How this calculation works
A lifetime distribution can describe both the fraction surviving to an age and the survival expected after a unit has already reached that age. Those are different probabilities. Weibull shape determines whether the modeled hazard falls, stays constant or rises. This page evaluates a two-parameter Weibull distribution with entered parameters. It reports a failure-time quantile, survival at age, conditional survival over an extra interval and the instantaneous hazard.
S(t)=exp[−(t/eta)^beta]; quantile = eta[−ln(1−p)]^(1/beta); conditional survival = S(age+extra)/S(age)A worked example
For shape one and scale one hundred, the model is exponential. Median lifetime is one hundred times ln two, or about 69.3147 time units. A unit already aged five hundred has conditional survival exp(−20/100), about 0.818731, through another twenty units. That memoryless result is specific to shape one, not every Weibull shape.
Good to know
This is distribution evaluation with fixed inputs, not a Weibull parameter fit or an uncertainty interval. Repair, competing failure modes and mixed populations are absent. A high hazard does not by itself select a replacement age. Costs, consequences, model validation and parameter uncertainty need separate consideration before using a lifetime scenario operationally.
Reference: NIST — Weibull reliability and hazard (opens in a new tab) ↗
What this calculation answers
A lifetime distribution can describe both the fraction surviving to an age and the survival expected after a unit has already reached that age. Those are different probabilities. Weibull shape determines whether the modeled hazard falls, stays constant or rises.
This page evaluates a two-parameter Weibull distribution with entered parameters. It reports a failure-time quantile, survival at age, conditional survival over an extra interval and the instantaneous hazard.
The Weibull age replacement expected cost rate calculator evaluates a replacement-cost policy under an entered Weibull lifetime model. A distribution quantile alone does not choose the maintenance age.
Use the right input conventions
All ages, intervals and scale values must use the same time unit. The location parameter is fixed at zero. The cumulative probability field defines a failure-time quantile, so 0.5 asks for the median.
Shape and scale are supplied assumptions, not fitted from the data on this page. Large shape and time-scale contrasts that exceed finite numeric limits are rejected rather than silently displayed as valid infinite quantities.
Interpret the result
Conditional survival divides future survival by survival already achieved. Evaluating that relationship in logarithmic form avoids dividing two tiny probabilities directly. It still depends on the model being appropriate for surviving units.
Hazard is an instantaneous rate per time unit, not a probability. A value above one is therefore possible and must not be read as more than one hundred percent failure probability.
Work through the example
For shape one and scale one hundred, the model is exponential. Median lifetime is one hundred times ln two, or about 69.3147 time units.
A unit already aged five hundred has conditional survival exp(−20/100), about 0.818731, through another twenty units. That memoryless result is specific to shape one, not every Weibull shape.
Where the model stops
This is distribution evaluation with fixed inputs, not a Weibull parameter fit or an uncertainty interval. Repair, competing failure modes and mixed populations are absent.
A high hazard does not by itself select a replacement age. Costs, consequences, model validation and parameter uncertainty need separate consideration before using a lifetime scenario operationally.
A few questions, answered
Is hazard the same as failure probability over the next interval?
No. Hazard is an instantaneous rate; interval failure probability follows from the integrated survival relationship.
What changes in the comparison?
Increase shape while keeping scale and ages fixed to compare aging-dependent hazard.