Probability plotting

The formula

Step one is deciding what cumulative fraction to assign the i-th ordered data point out of n. The workhorse is Benard's median-rank approximation:

F_i ≈ (i − 0.3) / (n + 0.4)

Read: the i-th failure out of n sits, with 50% confidence, at about this percentile of the population. The exact value is the median of a Beta(i, n−i+1) distribution; Benard's shortcut lands within about 1% of it for practical sample sizes. For the first failure the exact answer has a closed form, 1 − 0.5^(1/n) — for n = 10 that's 0.0670 against Benard's 0.0673.

Step two is the straightening transform. For a Weibull population, F(t) = 1 − exp(−(t/η)^β) rearranges to:

ln( −ln(1 − F) ) = β·ln t − β·ln η

Read: on axes of ln(−ln(1−F)) versus ln t, Weibull data is a line whose slope is the shape parameter β, and which crosses F = 63.2% at t = η. That 63.2% is 1 − 1/e — it falls out of setting t = η, not out of anyone's preference.

For a normal population the transform is the inverse normal CDF. Plot the ordered data against z_i = Φ⁻¹(F_i):

x_(i) vs z_i    →    slope ≈ σ,  intercept ≈ μ

Read: location and scale parameters of the fitted distribution are the intercept and slope of the fitted line — that identity is what makes the whole method more than a picture ^[1]. Q-Q plot is the general name for the same move: quantiles of your data against quantiles of a reference distribution, straight line meaning agreement ^[5].

Most plotting positions in use are members of one family:

F_i = (i − a) / (n + 1 − 2a)

Read: one tuning constant covers the field — a = 0 gives the mean rank i/(n+1), a = 0.3 gives Benard's median rank, a = 0.375 gives Blom, a = 0.5 gives Hazen's original (i − 0.5)/n ^[5]^[7]. The differences only matter in the tails and at small n — which is unfortunately where reliability work lives.

Where you meet it

At the fatigue bench with eight coupons. Eight failures is not enough for fancy statistics, but ranked on Weibull paper it's enough to see a slope. β below 1 says infant mortality, β near 1 says random-in-time failures, β well above 1 says wearout. A quick calibration: if the line climbs from 10% to 90% failed across one decade of life, β is about 1.34 — steeper than that and you're looking at a genuine wearout mechanism.
At the reliability review board, defending a B10 number. The 10%-failed life of a bearing or actuator lot is read straight off the fitted line at F = 0.10. When the customer asks where the number came from, the answer is one chart with the raw failure times visible on it — which is precisely why review boards like the method more than a black-box MLE printout.
At the burn-in decision. Environmental stress screening only pays if early life is where the failures live. A Weibull plot of field returns with β < 1 justifies the screen; β > 1 means burn-in consumes good life and screens out nothing. That one slope settles a budget argument.
Before quoting Cpk. Process capability math assumes normality. A normal probability plot of the measurement data is the two-minute check that keeps a capability index from being fiction — curvature at the ends means the ±3σ tail math is wrong before you start.

How it works

The core idea is a change of variables. Any CDF is a monotone curve from 0 to 1; pick the distribution family, warp the probability axis by its inverse CDF, and members of that family become straight lines. Historical probability paper was exactly this warp pre-printed on a grid; the modern version is the same arithmetic done in software ^[1]^[2]. The judgment call — straight or not — stays human, which is a feature: the eye catches doglegs and outliers that a single fit statistic averages away. When you do want a number, the correlation coefficient of the fitted line serves as the goodness-of-fit score, and comparing it across candidate distributions picks the best model ^[1].

The plotting position is where the craft lives. The naive estimate F_i = i/n fails outright: the last point gets F = 1, and ln(−ln(0)) is infinite — your biggest data point leaves the paper. The mean rank i/(n+1) fixes that, but for skewed life distributions the median rank is the standard, because the i-th failure's true percentile is itself a random variable with a skewed distribution, and the median is the fair bet. For n = 10, the smallest failure plots at 6.7% by median rank versus 9.1% by mean rank — a visible difference in the tail you're usually extrapolating from. Benard's (i − 0.3)/(n + 0.4) is the default in essentially every reliability package for this reason ^[7]; Blom's (i − 0.375)/(n + 0.25) plays the same role for normal plots ^[5].

