Two portfolios show identical 8% average annual returns. One had its worst year at −3%, the other at −42%. The mean lies, and statistical characteristics tell you which return stream you can actually live with.
Central tendency is where a return distribution sits. You calculate more than one measure because each treats compounding and outliers differently. Example 1 works the means and Example 3 works the outlier-resistant measures on one ten-year sample.
The arithmetic mean sums observations and divides by count. Simple, but it inflates multi-period return estimates because it ignores compounding.
The geometric mean is the correct measure of compounded historical return.
The harmonic mean equals n divided by the sum of reciprocals. You calculate it for cost averaging, where it equals the average price paid per share. Spend $1,000 at $40 and $1,000 at $50, and you own 25 + 20 = 45 shares for $2,000, an average cost of $44.44.
Common mistakes
- Using arithmetic mean for compounded growth. A series of +50%, −50% averages to 0% arithmetically but ends at 0.75 (geometric = −13.4%). Trap: choosing arithmetic when the question asks for the annualized return.
- Dividing by N for sample statistics. Sample variance, covariance, and standard deviation use n − 1. Trap: a calculator left in population mode silently understates dispersion.
- Using n for a quantile position. In Example 3, the first quartile sits at position 11 × 0.25 = 2.75, giving −2.5%. Trap: 10 × 0.25 = 2.5 gives −4.0%.
Bottom line
- Geometric mean correctly compounds period-over-period returns; arithmetic mean overstates growth; for positive values (prices, or growth factors 1 + R), harmonic ≤ geometric ≤ arithmetic
- Trimmed and winsorized means, the median and the IQR all resist outliers; a mean far above the median signals positive skew
- Percentile position on sorted data, interpolating between neighbors; IQR = Q3 − Q1
- Sample variance, covariance, and standard deviation divide by n − 1, not n; MAD averages absolute deviations and divides by n
Exam shortcut
When a question says "annualized return" or "compound growth", reach for the geometric mean, never arithmetic. When ranking assets with different positive mean returns, compute CV (σ/mean), not raw σ. For any quantile, sort first, then use (n + 1) × y/100 for the position and interpolate.
The full lesson (about 3,165 words, 21 min read) adds 3 worked examples, all 7 common mistakes, a self-check, free in the app.
Learning objectives
- statistical characteristics of asset returns
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