Data distribution

(The author of this entry is Henrik von Wehrden.)

Data distribution

Data distribution is the most basic and also fundamental step of analysis for any given data set. On the other hand, data distribution encompasses the most complex concepts in statistics, thereby including also a diversity of concepts that translates further into many different steps of analysis. Consequently, without understanding the basics of data distribution, it is next to impossible to understand any statistics down the road. Data distribution can be seen as the fundamentals, and we shall often return to these when building statistics further.

Types of distribution

The normal distribution

How wonderful, it is truly a miracle how almost everything that can be measured seems to be following the normal distribution. Overall, the is not only the most abundantly occurring, but also the earliest distribution that was known. It follows the premise that most data in any given dataset has its majority around a mean value, and only small amounts of the data are found at the extremes.

Just as the normal distribution, most phenomena we can observe follow a normal distribution. The fact that many do not want this to be true is I think associated to the fact that it makes us assume that the world is not complex, which is counterintuitive to many. While I believe that the world can be complex, there are many natural laws that explain many phenomena we investigate. Linear patters describe such a natural law. Quite often if you increase one thing, an associated phenomena increases as well. If you make the oven more hot, your veggies in there will be done quicker. However, this linearity only works within a certain reasonable section of the data. It would be possible to put the oven on 100 °C and the veggies would cook much slower than at 150 °C. However, this does not mean that a very long time at -30 °C in the freezer would boil your veggies in the long run as well. Frost is a good example of a non-linear phenomena, actually on elf the few examples I know. However, within a certain reasonable data section, you can often observe that one phenomena increases if another phenomena increases, and this relation follows a linear pattern. This is clearly logical, as for instance if you bike faster, you arrive faster. Many phenomena follow such a linear pattern. A prominent example is the Island Theory from Mac Arthur and Wilson. They counted specs on island, and found out that the larger islands are, the more species they host. While this relation has a lot of complex underpinnings, it is -on a logarithmic scale- totally linear. Larger island contain more species, with a clearly mathematical beauty. Phenomena that you can count are often linear on a log scale. Among coffee drinkers most coffee drinkers drin 1-2 cups per day, but few drink 10, which is probably good. Counting cup consumption per day in coffee drinker follows a log-linear distribution. The same hold for infections within an outbreak of a contagious disease. The Western African Ebola crisis was an incredibly complex and tragic event, but when we investigate the early increase in cases, the distribution is exponentially increasing -in this case-, and this increase follows an almost cruel linear pattern.

Take height. Most people have an average height, but only a few people are very tall, and a few people are very short. The majority of people have clearly an average height. Many such natural phenomenon follow the normal distribution. Just measure the weight of some spaghetti with a very precise balance. The majority will resolve around a mean value, and only some few will be much heavier or much lighter. While it may seem like a magic trick, it is actually true that many phenomena that can be measured will follow the normal distribution, at least when your sample is large enough. Consequently, much of the probabilistic statistics is built on the normal distribution.

See Tests for normal distributionto learn how to check if the data is normally distributed.

The Poisson distribution

Things that can be counted are often not normally distributed, but are instead skewed to the left. While this may seem curious, it actually makes a lot of sense. Take an example that coffee-drinkers may like. How many people do you think drink one or two cups of coffee per day? Quite many, I guess. How many drink 3-4 cups? Fewer people, I would say. Now how many drink 10 cups? Only a few, I hope. A similar and maybe more healthy example could be found in sports activities. How many people make 30 minute of sport per day? Quite many, maybe. But how many make 5 hours? Only some very few. In phenomenon that can be counted, such as sports activities in minutes per day, most people will tend to a lower amount of minutes, and few to a high amount of minutes. Now here comes the funny surprise. Transform the data following a Poisson distribution, and it will typically follow the normal distribution. Hence skewed data can be often transformed to match the normal distribution. While many people refrain from this, it actually may make sense in such examples as island biogeography. Discovered by MacArtur & Wilson, it is a prominent example of how the log of the numbers of species and the low of island size are closely related. While this is one of the fundamental basic of ecology, a statistician would have preferred the use of the Poisson distribution.

The Pareto distribution

Do you know that most people wear 20 % of their clothes 80 % of their time. This observation can be described by the Pareto distribution. For many phenomena that describe proportion within a given population, you often find that few make a lot, and many make few things. Unfortunately this is often the case for workloads, and we shall hope to change this. For such proportions the Pareto distribution is quite relevant. Consequently, in is rooted in income statistics. Many people have a small to average income, and few people have a large income. This makes this distribution so important for economics, and also for sustainability science.