4. Descriptive statistics for failure characterization

Before any hypothesis test, control chart, or regression model can be trusted, the failure data itself must be understood. Descriptive statistics are the first analytical layer in root cause analysis: they compress a raw failure dataset into a small set of numbers and graphics that reveal where failures cluster, how widely they scatter, and what shape their distribution takes. In reliability and failure analysis, these three properties — central tendency, dispersion, and distribution shape — determine which subsequent methods are valid. A mean time to failure computed on a bimodal dataset, for example, describes no real failure mode at all.

Central Tendency: Locating the Failure

Central tendency answers "what is a typical value?" For failure data, the three common measures behave very differently:

• Mean (average): the arithmetic sum divided by count. It is the natural estimator for symmetric data but is strongly pulled by outliers — a single 4,000-hour failure in a set of 200-hour failures shifts the mean substantially.
• Median: the 50th percentile. It is robust to outliers and is the preferred location measure for skewed lifetime data, which is the norm in reliability.
• Mode: the most frequent value or peak of the distribution. In failure analysis the mode is diagnostically valuable because each distinct peak often corresponds to a distinct failure mechanism.

Consider a pump seal dataset with failure times (hours): 180, 210, 225, 240, 255, 260, 275, 290, 310, 1,850. The mean is 409.5 h, but the median is 257.5 h. The mean is inflated by one early-life outlier; the median better represents the bulk of the population. Reporting only the mean here would mislead a maintenance interval decision by roughly 150 hours.

Dispersion: Quantifying Spread and Risk

Two datasets can share an identical mean yet represent completely different reliability risks. Dispersion measures how much individual failures deviate from the center:

• Range: max − min. Simple but dominated by extremes.
• Variance (s²): the average squared deviation from the mean. Squaring makes it sensitive to outliers and gives it squared units.
• Standard deviation (s): the square root of variance, in the original units. For the seal data, s ≈ 490 h — larger than the mean itself, a strong signal of a heavy-tailed or contaminated distribution.
• Interquartile range (IQR): Q3 − Q1, the spread of the middle 50%. For the seal data, Q1 ≈ 217.5 h and Q3 ≈ 282.5 h, so IQR = 65 h. The IQR is robust and pairs naturally with the median.
• Coefficient of variation (CV = s / x̄): a unitless relative spread, useful for comparing components measured on different scales.

A practical rule: when s is comparable to or larger than the mean, suspect mixed failure modes, data entry errors, or a non-normal distribution before proceeding to parametric analysis.

Distribution Shape: Skewness and Kurtosis

Shape statistics describe the asymmetry and tail weight of the failure distribution:

• Skewness: positive (right-skewed) values indicate a long tail of large failure times; negative values indicate a long left tail. Most wear-out and fatigue data are right-skewed, which is why Weibull and lognormal models are standard in reliability.
• Kurtosis: measures tail heaviness relative to a normal distribution. Excess kurtosis > 0 signals heavy tails and a higher probability of extreme failures — critical for safety and warranty risk.

For the seal dataset, skewness is strongly positive (≈ 2.9) and excess kurtosis is high, confirming that the single 1,850-hour point is not a random fluctuation but a distinct population. This is the statistical fingerprint of a mixed failure mode: infant mortality plus wear-out.

Graphical Summaries: Seeing the Failure Structure

Numbers alone can hide structure. Graphical summaries expose it:

• Histogram: bins failure times and shows frequency. Bin width matters — too wide hides bimodality, too narrow creates noise.
• Box plot: displays median, quartiles, and outliers in one compact view. Points beyond 1.5 × IQR are flagged as potential outliers.
• Empirical CDF (ECDF): plots the fraction of failures below each time. It is the non-parametric foundation for reliability estimates and reads directly as "probability of failure by time t."
• Probability plot: transforms the data onto a Weibull or normal axis; a straight line indicates the assumed distribution fits. Curvature or two line segments reveal mixed modes.

Putting It Together: A Reproducible Characterization Procedure

The following procedure can be applied to any failure time dataset. It is written for a spreadsheet or a Python/R environment, but the logic is tool-independent.

1. Load and clean: record each failure time with its unit, censoring status (failed vs. suspended), and subsystem. Remove or flag obvious data-entry errors (e.g., negative times, duplicate timestamps).
2. Compute location and spread: calculate mean, median, standard deviation, IQR, and CV. If s > x̄, flag the dataset for shape inspection.
3. Compute shape: calculate skewness and excess kurtosis. Values of |skewness| > 1 or excess kurtosis > 2 warrant a distribution check.
4. Plot: generate a histogram with a defensible bin width (e.g., Sturges' rule or Freedman–Diaconis), a box plot, and an ECDF. Overlay a fitted Weibull or normal curve only after the empirical shape is understood.
5. Interpret and decide: if the plot shows a single mode and mild skew, proceed with a parametric model. If it shows two modes or a heavy tail, stratify by suspected cause (e.g., by production batch or operating condition) and re-analyze each stratum separately.

Interpreting the Seal Example

Returning to the seal data: the histogram shows a dense cluster between 180 and 310 hours and an isolated bar near 1,850 hours. The box plot flags 1,850 as an outlier beyond 1.5 × IQR. The ECDF rises steeply to about 90% by 310 hours, then flattens. These three graphics agree: the dataset is a mixture. The correct next step is not to compute a single mean but to separate the two populations — likely infant mortality (installation damage, material defects) and wear-out (normal service life) — and characterize each independently. Only then can a meaningful maintenance interval or supplier corrective action be derived.

Descriptive statistics are not a preliminary formality; they are the diagnostic instrument that tells the analyst which failure mechanisms are present and which analytical models are defensible. Central tendency locates the failure, dispersion quantifies the risk, shape reveals the mechanism, and graphics make the structure visible. Together they form the evidentiary basis for every subsequent step in root cause analysis.