The gotchas, in the order they cost money:

Suspensions are data. Units still running when the test stopped, or pulled for an unrelated failure, cannot be thrown away and cannot be counted as failures. They get handled by adjusted ranks: each survivor shifts the rank increments of the failures after it. Ignore ten survivors out of fifteen units and your plotted population fails five times faster than the real one. This is the single most common shop-floor Weibull error.

Curvature is a message, not noise. A plot that bows consistently means the model is wrong in a specific way. Concave-down on Weibull paper often means a failure-free period — subtract a threshold t₀ and replot. A dogleg with two distinct slopes means two failure mechanisms mixed in one dataset; fit them together and you get a β that describes neither. Separate by failure mode first, plot second.

The ends wiggle. Let them. The first and last order statistics have the largest sampling variance, so a small sample's extreme points scatter off the line even when the model is right. Rejecting normality because one point out of twelve strays at the tail is the classic overread. Judge the middle 80% first.

Small samples can't referee Weibull versus lognormal. Both are skewed two-parameter life models and both will plot acceptably straight on the other's paper at n = 15 or 20. They diverge in the far tail — exactly where the B1 or B0.1 extrapolation lives. Choose between them on failure physics, not on which correlation coefficient is 0.02 higher.

Q-Q and P-P are not the same plot. A Q-Q plot compares quantiles and stretches disagreement in the tails; a P-P plot compares cumulative probabilities and stretches disagreement in the middle ^[5]^[8]. Life-data work almost always wants Q-Q behavior, because the money questions — B10, B1, warranty exposure — are tail questions.

Rank regression is not MLE. The straight-line fit (median-rank regression) weights every plotted point roughly equally and behaves well for small, complete samples; maximum likelihood handles heavy censoring better and is what most software defaults to for big field datasets. They give different answers on the same data. Know which one produced the number you're signing.

History

The method starts with a water engineer. Allen Hazen — the sanitary-engineering pioneer whose name is also on the Hazen-Williams pipe formula — was plotting ranked streamflow data by 1913, and in the published discussion of Weston Fuller's 1914 ASCE paper on flood flows he presented probability paper for the first time, arguing that peak flows follow a lognormal law ^[9]^[10]. The idea of a percentile grid that turns a distribution into a straight edge you could extrapolate to the 100-year flood was born in reservoir design, decades before anyone applied it to hardware.

The distribution that made the technique a reliability tool arrived in 1951, when Waloddi Weibull, a Swedish engineer and ballistics researcher, published "A Statistical Distribution Function of Wide Applicability" in the ASME Journal of Applied Mechanics, demonstrating one flexible CDF fit to everything from steel yield strength to fatigue life ^[11]^[4]. Because that CDF linearizes under a double log, it was a natural match for Hazen-style paper, and Weibull plots became the shop-floor standard for failure data ^[3]^[4].

The plotting position got its modern form in 1953, when A. Benard and E. C. Bos-Levenbach published "Het uitzetten van waarnemingen op waarschijnlijkheidspapier" — the plotting of observations on probability paper — in Statistica Neerlandica, giving the (i − 0.3)/(n + 0.4) median-rank approximation that reliability software still defaults to ^[6]^[7].

The statisticians formalized the whole family late. Martin Wilk and Ram Gnanadesikan of Bell Labs published "Probability plotting methods for the analysis of data" in Biometrika in 1968, organizing the field around the quantile-quantile (Q-Q) and percent-percent (P-P) plots and giving the technique its modern vocabulary ^[8]^[5]. The engineers had been drawing the pictures for fifty years; Wilk and Gnanadesikan explained what the pictures were.

Related tools

/tools/bearing-life-l10 — the L10 rating is the 10%-failed point of a Weibull population; a probability plot is how you check a bearing lot actually behaves like its rating
/tools/arrhenius-rate — accelerated life testing ends at a probability plot per temperature; parallel lines confirm the acceleration model shifted life without changing the failure mechanism
/tools/half-life-decay — the exponential distribution is the Weibull with β = 1; the constant-hazard line every burn-in argument is measured against

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →