3. Data types and distributions in crash engineering

Crash engineering decisions rarely fail because the statistical formula was wrong. They fail because the data fed into the formula did not satisfy the assumptions the formula silently made. Before a t-test, an ANOVA, or a regression is applied to sled-test results, the analyst must know what kind of object each measurement is, how it is distributed, and whether the normality assumption that underpins parametric tests is defensible. This chapter establishes that foundation across the four data families that dominate occupant-safety work: acceleration pulses, force–displacement curves, injury criteria, and the derived scalar metrics used in decision-making.

Four Data Families and What They Actually Are

It is useful to separate crash data by structure rather than by sensor name, because structure determines which statistical tools are legitimate.

• Time-series signals — acceleration pulses from accelerometers on the vehicle, sled, or ATD (anthropomorphic test device). These are sampled at 10–20 kHz, filtered per SAE J211 (typically CFC 60 for occupant accelerations, CFC 180 or 1000 for structural response), and are not independent observations. Adjacent samples are strongly autocorrelated, so treating 5,000 samples as a sample size of 5,000 is a serious error.
• Functional curves — force–displacement responses from barrier faces, crush tubes, or belt load limiters. These are paired (x, y) records where x is often monotonic but y is not; energy absorption is the integral, and the curve shape (initial peak, plateau, densification) carries the engineering meaning.
• Injury criteria — scalars computed from signals: HIC36/HIC15, 3 ms clip acceleration, Nij, chest deflection, BrIC. Each is a deterministic functional of a filtered signal, so its distribution is inherited from the underlying pulse population, not from a simple additive error model.
• Design and categorical variables — restraint configuration, pretensioner firing, airbag type, occupant size (5th percentile female, 50th percentile male), impact angle. These are factors, not responses, and belong on the right-hand side of a model.

Acceleration Pulses: Shape, Filtering, and Derived Scalars

A crash pulse is characterized by its peak amplitude, duration, and shape (front-loaded, approximately triangular, or haversine-like). For statistical work, the pulse is reduced to a small set of features: peak positive acceleration, pulse duration above a threshold, delta-v (the integral of acceleration), and the centroid time. Delta-v is the most decision-relevant scalar because it correlates with injury risk and is the natural covariate in regression models.

Filtering matters for distributional work. A CFC 60 filter removes high-frequency content that inflates peaks; comparing HIC values computed from differently filtered signals is meaningless. When building a dataset, record the filter class alongside every scalar. A practical example: a frontal sled matrix with 24 tests at three delta-v levels (32, 40, and 48 km/h) yields 24 HIC15 values. The analyst should first inspect the distribution of HIC15 within each delta-v cell before pooling, because pooling across delta-v levels creates a mixture distribution that is often bimodal and will fail normality tests for a reason that has nothing to do with measurement error.

Force–Displacement Curves: Features, Not Samples

Force–displacement data from a crushable energy absorber typically shows an elastic rise to an initial peak, a plateau where plastic folding occurs, and a densification region where force rises steeply as the material compacts. The engineering quantities of interest are the initial peak force (which drives occupant deceleration), the mean crush force (which sets energy absorption), and the stroke efficiency. For statistics, extract these as scalars per test. A common mistake is to fit a regression to all (displacement, force) pairs across multiple tests; this ignores between-test variation and produces confidence intervals that are far too narrow.

Where curve-level comparison is genuinely needed — for example, comparing two absorber designs across five replicates each — use functional data analysis or compare the extracted features with a two-sample test. The feature-based route is usually sufficient and far more interpretable to a design team.

Injury Criteria and Their Distributions

Injury criteria are bounded below by zero and often right-skewed. HIC15 cannot be negative; chest deflection is non-negative; Nij is a ratio and can exceed 1. This boundedness means that a normal distribution is often a poor fit at the tails, exactly where injury-risk decisions are made. Two practical consequences follow. First, log-transforming positively skewed criteria (HIC, BrIC) frequently restores approximate symmetry and stabilizes variance, which improves the behavior of ANOVA residuals. Second, when the decision concerns the probability of exceeding a threshold (for example, AIS 2+ risk), a parametric model fitted to the bulk of the data may badly misestimate the tail; a bootstrap or a log-normal fit is often more defensible.

Normality Assessment in Practice

Normality is a property of the residuals of the model, not of the raw response, and it matters most for small samples. A workable sequence for a crash dataset is:

1. Plot before testing. A histogram and a normal Q–Q plot of the response within each factor cell reveal skew, outliers, and bimodality faster than any test statistic.
2. Test with the right null. Shapiro–Wilk is appropriate for n between roughly 3 and 50; Anderson–Darling is a reasonable alternative. With n above a few hundred, these tests reject trivial departures, so rely on effect size and plots instead.
3. Check homogeneity of variance with Levene's or Brown–Forsythe's test, because ANOVA assumes equal variances across cells. Crash data frequently violate this when delta-v levels differ.
4. Decide, don't just report. If residuals are skewed, try a log or Box–Cox transform. If variances are unequal, use Welch's ANOVA or a non-parametric alternative (Kruskal–Wallis) and state the choice.

The figure below summarizes how each data family flows from acquisition to the statistical object that actually enters a hypothesis test, and where normality assessment belongs in that chain.

Worked Example: Building a Defensible Dataset

Consider a frontal sled matrix with three delta-v levels (32, 40, 48 km/h) and two restraint configurations (baseline, pretensioner + load limiter), with four replicates per cell — 24 tests total. The response of interest is HIC15. The analyst proceeds as follows.

First, confirm that every HIC15 was computed from a CFC 60-filtered head resultant acceleration and that the integration window is consistent (HIC15 uses a 15 ms window). Second, plot HIC15 against delta-v, colored by restraint configuration. If the baseline configuration shows a steep rise with delta-v while the pretensioned configuration is flatter, the interaction is the engineering story, and a two-way ANOVA with interaction is the natural model. Third, fit the model and inspect residuals: a normal Q–Q plot of residuals, a residuals-versus-fitted plot, and Levene's test across the six cells. If residuals are right-skewed, refit on log(HIC15) and compare residual diagnostics. Report the model on the scale that satisfies the assumptions, and translate back to HIC units for the design team.

A reproducible check: with four replicates per cell, Shapiro–Wilk on each cell has very low power, so a non-significant result is weak evidence of normality. Rely on the pooled residual Q–Q plot and on the plausibility of the physical mechanism. If a single test shows an anomalously high HIC15, investigate the pulse and the ATD channel before deleting it; an outlier in crash data is often a real event (a bottoming-out restraint, a contact with the knee bolster) rather than a measurement error.

Practical Guidance Summary

The central discipline is this: reduce curves to physically meaningful scalars, verify that those scalars are approximately independent across tests, and assess normality and variance homogeneity on model residuals rather than on raw signals. When those conditions hold, the parametric machinery of the following chapters — t-tests, ANOVA, and regression — becomes a reliable decision tool rather than a source of confident but unfounded conclusions.