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.
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. Dollar-cost averaging produces the harmonic mean price paid per share.
KEY: For any non-constant return series, harmonic ≤ geometric ≤ arithmetic. They are equal only when every return is identical.
The trimmed mean drops the highest and lowest x% before averaging. The winsorized mean replaces extreme values with the nearest non-extreme percentile.
Median is the 50th percentile. Quartiles split data into 4 groups, quintiles 5, deciles 10, percentiles 100. The interquartile range (Q3 − Q1) captures the middle 50% and is robust to outliers.
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.
- Reversing skewness signs. Negative skew = long left tail = mean below median. Trap: reading "negative skew" as "returns are below average" rather than "the loss tail is longer".
Bottom line
- Geometric mean correctly compounds period-over-period returns; arithmetic mean overstates growth, with ordering harmonic ≤ geometric ≤ arithmetic for any non-constant series
- Sample variance, covariance, and standard deviation divide by n − 1, not n; population mode understates dispersion
- Skewness > 0 = right tail (occasional big gains); skewness < 0 = long left tail with severe losses, typical of equity returns
- Excess kurtosis > 0 = leptokurtic = fat tails = more extreme outcomes than a normal distribution predicts; a normal distribution has kurtosis 3 (excess 0)
Exam shortcut
When a question says "annualized return" or "compound growth", reach for the geometric mean, never arithmetic. When ranking assets with different mean returns, compute CV (σ/mean), not raw σ. When a distribution is described as "long left tail" or "more crashes than normal predicts", pick negative skew + leptokurtic, the answer pair the exam loves to test together.
The full lesson (about 1,883 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- statistical characteristics of asset returns
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