Central Tendency & Spread Diagnostic Framework
Core Formulas
- Mean μ = Σx / n
- Median = middle (odd) or avg(two middle) (even)
- Mode = highest frequency value(s)
- Range = max − min
Robustness Matrix
- Outliers present → prefer Median
- Skewed → report Median + IQR
- Multimodal → list all modes; avoid single “average” narrative
Distribution Signals
- Mean ≈ Median ≈ Mode → roughly symmetric
- Mean > Median → right skew | Mean < Median → left skew
- Separated peaks → possible mixture populations
Extended Means (Use When Appropriate)
- Weighted Mean = Σ(wx)/Σw (grades, portfolios)
- Geometric Mean = (Πx)^(1/n) (growth chains)
- Harmonic Mean = n / Σ(1/x) (rates & ratios)
Spread Essentials
- Variance σ² = Σ(x−μ)² / n (or / n−1 sample)
- Std Dev σ = √σ²
- IQR = Q3 − Q1 (resistant)
Reporting Template
n=…, Mean=…, Median=…, Mode(s)=…, Range=…, StdDev=…, IQR=… (note skew & outliers)
Checklist
- Inputs cleaned & typed correctly
- Units consistent
- Outliers flagged (not silently dropped)
- Sample vs population clarified
Common Pitfalls
- Using mean alone on skewed data
- Averaging existing averages without weights
- Using geometric mean for negative values
- Over‑precision beyond input quality
Insight: Pair a center metric with at least one spread metric— central tendency without variability can mislead risk perception.