In this article I discuss in detail on the real meaning of measures of central tendency and dispersion, more geared towards the CFA Level 1 Quantitative methods subject.
Here’s an article on Measures of Dispersion tailored to the CFA Level 1 exam syllabus. The article covers all required components: basics, examples, strengths, weaknesses, and use cases — presented in a structured format suitable for exam preparation and practical understanding.
Measures of Central Tendency and Dispersion- Meaning?
Measures of dispersion indicate how much variability or spread exists in a data set. While central tendency (mean, median, mode) tells us where the data is centered, dispersion tells us how spread out the data points are around that central value.
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Key Measures of Dispersion
a. Range
- Definition: The difference between the maximum and minimum values.
- Formula:Range=Xmax−Xmin
- Example: If returns over 5 days are 2%, 4%, 3%, 6%, 1%, thenRange=6%−1%=5%
b. Mean Absolute Deviation (MAD)
- Definition: The average of the absolute deviations from the mean.
- Formula:MAD=1n∑i=1n∣Xi−Xˉ∣
- Use: Simpler alternative to standard deviation; used when you want a basic sense of variability.
c. Variance
- Definition: The average of the squared deviations from the mean.
- Formula:σ2=1n∑i=1n(Xi−Xˉ)2
- Units: Squared units of the original data.
d. Standard Deviation
- Definition: The square root of variance; most commonly used dispersion measure in finance.
- Formula:σ=1n∑i=1n(Xi−Xˉ)2
- Interpretation: Expressed in the same units as the original data; shows typical deviation from the mean.
e. Semi-variance
- Definition: Measures dispersion of values that fall below the mean only.
- Use: More appropriate when focusing on downside risk.
f. Chebyshev’s Inequality
- Concept: Regardless of distribution, at least (1−1/k2) of the observations lie within k standard deviations of the mean.
- Example: For k=2, at least 75% of values lie within ±2σ of the mean.
Strengths and Weaknesses
| Measure | Strengths | Weaknesses |
|---|---|---|
| Range | Simple to compute | Sensitive to outliers, ignores all but two values |
| MAD | Easy to interpret, less sensitive to outliers than σ | Not used in many statistical procedures |
| Variance | Used in theoretical models | Units are squared, less intuitive |
| Standard Deviation | Widely accepted, intuitive interpretation | Assumes normality, sensitive to extreme values |
| Semi-variance | Focuses on downside risk | Ignores positive volatility (can be biased for total risk) |
Practical Use Cases in Finance
| Use Case | Preferred Measure | Reason |
|---|---|---|
| Risk assessment of portfolios | Standard deviation | Indicates total volatility |
| Downside risk (e.g., VaR, CVaR) | Semi-variance | Focus on losses rather than all fluctuations |
| Comparing investment consistency | MAD or standard deviation | Helps identify stable vs. volatile investments |
| Initial exploratory data analysis | Range | Quick check for spread and outliers |
CFA Level 1 Focus
The CFA Level 1 exam expects you to:
- Know formulas for variance, standard deviation, and MAD.
- Interpret standard deviation and its relation to risk.
- Apply Chebyshev’s Inequality in non-normal distributions.
- Use coefficient of variation (CV) to compare relative risk:CV=σXˉ
Summary
| Concept | Key Point |
|---|---|
| Dispersion | Measures the spread of data around the mean |
| Standard Dev. | Most common; used in risk and return analysis |
| MAD | Simpler, less sensitive to outliers |
| Semi-variance | Highlights downside risk |
| Chebyshev’s | Works for any distribution |
7. Example CFA-Style Question
Q: An analyst observes the following returns over five periods: 3%, -2%, 4%, 1%, and 0%. What is the standard deviation of returns?
Step 1: Calculate the mean:Xˉ=3+(−2)+4+1+05=1.2%
Step 2: Compute squared deviations and average:σ=(3−1.2)2+(−2−1.2)2+(4−1.2)2+(1−1.2)2+(0−1.2)25≈2.23%
Conclusion
Measures of dispersion are foundational for risk analysis and comparison of investment opportunities. CFA Level 1 candidates must understand both the mathematical formulas and practical implications of these measures in portfolio construction, risk assessment, and performance evaluation.
