# Failure (or hazard) rate

Operations · `Failure (or hazard) rate` · verification: source checked

Failure rate, also called hazard rate, is the instantaneous conditional rate at which survivors in a non-repairable population fail at time t. It is written h(t) and defined as f(t)/(1-F(t)) or f(t)/R(t). The document also defines average failure rate over an interval using H(t) and R(t).

## Specification

| Property | Value |
|---|---|
| Kind | metric |

## Formula

```
h(t) = f(t) / (1 - F(t)) = f(t) / R(t)
```

- `h(t)`: failure rate or hazard rate
- `f(t)`: f(t)
- `F(t)`: F(t)
- `R(t)`: R(t)

## Study notes

- The failure rate is defined for non-repairable populations as the instantaneous rate of failure for the survivors to time t.
- h(t) is the instantaneous conditional failure rate.
- H(t) = ∫_0^t h(t) dt.
- H(t) = -ln R(t).
- F(t) = 1 - exp[-∫_0^t h(t) dt] = 1 - e^{-H(t)}.
- h(t) = -d ln R(t) / dt.
- Average failure rate: AFR(T1,T2) = (H(T2)-H(T1))/(T2-T1) = (ln R(T1)-ln R(T2))/(T2-T1).
- AFR(0,T) = AFR(T) = H(T)/T = -ln R(T)/T.

## Sources

1. [8.1.2.3. Failure (or hazard) rate](https://www.itl.nist.gov/div898/handbook/apr/section1/apr123.htm) (accessed 2026-10-25)

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