Understanding statistics is hindered by our unwillingness to change.

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The study showed that people prefer complex methods because they are used to them.

The illegal accusation of Sally Clark in the murder of her two sons is a famous example of the misuse of statistics in court.

In 1999, the British solicitor Sally Clark went on trial for the murder of her two young sons. She claimed that both of them were victims of sudden infant death syndrome . The expert, the prosecution witness, Roy Meadow, argued that the chances that this syndrome would take the lives of two babies from a rich family were 1 in 73 million, which equated them with a chance to put on horse races with a factor of 80 to 1 for four years in a row and win all the time. A jury sentenced Clark to life imprisonment.

However, the Royal Statistical Society issued a statement after the announcement, stating that Meadow was wrong in his calculations, and that for the numbers he declared "there were no statistical grounds." Clark's sentence was canceled as a result of an appeal in January 2003, and this case became a canonical example of the consequences of incorrect reasoning based on statistics [ The sentence was canceled after it turned out that the pathologist had issued the wrong conclusion. Clark unjustly spent three years in prison, received a serious psychological trauma, and four years later died from an alcohol overdose / approx. trans. ].

The new study, published in the journal Frontiers in Psychology, examined the question of why it is so difficult for people to solve statistical problems, in particular, why we clearly prefer complex solutions to simple and intuitive ones. This property must be recorded at the expense of our resistance to change. The conclusion of the study says that the fault of all is unwillingness to change: we try to adhere to the well-known methods we studied in school, which does not allow us to see the existence of a simpler solution.

Approximately 96% of the population hardly solves problems related to statistics and probability. However, to be a well-informed citizen of the 21st century, you need to competently cope with these kinds of tasks, even if you do not encounter them in your professional field. “As soon as you pick up a newspaper, you are faced with a huge number of numbers and statistical calculations that need to be correctly interpreted,” says co-author Patrick Weber, a graduate student in mathematics teaching at Regensburg University in Germany. And most of us do not reach this level much.

Part of the problem is the counterintuitive method of presenting such problems. Meadow presented his testimony in the so-called. “Natural frequency format” (for example, “one out of ten people”), and not in percentages (“10% of the population”). It was a smart decision, because “1 out of 10” is more intuitive [ that it’s more understandable, for now just a hypothesis / approx. trans. ] and clearer to the jury. Recent studies have shown that indicators for solving statistical problems increase from 4% to 24% when tasks are presented in the natural frequency format.

This makes sense, since it is rather difficult to calculate probabilities; this requires three multiplication and one division, according to Weber, after which you need to divide the two resulting terms of the equation. And for the format of the natural frequency requires only one addition and one division. “With natural frequencies, you have data that you can clearly imagine,” says Weber. The probability format is more abstract and less intuitive.

Bayesian Challenge

What about the remaining 76% of people who are not able to solve such problems? Weber and colleagues tried to figure out why. They took 180 university students and gave them two test tasks on the so-called. Bayesian output , issued either in the format of probabilities, or in the format of the natural frequency.

The tasks included Bayesian statistics — say, the probability that a 40-year-old woman would find breast cancer (1%) —with a sensitivity element (for women with breast cancer, a mammogram test would give 80% of cases) and the number of false positive (women without cancer have a chance at 9.6% to get a positive result). Question: If a 40-year-old woman gets a positive test for breast cancer, what is the likelihood of having a real disease (assessment of "a posteriori" probability)?

In one of the pilot tasks, participants were asked to calculate the probability that a randomly selected person with fresh marks of injections on his arm would be a heroin addict.

The mammogram task is too well known, so Weber and his colleagues invented their tasks. For example, the probability that a randomly selected person from a given population is a heroin addict is 0.01% (base value). If the chosen person is an addict, then there is a 100% chance that he will have fresh marks from the needles on his arm (an element of sensitivity). However, there is a probability of 0.19% that a randomly selected person will have fresh marks from the needles on his arm, but he will not be a drug addict (the probability of a false positive response). What is the probability that a randomly selected person with fresh marks from the needles on his hand will be a heroin addict?

Here is the same task in the natural frequency format: 10 out of 100,000 people are heroin addicts. 10 people out of 10 drug addicts have fresh marks from the needles on their hands. At the same time, 190 out of 99,990 non-drug addicts have fresh needles marks. What percentage of people with fresh needle marks will be addicts?

In both cases, the answer will be 5%. But the process of getting a response in the natural frequency format is much simpler. A set of people with traces of injections on the arm is the sum of 10 drug addicts and 190 non-drug addicts. 10/200 gives us the right answer.

Thinking inertia

Students needed to demonstrate calculations to make it easier to follow their thinking process. Weber and his colleagues were surprised to find that even after receiving tasks in the natural frequency format, half of the participants did not use a simpler method for solving them. They translated the problem into a more complex format with percentages and with all the additional steps, since such an approach was more familiar to them.

This is the essence of the inertia of thinking, also known as the effect of customization. “We build our previous knowledge into our solutions,” says Weber. This can be helpful and help us make decisions faster. But this may not allow us to see new, simpler solutions to problems. Even chess chess experts are subject to this. In response to the opponent’s move, they choose a tried and known strategy, well known to them, while there may be a simpler solution for setting a mat.

Weber suggests that one of the reasons for this is that students are too often confronted with the format of probabilities in mathematics lessons. This is in particular the problem of the standard educational program, but he also believes that among teachers there may be bias towards natural frequencies, to their seeming mathematical weakness. But actually it is not. "You can completely strictly determine these natural frequencies mathematically," - Weber insists.

To change this approach is quite difficult - first, it is necessary to revise the program of teaching mathematics, including the format of the natural frequency. But this will not affect the situation so much if teachers do not feel comfortable using this format, so universities will also have to include it in the teacher training program. “This will provide students with a useful tool to help them understand the concept of uncertainty, complementing standard probabilities,” says Weber.

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