Weibull Median⚠ unverified

Understanding the Parameters

• Scale parameter $\eta$ — the characteristic life, setting the timescale; the median is $\eta$ times a shape-dependent factor $(\ln 2)^{1/\beta}$. Because $\ln 2 \approx 0.693 < 1$, the median is always less than the characteristic life (which is the $63.2\%$ point) — sensible, since $50\%$ is reached before $63.2\%$.

• Shape parameter $\beta$ — controls how the factor $(\ln 2)^{1/\beta}$ behaves. For $\beta = 1$, $t_{50} = \eta\ln 2 \approx 0.693\,\eta$ (the exponential median). As $\beta$ grows, the exponent $1/\beta$ shrinks toward $0$ and $(\ln 2)^{1/\beta} \to 1$, so the median rises toward $\eta$ — the distribution tightens and its centre moves toward the characteristic life.

• The output $t_{50}$ — the median lifetime: half the population fails before it, half after. It is the natural "half-life" of the population and a common warranty and design-life anchor.

• Median vs mean vs mode. For a symmetric distribution these coincide, but the Weibull is skewed (except near $\beta \approx 3.4$, where it is nearly normal), so the three genuinely differ. The median's virtue is robustness: unlike the mean, it is not pulled by the long upper tail, making it a stable measure of typical life.

Derivation (Approaching a Proof)

The median is the time at which the reliability has fallen to one half — equivalently, where half the population has failed. Setting the Weibull survival function (Weibull Reliability) equal to $0.5$:

$$R(t_{50}) = e^{-(t_{50}/\eta)^\beta} = \frac12.$$

Take logarithms of both sides:

$$-\left(\frac{t_{50}}{\eta}\right)^\beta = \ln\frac12 = -\ln 2 \quad\Longrightarrow\quad \left(\frac{t_{50}}{\eta}\right)^\beta = \ln 2.$$

Raise to the power $1/\beta$ and solve:

$$t_{50} = \eta\,(\ln 2)^{1/\beta}. \qquad\blacksquare$$

This is a special case of the general percentile (B-life) formula $t_p = \eta\,[-\ln(1-p)]^{1/\beta}$ with $p = 0.5$; the B10 Life is the same formula with $p = 0.10$. The whole family of Weibull life percentiles follows this single inversion of the survival function.

Dimensional check. $(\ln 2)^{1/\beta}$ is a dimensionless number; $\eta$ carries the time units, so $t_{50}$ is a time. $\checkmark$

History and Development

• B-life notation. Reliability engineering labels percentile lives as B-lives (or L-lives in bearings): B10 for $10\%$, B50 for the median, etc. The median (B50) is a standard "characteristic" life quotation, especially where the mean is distorted by a heavy tail.

• Robustness to skew. The preference for the median over the mean in life data mirrors its use throughout statistics for skewed distributions (incomes, response times): it reports the "typical" value without being dragged by outliers or a long tail.

• Design and warranty anchors. Median life is a common target for consumer-product design life and warranty setting, while safety-critical applications use lower percentiles (B10 Life or below) to bound early failures.

Related Concepts: Weibull Mean, Weibull Mode, B10 Life, Weibull Reliability, Weibull Distribution PDF, Weibull Failure Rate

Notes: Registry calculator weibull-median (unverified). $t_{50} = \eta(\ln 2)^{1/\beta}$ — correct as shipped; the B50 life. Always $< \eta$ (the $63.2\%$ point) since $\ln 2 < 1$. Special case of the percentile formula $t_p = \eta[-\ln(1-p)]^{1/\beta}$ (cf. B10 Life, $p=0.1$). Robust to skew, unlike the mean. Registry labels $\eta$ dimensionless (is a time).