Exceedance probability, recurrence interval, frequency factor

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.