Back of the envelope statistics

Back of the envelope statistics

Back of the envelope calculations give you a first impression about your idea and where it can go to.

Back of the envelope calculations are rough estimates that are made on the small piece of paper, hence the name. These are extremely helpful to get a quick estimate about the basic numbers for a given formula of principle, thus enable us to get her quick calculation with either the goal to check for the plausibility of the assumption, or to derive a simple explanation of the more complex issue. Back of the envelope calculations can be for instance helpful when you want to get a rough estimate about an idea that can be expressed in numbers. Prominent examples for back of the envelope calculations include the dominant character coding of the World Wide Web and the development of the laser. Back of the envelope calculations are fantastic within sustainability science, I think, because they can help us to illustrate complex issues in a more simple form, and they can serve as a guideline for a quick planning. Therefore, they can be used within other more complex forms of methodological applications, such as scenario planning. By quickly calculating different scenarios we can for instance make a plausibility check and focus our approaches on-the-fly. I encourage you to learn back of the envelope calculations in your everyday life, as many of us already do. I learned to love "Tydlig", which is one of the best apps I ever used, but unfortunately I only know her version for my Apple devices. It can however be quite helpful to break numbers down into overall estimates.

Examples of back of the envelope calculations

Simple calculations

Adding, subtracting, multiplying and dividing are part of the everyday language of mathematics, and while we all should have learned these in school, most of us are not versatile in applying them regularly. How do we divide the bill by three? What is 20 % of the bill to add as a tip? How many pieces of cake remain? How can I double this recipe? There is ample evidence that this can be intuitive to some, yet most of us struggle. However, I think it is extremely important to regularly practice, and there are apps such as brilliant and games like Sudoko to improve our skills with numbers. Only if you learn the basics will you be able to master the supreme mathematics. In the following I provide you some simple examples of back of the envelope calculations, which may help you to gain some understanding on why these could be valuable.

Probability

We often hear numbers about probability and chances. While many of these are arbitrary, such as the chances to win the Lottery, there are others probabilities that matter in our day to day life. For instance, the chances of catching a Covid-19 infection outside is 19 times lower than inside. One common misconception is that the chances are not 0 - the chances are just lower. Another misconception is the question of how low your chances are in general. Quite often we cannot stop computing a chance calculation where our chances are actually low. Will I win the Lottery (probably not), or will I be diagnosed with a rare condition (probably not). Still, we think more about such things than are actually our chances. Being more clear about the probability of a certain event to happen may actually help us to compute the chances once and for all, and then act accordingly. The Corona crisis is a good example, where a combination of several modes of action, such as social distancing, wearing masks or washing your hands, can substantially lower your chances of catching the disease. However, it will be next to impossible to lower your chances to 0, at least not to the amount to creating harm to others or yourself that may outweigh the low chances of catching the disease. Calculating probabilities can be a good exercise to be more clear about your chances.

However, sometimes it is not the chances or probabilities that you are interested in, but the actual numbers, and these two differ. A chance of 1:100000 seems rather smallish, yet if I tell you that this is the chance of a popular touristic attraction - a scenic cliff - that a tourist falls of the cliff while taking a selfie, then you would agree to create safety measures. Numbers count. There is a difference between proportions or chances on the one end, and absolute numbers on the other end, and sometimes we need to choose one over the other. When you are for instance afraid of flying in a plane, because the plane might crash, knowing the chances of actually crashing will hardly help you to get over your fear. On the other hand are many smokers aware of the risks and seemingly do not care. I propose that we should more often calculate our chance for in other words the probability, as this might give us an actually more accurate picture of our conditions and circumstances. Based on this information, we can then decide what to do, and how to act.

Trend statistics

Another prominent example of back of the envelope calculations is knowledge about predictions. Here, several misconceptions are at stake, and it may be beneficial to debunk these. First of all, we have to conclude that many predictions that we make based on rather small samples can be wrong. There are prominent examples of recent elections where much was at stake, and it was ambiguous what would come out in the end. The US election in 2016 and 2020 were most relevant examples, which showcase how different samples lead to changes in the outcome. For instance, in 2020, over the first day after the election, the counting was leaning towards Trump, yet with more and more counts coming in from the larger cities, many states tilted towards Biden. Here, the pre-counts were biased towards a more positive outcome for the Republicans. However, looking at the counties, and checking where most counts were still being counted, gave a clear picture early on towards a shift to Biden. Such trend statistics can hence be rather advanced, and knowledge about the context goes a long way. More often than not, this is however not the case. The case statistics from Wuhan showed an early picture of the priors in terms of the spread of this disease, and already in late January were the priors of the disease rather clear. With this rate of increase, and the rate of mortality evolving with a few weeks lack, the information that patients were spreading the disease before showing symptoms sealed the trend. From then on, all trend calculations could be probed based on the initial development in Wuhan, and calculating in counter measure or vaccines later were mere modifications of the same trend function. Calculating percentage growth and then extrapolating further up until a maximum system saturation is thus a common tool to understand what might happen. Looking at temporal trend data and understanding the patterns is one of the most essential tools in these times of global change. Become versatile in reading bar plots, and learn to calculate trends both in absolute numbers as well as in percentage growth, then many developments will not come as a surprise to you.

Group differences

While there are advanced statistical tests to compare groups and calculate differences between individual groups, it is clear that more often than not, comparing mean values between groups is the most relevant calculation. While there is for instance a high variance in lactose intolerance across the globe, people from certain regions should avoid milk probably more than people from other groups. Equally are some groups more prone to specific diseases than others. This may not say much about your individual risk, but is still an initial prior that can translate into a different chance calculation (see above). Variance and mean values are not the same. Consider the case of the enzyme breaking down alcohol, which is missing in some people of Asian heritage. While the mean values in terms of Alcohol intolerance are clear, this may not help you in a drinking game, since the variance regarding this enzyme is quite high. Another example is the soapy taste some people experience when eating corianthe. While indeed more Europeans as compared to some Asians experience a soapy taste, this difference is remarkably small, and based on a study that samples many Europeans, and few participants of Asian heritage. Hence back of the envelope calculations can not only help you to know certain patterns better but can equally help you to recognise flaws in other peoples assumptions or calculations. It is indeed quite often the case that research highlight results that may be statistically significant, but when you check out the actual patterns, then these are quite small.

System dynamics

The last calculation I want to highlight are complex system calculations where several calculations are broken down based on various assumptions. A prominent calculation is the carbon footprint of a person. While this can be a quite advanced endeavour demanding many calculations of supply chains and circular dynamics, there are proxies that can allow you to make a rough estimation. The combination of your food choices, travel distance and frequency -including commute habits- as well as heating and electricity usage can lead to severe changes in the carbon footprint of a person. Approximating such crude measures can often be more transformative than advanced calculation we better leave to the professionals. Building your life choices on simple heuristics can be helpful for many people, and back of the envelope calculations can support such approaches.

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The author of this entry is Henrik von Wehrden.