---
title: "Exceedance probability, recurrence interval, frequency factor"
subject: "Engineering Hydrology"
chapter: 6
key: "semester_5/engineering_hydrology:define-exceedance-recurrence-freqfactor"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/define-exceedance-recurrence-freqfactor
site: BCT Engineering
---

# Exceedance probability, recurrence interval, frequency factor

Engineering Hydrology · Chapter 6: Flood Hydrology

- Asked in: 2078 Bhadra, 6 marks
- Video: [Question on Return Period: Hydrology Lecture 18.a](https://www.youtube.com/watch?v=3noSDtcrxc4) — Ankit
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/define-exceedance-recurrence-freqfactor
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-6

## Question

Define the terms exceedance probability, recurrence interval and frequency factor.

## Answer

#### Exceedance probability
The **exceedance probability** $P(X\ge x_T)$ is the probability that an event of magnitude $x_T$ is equalled or exceeded in any one year. For example, a 100-year flood has an exceedance probability of $P=0.01$ (1%) each year.

#### Recurrence interval (return period)
The **recurrence interval** $T$ is the average number of years between events equal to or greater than a given magnitude. It is the reciprocal of the exceedance probability:

$$
T=\frac{1}{P}
$$

It is an average over a long record, not a fixed period: a 50-year flood may occur in two consecutive years. The probability that it occurs at least once in n years is $1-(1-1/T)^n$.

#### Frequency factor
The **frequency factor** $K_T$ is the number of standard deviations by which the event of return period T differs from the mean, so that

$$
x_T=\bar x + K_T\,\sigma
$$

It depends on the distribution used and on T (and, for Pearson III, on the skewness). For Gumbel's distribution $K_T=\dfrac{y_T-\bar y_n}{S_n}$ with $y_T=-\ln[\ln(T/(T-1))]$, and for the normal distribution it is the standard normal variate $z$. It is the common basis of Chow's general equation for frequency analysis.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A 100-year flood has an annual exceedance probability of:
   A. 0.1
   B. 0.001
   C. 0.01
   D. 1.0
   Answer: C. P = 1/T = 0.01. (Key point: Exceedance probability)

2. The recurrence interval is:
   A. The exact number of years between floods
   B. The time of concentration
   C. The duration of the flood
   D. The average number of years between events equal to or greater than a magnitude
   Answer: D. It is an average over a long record, so a 50-year flood can happen in consecutive years. (Key point: Recurrence interval)

3. The frequency factor K_T is:
   A. The number of standard deviations the T-year event is from the mean
   B. The return period
   C. The skewness
   D. The sample size
   Answer: A. x_T = mean + K_T sigma. (Key point: Frequency factor)

4. For Gumbel, K_T equals:
   A. y_T x S_n
   B. (y_T - y_n)/S_n
   C. (y_T + y_n)/S_n
   D. y_T/ln T
   Answer: B. Using reduced mean and standard deviation for the sample size. (Key point: Gumbel K)

## Related in Engineering Hydrology

- [Gumbel exceedance probability and 0.1 flood](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-exceed-10000-3000) — Engineering Hydrology, chapter 6
- [Flood statistics and exceedance probability, 2000-2009](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aflood-stats-2000-2009) — Engineering Hydrology, chapter 6
- [Gumbel's distribution and frequency factor derivation](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-distribution-frequency-factor) — Engineering Hydrology, chapter 6
- [Gumbel design flood, safety factor, confidence limits](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-20yr-life35-rel87) — Engineering Hydrology, chapter 6
- [Define return period, frequency and risk](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Adefine-return-period-frequency-risk) — Engineering Hydrology, chapter 6

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