page_title: Cumulative frequency analysis
section_title: Confidence intervals
hierarchical_section_title: Cumulative frequency analysis / Prediction / Confidence intervals
caption_attribution_description: English: Binomial confidence belts
context_page_description: Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The phenomenon may be time- or space-dependent. Cumulative frequency is also called frequency of non-exceedance.
  Cumulative frequency analysis is performed to obtain insight into how often a certain phenomenon is below a certain value. This may help in describing or explaining a situation in which the phenomenon is involved, or in planning interventions, for example in flood protection.
  This statistical technique can be used to see how likely an event like a flood is going to happen again in the future, based on how often it happened in the past. It can be adapted to bring in things like climate change causing wetter winters and drier summers.
context_section_description: Probability theory can help to estimate the range in which the random error may be.
  In the case of cumulative frequency there are only two possibilities: a certain reference value X is exceeded or it is not exceeded. The sum of frequency of exceedance and cumulative frequency is 1 or 100%. Therefore, the binomial distribution can be used in estimating the range of the random error.
  According to the normal theory, the binomial distribution can be approximated and for large N standard deviation Sd can be calculated as follows:
  Sd =√Pc(1 − Pc)/N
  where Pc is the cumulative probability and N is the number of data. It is seen that the standard deviation Sd reduces at an increasing number of observations N.
  The determination of the confidence interval of Pc makes use of Student's t-test (t). The value of t depends on the number of data and the confidence level of the estimate of the confidence interval. Then, the lower (L) and upper (U) confidence limits of Pc in a symmetrical distribution are found from:
  L = Pc − t⋅Sd
  U = Pc + t⋅Sd
  This is known as Wald interval.
  However, the binomial distribution is only symmetrical around the mean when Pc = 0.5, but it becomes asymmetrical and more and more skew when Pc approaches 0 or 1. Therefore, by approximation, Pc and 1−Pc can be used as weight factors in the assignation of t.Sd to L and U :
  L = Pc − 2⋅Pc⋅t⋅Sd
  U = Pc + 2⋅(1−Pc)⋅t⋅Sd
  where it can be seen that these expressions for Pc = 0.5 are the same as the previous ones.
  Notes
  The Wald interval is known to perform poorly.
  The Wilson score interval provides confidence interval for binomial distributions based on score tests and has better sample coverage, see and binomial proportion confidence interval for a more detailed overview.
  Instead of the "Wilson score interval" the "Wald interval" can also be used provided the above weight factors are included.
caption: 90% binomial confidence belts on a log scale.
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