.2(Σfd' ÷ N) = 12 ÷ 20 = 0.6(Σfd' ÷ N)² = (0.6)² = 0.36- Difference:
1.2 - 0.36 = 0.84 - Square Root:
√0.84 = 0.9165 - Multiply by
i:
σ = 0.9165 × 10 = 9.17
3. What Is Variance?
Variance is simply the square of the Standard Deviation.
- Formula:
Variance = σ² - Conversely:
Standard Deviation = √(Variance) - Example: If the Variance of a portfolio is
64, its Standard Deviation is√64 = 8.
4. Coefficient of Variation (CV) & Comparing Consistency
In management, we constantly compare two candidates, machines, sales teams, or sports players. To answer “Who is more consistent?”, we use the Coefficient of Variation (CV).
The Managerial Decision Rule (Crucial for Exams!):
- Lower CV: Indicates Lesser Variability → Greater Consistency, Higher Stability, Higher Uniformity.
- Higher CV: Indicates Greater Variability → Lower Consistency, Less Stability, Higher Risk.
5. Practical Activity 1: Comparing Consistency in Sports
Scenario: A cricket team manager wants to select one batsman for the upcoming tournament. Scores over the last 10 innings:
| Parameter | Batsman Rohit | Batsman Virat |
|---|---|---|
| Mean Score (x̄) | 50 runs | 48 runs |
| Standard Deviation (σ) | 12 runs | 6 runs |
Decision Analysis:
- Comparing Average Run Scorer:
Rohit has a slightly higher average (50 > 48), so Rohit is a higher run scorer on average. - Comparing Consistency via CV:
CV of Rohit = (12 ÷ 50) × 100 = 24%CV of Virat = (6 ÷ 48) × 100 = 12.5%
Managerial Conclusion:
Virat’s Coefficient of Variation (12.5%) is significantly lower than Rohit’s (24%). Therefore, Virat is far more consistent and reliable, while Rohit shows greater erratic fluctuations in performance!