Calculate average recurrence interval from record years and events observed, or estimate the risk of at least one event over a design life.
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Average Recurrence Interval Formula
The average recurrence interval, or ARI, is the estimated average time between events of a specified size or greater. The calculator has two modes: estimating ARI from a record of observed events, and estimating the chance of at least one event during a design life.
ARI = Y/E
- ARI = average recurrence interval, in years
- Y = years of record
- E = number of qualifying events observed during the record
AEP = 1 - e^(-1/ARI)
- AEP = annual exceedance probability, as a decimal
- e = Euler’s number, approximately 2.71828
- ARI = average recurrence interval, in years
AEP_simple = 1/ARI
- AEP_simple = simple annual probability estimate
- ARI = average recurrence interval, in years
Risk = 1 - (1 - 1/ARI)^n
- Risk = probability of at least one event during the design life
- ARI = average recurrence interval, in years
- n = design life, in years
In From Record mode, you enter the length of the record and the number of events observed. The calculator divides years by events to estimate ARI, then also reports the annual exceedance probability.
In Design Risk mode, you enter an ARI and a design life. The calculator estimates the probability that at least one event of that size or greater occurs during the design life.
Common ARI and Annual Exceedance Probability Values
ARI is often described as a “return period,” but it does not mean the event happens exactly once every stated number of years. It is a long-term average.
| ARI | Simple AEP | Common description |
|---|---|---|
| 2 years | 50% | Frequent event |
| 10 years | 10% | Common design event |
| 25 years | 4% | Moderate event |
| 50 years | 2% | Major event |
| 100 years | 1% | Often called a 100-year event |
| 500 years | 0.2% | Rare event |
Typical Design Life Values
| Structure or project type | Typical design life |
|---|---|
| Small culvert | 10 years |
| Minor road | 25 years |
| Bridge | 50 years |
| Levee | 75 years |
| Major dam | 100 years |
Average Recurrence Interval Examples
Example 1: Calculate ARI from a record
You have 50 years of data and 5 qualifying flood events.
ARI = 50/5 = 10 years
The estimated average recurrence interval is 10 years. The simple annual exceedance probability is:
AEP_simple = 1/10 = 0.10 = 10%
Example 2: Calculate risk over a design life
You want the chance of at least one 100-year event during a 50-year design life.
Risk = 1 - (1 - 1/100)^50
Risk = 1 - 0.99^50 = 0.395 = 39.5%
The chance of at least one 100-year event in 50 years is about 39.5%.
Average Recurrence Interval FAQ
Does a 100-year event happen once every 100 years?
No. A 100-year event has about a 1% chance of being equaled or exceeded in any single year. It can happen in back-to-back years, or it might not happen for much longer than 100 years. The 100 years is a long-term average interval, not a schedule.
What is the difference between ARI and AEP?
ARI is expressed as a time interval, such as 10 years or 100 years. AEP is expressed as an annual probability, such as 10% or 1%. They describe the same idea in different ways. A larger ARI means a smaller annual probability.
Why is the design-life risk higher than the annual probability?
The annual probability applies to one year only. Over many years, the chance of seeing at least one event accumulates. For example, a 1% annual event has about a 39.5% chance of occurring at least once in 50 years, assuming each year’s occurrence is independent.