Sample Quantile Estimators and Plotting Positions

Ask ten different statistics packages for the 95th percentile of the same 40-day P&L series and, in the tails, you can get ten slightly different numbers. That's not a bug — there are at least nine officially recognized ways to estimate a quantile from a finite sample, and they disagree by construction because a quantile is a property of a smooth underlying distribution, while your data is a finite, lumpy set of points. Sample quantile estimators are the rules for bridging that gap, and plotting positions are the closely related rules for deciding what quantile a given sorted observation "represents" on a probability plot.

An analogy: measuring where the 9th runner "really" finished

Imagine 12 runners cross a finish line and you want to know the finishing-time value that "90% of runners are faster than." With 12 runners, no one runner sits exactly at the 90th percentile — the 90% mark falls somewhere between the 10th and 11th finisher. Do you just take the 11th runner's time? Average the 10th and 11th? Weight the average toward whichever is closer to the true 90% mark? Every reasonable choice is a different quantile estimator, and with only 12 runners, the choice can swing your answer meaningfully. With 12,000 runners, the different rules nearly agree, because there's a runner right there at (or very near) the 90% mark.

The mechanics, one piece at a time

Sort your $n$ observations to get order statistics $x_{(1)} \le x_{(2)} \le \dots \le x_{(n)}$. A plotting position assigns each sorted observation $x_{(i)}$ a probability $p_i$ — the "quantile level it represents" — using a formula like

$$p_i = \frac{i - a}{n + 1 - 2a},$$

where $a$ is a constant between 0 and 1 that differs by convention: $a = 0.5$ gives the classic $(i - 0.5)/n$ rule, $a = 0.375$ gives the Blom formula favored for normal-probability plots, and $a = 0$ gives $i/(n+1)$, the Weibull formula. In plain English: each sorted data point is assigned a "share" of the 0-to-1 probability line, and the different formulas just disagree, mostly at the edges, about exactly how much share the smallest and largest points should get. To get a quantile estimate for a specific target level $q$ (say the 95th percentile), you invert this — find where $q$ falls between two $p_i$'s and interpolate between the corresponding $x_{(i)}$ values, usually linearly.

Worked example 1: the 90th percentile of 10 P&L observations

Take 10 sorted daily P&L figures (in $000s): −8, −5, −3, −1, 0, 1, 2, 4, 6, 9. Using the common $(i - 0.5)/n$ plotting position with $n = 10$: the 10th point ($x_{(10)} = 9$) represents $p_{10} = (10 - 0.5)/10 = 0.95$, and the 9th point ($x_{(9)} = 6$) represents $p_9 = 0.85$. To estimate the 90th percentile ($q = 0.90$), interpolate linearly between $p_9 = 0.85$ (value 6) and $p_{10} = 0.95$ (value 9): the target is halfway between, so the estimate is $6 + 0.5 \times (9 - 6) = 7.5$, i.e. $7,500. A different convention (say $i/(n+1)$) would place $x_{(9)}$ at $p_9 = 9/11 \approx 0.818$ and $x_{(10)}$ at $p_{10} = 10/11 \approx 0.909$, giving a slightly different interpolated answer near $7,900 — a difference of roughly $400 purely from the choice of formula, on identical data.

Worked example 2: why the choice barely matters at scale

Now repeat the exercise on 1,000 sorted daily P&L observations. The gap between adjacent order statistics near the 90th percentile is tiny — often a few dollars — so whichever plotting-position convention you use, the interpolated 90th-percentile estimate lands within a rounding error of the others. With $n = 1{,}000$, $(i-0.5)/n$ and $i/(n+1)$ differ by at most $0.5/1000 = 0.05\%$ in probability terms, which barely nudges the interpolation. The lesson: estimator choice is a real source of disagreement in small samples and matters most exactly where risk desks operate — thin tails, short lookback windows — and becomes nearly irrelevant with large, dense samples.

What this means in practice

Quantile estimator choice quietly affects VaR estimates, backtesting thresholds, and any percentile-based risk limit computed from a short historical window — see Value at Risk. Most statistical software defaults to one convention (R's default is type 7, roughly the linear-interpolation rule used above) without telling you, so two teams computing "the same" 99th-percentile VaR from the same data can get different numbers just from differing software defaults. When precision at the tail matters, state which quantile estimator you used, and prefer methods built for tail estimation, like extreme value theory, over naive sample quantiles when the sample is small.