Three Key Statistical Concepts

 

 

1. Cumulative probability (Fi) is probability that precipitation takes a value less than or equal to a given amount.  To calculate the cumulative  probability function  (e.g., annual rainfall equal to or less than a certain amount):

                   - rank annual rainfall from lowest to highest

                   - rank is m, where 1 is the rank given to the lowest precipitation

                   - n is number of data points (number of years in this case)

                                     

                   Fi = m/(n+1) * 100%                                   

         

                   Fi = cumulative probability, i.e. the probability that that amount of rainfall or less will occur in a given year. It is key to note that for cumulative probability the lowest precipitation quantity receives a rank of 1.  This is a useful statistic for drought prediction, where we are interested in probability of a certain amount of precipitation or less occurring.                  

 

2. Exceedence probability (p) is the probability that rainfall of that much or higher will occur in a given year.  Although simple in concept, no end of confusion is created by the fact that the same equation is used as for the cumulative probability frequency.  The big difference is that m in this case is rank where the highest value receives a rank of 1 (m = 1).  

 

 p = m/(n+1)

 

This is a useful statistic for flood prediction, where we are interested in the  probability of a certain amount of precipitation or more that might cause flooding.

 

3.  The Recurrence Interval or Return Period (T) is the average interval in years between events equaling or exceeding a certain magnitude.  The return period is calculated from:

 

                   T = 1/p = (n+1)/m

 

Where m is again ranked so that the highest precipitation value gets a rank of 1.  The return period is a favorite tool of the press and planning agencies, because at first glance, people can readily grasp the concept of an event that occurs every x years on average.  The problem is that people assume this means an event of this size occur once every 20 years, when in fact it may occur 5 years in a row, then not occur for another 95 years (which gives an average of once every 20 years).

 

4.  A working example with 10 years of precipitation data from Kenya.  Therefore n = 10 and  n + 1 = 11.

 

 

 

 

Year

 

 

Precip (mm)

Cumulative Probability Rank

(1 is lowest to 10 is highest)

 

Cumulative

Probability

(%)

Exceedence Probability Rank

(1 is highest to 10 is lowest)

 

 

Exceedence Probability

 

Recurrence Interval

(years)

1931

1306

4

4/11*100 = 36.4%

7

7/11*100 = 63.6%

11/7 = 1.57

1932

1345

5

5/11*100 = 45.5%

6

6/11 *100 = 54.5%

11/6 = 1.83

1933

1032

1

1/11*100 = 9.1%

10

10/11*100= 90.9%

11/10 = 1.1

1934

1580

9

81.8%

2

18.2%

5.5

1935

1293

3

27.3%

8

72.7%

1.37

1936

1497

8

72.7%

3

27.3%

3.67

1937

1469

7

63.6%

4

36.4%

2.75

1938

1392

6

54.5%

5

45.5%

2.2

1939

1037

2

18.2%

9

81.8%

1.22

1940

1633

10

90.9%

1

9.1%

11