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Failure rate
Definition
Failure rate, in lifetime analysis, is the instantaneous rate at which units fail conditional on having survived to a given time. It is also called the hazard rate. For a continuous lifetime with density f(t) and survival probability S(t), the rate is h(t) = f(t)/S(t) wherever S(t) is positive. It has inverse-time units and is not itself a probability.
What it means
A unit that has already failed is no longer among the survivors whose next failure is being considered. This conditioning distinguishes the hazard from the overall density of failure times and from the proportion that has failed since observation began.
In everyday reporting, “failure rate” may also mean failures per request, failures per operating hour, or another ratio. State the numerator, denominator, time basis, and whether failures can recur before comparing figures.
How it works
Define what counts as failure and when a unit enters observation. Decide whether time is calendar age, operating hours, cycles, or another exposure measure. Different time scales answer different questions, and mixing them can distort comparisons.
For a continuous lifetime T, the hazard is the limit of P(t ≤ T < t + Δt | T ≥ t)/Δt as the interval shrinks to zero. Over a sufficiently short interval, h(t)Δt approximates the conditional failure probability if the hazard changes little. The approximation is not an identity for long intervals.
Integrate the hazard to obtain cumulative hazard H(t) = ∫₀ᵗ h(u)du. For a nonnegative continuous lifetime starting with survival one, S(t) = exp[−H(t)] and F(t) = 1 − exp[−H(t)]. Cumulative hazard can exceed one; the cumulative failure probability cannot.
A constant hazard λ gives the exponential survival function exp(−λt). Other distributions can represent increasing or decreasing hazard. A bathtub-shaped pattern is a useful description for some products, combining early failures, a flatter period, and wear-out, but it is not a universal law.
Estimate survival or hazard using all relevant observation histories, including right-censored units that have not failed when observation ends. Suitable noninformative-censoring or conditional assumptions are needed for standard methods. Simply deleting those units selects failures and can distort the analysis.
Distinguish first-failure analysis from repairable-system event counts. A system can fail, be repaired, and fail again. Its rate of occurrence of failures concerns an event-counting process, not necessarily the first-lifetime hazard. Repair quality and aging can affect that process.
When reporting an empirical failures-per-exposure ratio, explain its assumptions and aggregation. It can estimate a constant intensity under an appropriate model, but does not automatically reveal how risk varies with age, operating conditions, or time.
Three practical examples
- An engineer studies whether a component's first-failure hazard rises with operating age.
- A maintenance team reports recurrent equipment failures per operating hour, explicitly using an event-rate interpretation.
- An AI service team reports failed requests per total requests and avoids confusing that probability-like proportion with a lifetime hazard.
Think of it this way
Think of hazard as a speedometer for failure among the units still operating. The speedometer reading is a rate at a moment; it is not the total distance already traveled or a probability over an arbitrary period.
Worked example
Assume an invented component lifetime has constant hazard λ = 0.002 per hour. Over 100 hours, cumulative hazard is 0.002 × 100 = 0.2.
Survival is exp(−0.2) ≈ 0.818731, so the probability of failing by 100 hours is about 0.181269, or 18.13%. It is not exactly 20%; that would use the short-interval approximation over a relatively long interval.
For a component that has survived to any age, this constant-hazard model gives a next-ten-hour failure probability of 1 − exp(−0.002 × 10) ≈ 0.019801, or 1.98%.
The mean lifetime under this exponential model is 1/λ = 500 hours. That reciprocal relationship depends on the constant-hazard model and must not be applied indiscriminately to an age-varying hazard. The parameters are assumed for illustration, not estimated from observed components.
Illustrative case study
Fictional case: Ridge Systems compares two equipment fleets using the number of breakdowns. Analysts discover that one fleet operated twice as many hours and included older machines. They separate recurrent event rates from first-failure survival analysis and report exposure and age distributions. This invented case illustrates why a raw failure count is insufficient for comparing reliability.
Common misconceptions and mistakes
Do not interpret a hazard of 0.2 per hour as a 20% failure probability over every one-hour interval. The exact conversion depends on the integrated hazard.
Do not equate the failure density f(t) with the conditional hazard h(t). They have different denominators and meanings.
Do not assume decreasing population hazard proves individual components improve with age. Population heterogeneity and survival selection can contribute to the pattern.
Business use
Clear failure-rate definitions support maintenance planning, reliability comparisons, and operational monitoring. AI predictive-maintenance models should use the appropriate event definition, exposure, and censoring information. Business decisions also need uncertainty and consequences: two systems with the same event rate can have very different downtime or replacement costs.
People also ask
Can a hazard exceed one?
Yes; it is a rate with units, not a probability.
Is failure rate always constant?
No.
Is a 2% failed-request proportion a hazard?
Usually it is a per-request proportion; call it that unless a time-to-event model is intended.
Check your understanding
Who is in the hazard's risk set?
Answer: Units still surviving at that time.
What connects cumulative hazard and survival?
Answer: S(t) = exp[−H(t)].
Why is the example's 100-hour failure probability below 20%?
Answer: The exact exponential conversion is 1 − exp(−0.2), rather than its linear approximation.
Related terms
- Survival function
- Censoring
- Reliability
- Hazard function
- cumulative hazard
- exponential distribution
- bathtub curve
- event rate
Learn next
Learn to distinguish density, survival, and hazard. Then study censored lifetime data and the separate modeling of recurrent failures in repairable systems.
More in Statistics and probability
- Factor range - Factor range is the difference between the highest and lowest values of a quantitative factor used in an experimental design.
- Factorial design - A factorial design is an experimental arrangement that studies two or more factors together by combining their levels.
- Failure bias - Failure bias is an informal label for drawing general conclusions from visible failures while overlooking relevant successes or other cases that did not fail.
- False discovery rate - The false discovery rate, or FDR, is the expected proportion of rejected null hypotheses that are actually true.
- Familywise error rate - Familywise error rate, or FWER, is the probability of rejecting at least one true null hypothesis within a specified family of hypothesis tests.
- Fan chart (time series) - A fan chart is a time-series visualization that displays a central forecast together with nested probability bands for future outcomes.
Sources
NIST/SEMATECH, “Failure (or hazard) rate”: https://www.itl.nist.gov/div898/handbook/apr/section1/apr123.htm NIST/SEMATECH, “Repair rate or ROCOF”: https://www.itl.nist.gov/div898/handbook/apr/section1/apr125.htm NIST/SEMATECH, “Bathtub curve”: https://www.itl.nist.gov/div898/handbook/apr/section1/apr124.htm
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