An example of the use of statistics

Gumbel distribution.

The distribution of extreme values can be described through various methods. This example is about the high water level extreme per year in a non-existent river. Over a very long period of years, for each year the highest level is taken. From this long time series you determine how often certain water levels have occurred. If you divide the number of occurrences by the total number you get the probability. If you make a graph with the probability of the occurrence of water level heights, you would get the graph you see to the right. This distribution is called “Gumbel distribution”. On the horizontal axis you see the height of the water level extreme and on the vertical axis the probability.

This distribution is not symmetrical. You need a lot of data to determine the shape of the graph. The high extreme values that appear rarely are distributed over a long tail. And those values are the most interesting for flood risk management. Now how high is the probability of a water level extreme that is dangerous? And the other way around: what if the government decides to have a dike along this river that can withstand the water level with a 1 in 100 per year probability? How do you know what that water level is?

Let’s assume that the dangerous water height in this case is 70. The probability or this or an even higher level occurring, is equal to the surface under the graph from 70 onwards. Now we will not do this calculation, but you can image that it would be a small number. The probability of a water level of 70 or higher is very low. If through climate change, or upstream measures, the whole graph would shift to the right, the probability of having a water level of 70 or higher would increase. So, all other things being equal, the risk of flooding would increase.

The discharge of the river Rhine for instance is expected to rise as a result of the climate change. The present I in 1250 per year discharge will occur more frequently. So the risk of floods increases. If you want to keep the risk the same as now, you will have to take measures.

The authorities responsible for the safety of the people along the river, may decide that measures need to be taken to make sure that dangerous water levels can only occur with a probability of 1 in 1000 per year. What would that water level be? If you would have a sufficiently accurate graph you could draw a horizontal line from 0.001 and see where it crosses the graph for the second time. Here in this graph you see that, for those values of probability, the graph is not good enough. Other, more complicated, statistical methods are required.