Who this is for — Readers who want to examine a result set beyond its average by separating frequency, spread, skewness, tails and sample limitations.
A return distribution describes how frequently the different outcomes in a series occur. Before constructing it, specify the observation unit — trade, day or month — horizon, currency or R unit, time window and treatment of costs. Series expressed in different units are not directly comparable.
In plain terms — Two strategies can have the same average and completely different risk paths. The distribution shows how that average is composed; by itself, it does not show the order in which outcomes arrive.
Five separate readings
| Aspect | Question | Measures or tools |
|---|---|---|
| Centre | Which value summarises the series? | Mean, median and mode, which need not coincide |
| Spread | How far apart are outcomes? | Variance, standard deviation, interquartile range |
| Skewness | Is one tail longer than the other? | Skewness and quantile comparison |
| Tails and outliers | How much do extreme events matter? | Kurtosis, extreme quantiles, expected shortfall |
| Shape | Are several groups or regimes present? | Histogram, empirical density, QQ plot |
Skewness describes lack of symmetry; it is not a synonym for “heavy tail”. Kurtosis concerns tail weight relative to a reference model, and software uses more than one convention. The mean is also not necessarily where the largest number of observations is concentrated.
Same mean, different shape — Series A `[0.5R, 0.8R, 1R, 1.2R, 1.5R]` and B `[−4R, 0R, 0R, 0R, 9R]` both average +1R. The second concentrates its result in one positive outlier and includes a much larger loss. Five observations cannot identify the population; the example only shows why the mean does not describe shape.
What a histogram does not show
A histogram discards time order. Two series with the same empirical distribution may alternate outcomes or cluster them into long runs. Drawdown and dependence also require a cumulative curve, autocorrelation, run analysis and segmentation by regime.
A trend-following strategy may show many small losses and a few large gains; a mean-reversion strategy may show the opposite profile. These are common patterns, not necessary identities. Markets, rules and execution can change the observed shape.
Stop limitation — A stop defines an instruction or rule, not a certain cap on realised loss. Gaps, liquidity and slippage can produce outcomes beyond the intended level; the left tail must not be drawn as automatically truncated.
Sources
- NIST/SEMATECH, Histogram, centre, spread, skewness, outliers and modes.
- NIST/SEMATECH, Measures of Skewness and Kurtosis, definitions, conventions and sensitivity to extremes.
- CFTC, Commodity Trading Systems Sold on the Internet, limitations of hypothetical results and execution assumptions.