# How Statistics Can Be Manipulated: The Base Frequency Fallacy

Can statistics be manipulated? What are statistical fallacies? What is the base frequency fallacy? In this article we are going to learn some key aspects of statistics, in the same way that we are going to learn the lesson that, in statistics, not everything is what it seems.

It is very common that, reading the economic press, we see headlines that say that 90% of users recommend something, that 72% of citizens prefer a certain thing, or that 67% of the population prefer Apple laptops compared to HP computers, for example.

However, it should be noted that when we talk about statistics, the numbers can be easily manipulated in their interpretation. In this sense, let’s look at the latest example of laptops.

In this case, we say that 67% of the population prefers Apple laptops. However, does the entire population have or want laptops? What are the profiles surveyed? Have other options been proposed to the respondents?

When interpreting the data, certain fallacies allow us to say that 67% of the population prefers Apple. But that population analyzed might not be representative, as it could be a survey of 100 people with high purchasing power that allows them to buy it.

Another fallacy to highlight could be the gambler’s fallacy.

Have you ever played roulette? The gambler’s fallacy leads us to mistakenly think that past events affect future events when it comes to random activities. An example is found in roulette. If we have bet on red and black has come out four times in a row, even though the probability remains the same, we believe that it is more likely that red will come out on the next spin.

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As we can see, in statistics there are many fallacies in which we can fall when defending our arguments and that, therefore, make statistics can be manipulated, whether consciously or not. But this time we will focus on one: the base frequency fallacy.

This fallacy is widely used when you want to manipulate statistical information. In this way, it is intended that the recipient focuses on the specific data and generalizes it, forgetting that there was already a general data that, many times, is different from its deduction.

## The base frequency fallacy and Bayes’ theorem

Before entering fully into the analysis of this fallacy, as well as putting it in context and relating it to the world of cryptocurrencies, we must briefly talk about Thomas Bayes and his famous theorem.

What Bayes is saying in his theorem is that the probability of having a car accident depends on whether it rained yesterday or not. We are talking about an example, but it is very easy to understand.

Bayes’ theorem is used to calculate the probability of an event, having information in advance about that event. Well, classical probability has always focused on those events a posteriori. But Bayes, contrary to classical statistics, takes into account what happens a priori. In this way, what he proposes is that we take into account those probabilities that, a priori, condition those that we analyze.

Therefore, based on the classic statistics, we could say that the probabilities of a car having an accident depends on the number of times this car passes through a certain dangerous place, as well as other aspects to take into account. But Bayes also teaches us that the probability of having an accident (a posteriori) is also conditioned by other events, such as the fact that it rained the night before (a priori).

But do not worry if this example is not clear to you, because in the next section we will see another example, this time related to the pandemic caused by COVID19. In this case, we will focus on vaccination campaigns.

## The Base Frequency Fallacy and COVID19

Next, we are going to see an example of this fallacy and how it can be related to the COVID-19 pandemic. To do this, we are going to locate ourselves in a specific region of Spain, and with data from its own health department.

In the first place, we start from a premise, which is that in that region, 80% of the population is vaccinated, while the remaining 20% ​​is not.

Based on this premise, we have observed that, recently, the Ministry of Health offered some health data that should be analyzed to understand the example.

They showed that there were approximately 80 vaccinated and 20 unvaccinated admitted people. Thus, at a glance, we could deduce that most of the admitted people are vaccinated, being able to follow this statement that the vaccines, therefore, cause your admission to the ICU. But we must know that, based on the main studies, what the experts tell us is that vaccines prevent admission to the ICU.

So what happens?

Well, what happens in this case is that we would be facing the fallacy of the base frequency. At no time are we taking into account that the probability that an admitted person is vaccinated is greater than one that is not vaccinated. In this case, it would be 80% compared to 20%, since 8 out of 10 citizens in this region are vaccinated.

As we can see, in statistics nothing is what it seems and a deeper analysis would have to be carried out to obtain a conclusion, since the base frequency fallacy can lead us to erroneous conclusions, such as that for vaccinating you will be admitted to the hospital.

## The fallacy of base frequency and cryptocurrencies

To end this fallacy, let’s look at another example, this time more advanced, that allows us to identify what the base frequency fallacy is, and how this fallacy has been used in the world of cryptocurrencies.

In this last example, we are going to look at some data for various cryptocurrencies or cryptocurrencies. Specifically, we will focus on the price, through the variation rates, of Bitcoin, Litecoin or Dogecoin of a selection of ten days. Of course, stay tuned to the details.

In the graph we show, what we can see is that all three seem to follow a similar trend. In other words, all the analyzed cryptocurrencies present a very similar trend, as shown in the graph we are analyzing.

Thus, once we give the general data, which is that cryptocurrencies all follow a similar trend throughout the 10 days analyzed, we focus on Litecoin specifically.

Well, focusing on each of them and, specifically, Litecoin, we can see how it experienced a very pronounced decrease in its rate of change between the fourth and fifth day. The general data shows us that all cryptocurrencies follow this trend, but the specific data indicates that Litecoin specifically is a cryptocurrency with great volatility, since very pronounced ups and downs are observed during the period analyzed.

Thus, what this fallacy of the base frequency raises is what we discussed. Well, this, the specific information, will be the information that prevails if we attend to this fallacy.

## Is Litecoin Really More Volatile?

The fallacy of the base frequency can be encompassed in another, the neglect of extension. This consists of reaching a conclusion without taking into account the size of the observed sample. As we see, our fallacy would be a particular case of this.

Thus, if we focus our attention only on the two days in which Litecoin presents such a high negative variation rate (-12.70% and -11.80%), what we would say and what indicates that our intuitive thinking would be that Litecoin has a lot of volatility, but …

If we now turn our attention to the trend lines, we see that this cryptocurrency has a much higher positive slope than the other two. In other words, we could even say that we are talking about a cryptocurrency that has presented a better performance and that could lead us to obtain more income, although the fallacy that concerns us here may drive us out of the market for making highly biased claims. Now, it should be noted that this, that is, the positive slope has a lot to do with that volatility experienced during the fourth and fifth days.

Thus, if we are a conservative investor profile and we want to invest in more stable cryptocurrencies, both Bitcoin and Dogecoin are currencies that, according to the graph, remain more stable over time.

So which one should you invest in? A priori, in none.

## Investing in cryptocurrencies with the base frequency fallacy

You are probably wondering how our answer to the question can be that, considering that we have said that Bitcoin and Dogecoin are more stable currencies, but I must insist that I have not been wrong, because it is this fallacy that could have led me to lose it all in the markets.

But why? The answer is quite simple, because in our analysis, among other issues, we have made the aforementioned error of neglect of extension. That is, when we analyze a trend over time, the periods must be very wide, while those observed in this case do not exceed 10 days.

On the other hand, we observe that the coefficients of determination (R squared) that indicate whether the line is valid or not are close to zero. This means that these trend lines do not predict anything in this case. And it is that, although you probably have not realized it, we have fallen into the same fallacy. We have refocused on the specific data, which is a favorable trend for these cryptocurrencies, despite the fact that the general tells us that these lines, in reality, are absolutely useless.

Therefore, to invest in cryptocurrencies and regardless of technical analysis, simple statistics can help, but with the correct analysis. The fallacy of the base frequency teaches us that we should not let ourselves be carried away by appearances when it comes to analyzing data, since these, as he would say, are loaded by the devil.

And you do you think? Are you going to continue believing everything you see?