When presented with a new data sample, the first thing many of us scientists do is to characterise it in terms of two numbers: the average or *mean* value of the sample, and the spread or *variance* of the sample values around the mean. This has become second nature and we rarely stop to think twice about it. Yet it is indeed quite remarkable that data as different as Reading summer temperatures, the chest circumference of Scottish soldiers, or the sum of points obtained by rolling several identical dice can all be characterised by just these two numbers. Essentially, this is a consequence of the *Central Limit Theorem* in statistics, which states that in the examples above and many other situations, where the data arise as an average of more elementary data (for example tossing individual dice, averaging temperature throughout a season), the samples will tend to follow a *Gaussian* or *normal* distribution. The bell-shaped curve of this distribution is ubiquitous in all areas of quantitative science and may be the only mathematical function that has made it onto a bank note (Figure 1). The curve is described by two numbers, the mean determining the location of the bell, and the variance determining the width of the bell.

**Figure 1.** Carl-Friedrich Gauß (1777-1855) and the distribution named after him on the former 10 Deutsche Mark note (source: Wikipedia).

In meteorology we are often interested in extreme events such as strong windstorms, rain and flooding, heatwaves or drought. When we want to describe extreme behaviour, we have to change the way we collect data samples and characterise them. One option is to collect samples that comprise all strongest events in a block of data: the example I am presenting here is maximum daily winter precipitation (rain and snow) that falls over a river basin in each year. Unfortunately, such data samples can no longer be described by the tried and tested Gaussian distribution and its mean and variance. But mathematical statistics comes to the rescue in this situation too: there is an analogue of the Central Limit Theorem, called *Extremal Types Theorem*, telling us that we can replace the familiar Gaussian bell with a different function called the *Generalized Extreme Value (GEV)* distribution. We now need three numbers (or parameters) to characterise the GEV. They are called *location* μ, *scale* σ, and *shape* ξ, and their meaning is best illustrated graphically by so-called *Gumbel diagrams* shown in Figure 2. The vertical axis of these diagrams shows *return values* indicating the strength of an event (here daily river basin precipitation) and the horizontal axis shows *return times*, which tell us about the frequency of an event. The bold lines in the diagrams show different GEV distributions and they tell us how to relate a return time to an expected return value. For example, the brown curve in the top panel of Figure 2 shows that the expected return value for a return time of 20 years is 21 mm. We have to wait 20 years on average for a precipitation event of this amount to occur. The location parameter μ determines the vertical position of the GEV curve in the diagram – increasing it to μ=15 mm yields the green curve and the 20-year return value increases to 27 mm. The scale parameter σ determines the slope of the GEV curve in the Gumbel diagram as illustrated in the middle panel of Figure 2. The greater the σ, the more maximum precipitation will vary from year to year, and the more return values will increase with an increase in return time. Finally, the shape parameter ξ describes the curvature of the GEV curve (Figure 2, bottom panel).

**Figure 2.** Illustrative Gumbel diagrams showing GEV distributions with different values for the location parameter (top), for the scale parameter (middle), and for the shape parameter (bottom).

What is all this good for? One application is model evaluation, the process where we assess how realistically numerical models simulate the observed weather and climate. Here, I am interested in how well two versions of a climate model, a low-resolution version (named N96 in Figure 3) and a high-resolution version (N512, also in Figure 3) simulate the extremes of daily winter precipitation over European river basins. To obtain a summary assessment of this performance, I estimate the three GEV parameters for each of the models (N96, N512) and for a reference dataset (E-OBS) based on observed precipitation data from rain gauges. The results are shown in Figure 3. The top row shows the location, scale and shape values for the observations, and the middle and bottom rows show differences between the two models and the observations. We see that both models tend to produce too high precipitation extremes over large parts of Europe, especially over the northern European plains from the Loire river basin in the west to the Vistula basin in the east (greenish colours for the model-observation differences for the location and scale parameters). We also see that this problem is alleviated in the high-resolution (N512) model, where these differences are smaller than in the coarse (N96) model.

The statistical summary assessment shown here is only the first step in model evaluation and many questions remain. How do our two models represent rain-producing Atlantic storms, and how do these storms interact with the European landmass and, in particular, major mountain chains, such as the Alps? Trying to answer such questions is called process-based model evaluation and is an important part of the meteorological research here at Reading. But we will have to leave that for another blog.

**Figure 3.** Estimated GEV parameters for daily winter precipitation over European river basins. Top: precipitation observations (E-OBS), middle: difference between coarse model simulation (N96) and E-OBS, bottom: difference between high-resolution model simulation (N512) and E-OBS. Left: location parameter μ, centre: scale parameter σ, right: shape parameter ξ. Stippling shows statistically significant differences between N96 and E-OBS (middle row) and between N512 and N96 (bottom row).