Statistics are adorned with a scientific aura that elevates them to the rank of absolute truth. However, the traps they set are numerous. The subtlety of some of them delights us.

Lucid, Mark Twain wrote: “The facts are stubborn, it is easier to deal with the statistics. He added:”There are three kinds of lies: lies, the sacred lies and statistics. The field of calculations and statistical reasoning is full of traps, misleading evidence, and even scams: let us be on our guard, because intuition is often a bad advisor.

In Statistics. Beware !, the mathematician and psychologist Nicolas Gauvrit, from the Human and Artificial Cognitions laboratory, lists all the errors, tricks and misconceptions you should be aware of. We will borrow several examples from him. Let’s start with the percentages that we believe we have mastered since elementary school. Let’s check with a few questions (no machine or pencil needed).

Question 1. If the price of gasoline increases by 25%, it will return to its original price after decreasing by: (a) 25%, (b) 20% or © 16.67%?

Question 2. Two products A and B have the same price. The price of A increases by 12%, then decreases by 23%. B’s drops 23%, then increases 12%. In the end, (a) A is more expensive than B, (b) B is more expensive than A or © A and B are worth the same price again.

Question 3. We increase your salary by 2% per year. In ten years, it has increased by: (a) 21.90%, (b) 20%, or © 18.62%?

Question 4. A member of the government assures us that “the increase in debt which was 15% last year has been reduced to 14% this year”. The opposition claims, however, that “the deficit which was 15 billion euros last year has further increased this year by more than one billion euros”. Is one of them lying (a) or is it possible (b)?

Question 5. In a university, on the biology license examination, girls did better than boys and on the physics license examination, girls again performed better than boys. However, by combining the results of the two licenses, we discover that the boys were more successful than the girls. Has there been any gender-based wrongdoing?

To answer the first three questions correctly, a multiplicative perception of the percentages is necessary.

When the price of gasoline increases by 25%, rather than adding P to 25P / 100, multiply P by 1.25 (1 + 0.25) to calculate the new price. In order for it to return to its initial price, it must be multiplied by the inverse of 1.25, i.e. 0.80 or (1–0.20). The price must therefore be lowered by 20% to bring it back to its previous value. The correct answer to question 1 is therefore (b).

For question 2, the correct answer is ©, because the price of A has been multiplied by 1.12 then 0.77, which is equivalent to a multiplication by 0.77 then by 1.12, the operation being commutative. More generally, if you have increases and decreases to calculate, the order of calculation does not matter for the end result. On the other hand, we should not simplify by saying that + 12% and + 23% is + 35%!

For question 3, ten consecutive increases of 2% equals a multiplication by (1.02) 10. Without doing any calculations, we know that the total increase is greater than 20%, because (1 + x) n> 1 + nx. The correct answer is (a).

The last two questions of the test are more subtle. Both statements in question 4 may be true simultaneously. Last year’s € 15 billion deficit corresponds to 15% of the original debt (from two years ago). It was therefore 100 billion euros. Last year, the debt went from 100 billion to 115 billion. If, as the first statement indicates, the increase in the debt, that is to say the deficit, was 14%, this year, then the increase reached 14% of 115 billion, or 16 , 1 billion. This is in keeping with the second claim that the deficit has increased by over $ 1 billion. The two statements are perfectly compatible: the increase in debt can decrease in percentage each year at the same time as it increases in absolute value.

# The simpsons at school

Question 5 corresponds to a remarkable situation which offends intuition and can distort the analysis of certain real figures. Indeed, it is quite possible that girls do better than boys in biology bachelor’s degree and better in physics bachelor’s degree, and that, overall, boys do better than girls!

Take the (fictitious) example of a situation where 100 female candidates and 100 male candidates sit for the biology or physics license exams.

Physics biology cumulative G F G F G F PASS 80 10 4 50 84 60 FAIL 10 0 6 40 16 40 TOTAL 90 10 10 90 100 100

Girls do better in physics since 100% graduated compared to only 88% of boys. Likewise, in biology, 55.5% of girls and only 40% of boys are successful. Overall, however, 84% of boys have a diploma, compared to 60% of girls. How to explain it?

Girls are more numerous in biology license and boys more numerous in physics license. However, the success rate is better in physics than in biology. Girls therefore attempt a more difficult exam on average than boys. They only win, in total, because they opt for the easy way! Ninety out of 100 girls attempt the biology license which has a success rate of 55% while 90 out of 100 boys attempt the physics exam which has a 90% success rate. This situation is called the Simpson paradox or the Yule-Simpson effect. It was described by Edward Simpson in 1951 … and by George Yule in 1903.

This effect can have annoying consequences, as shown in a comparative study on the effectiveness of two different treatments for kidney stones, conducted in 1986 by C. Charig, D. Webb, S. Payne and O. Wickham.

Overall, treatment A had led to 273 successes (78%) out of 350 cases while treatment B in 350 cases gave 289, or 83%. Treatment B therefore seemed better than A. However, on closer inspection, it was found that in the case of small kidney stones, treatment A was better than treatment B: A obtained 80 of 87 successes, or 93%, against 234 successes out of 270, or 87%, for treatment B. And it was the same for the large kidney stones where A obtained 73% of successes (192 successes out of 263) whereas B only obtained 69% (55 success out of 80)!

Another real case of Simpson’s paradox was reported in 1975 by P. Bickel, E. Hammel and J. O’Connell regarding the admission of students to the various faculties of the University of Berkeley, USA. . Overall analysis of the data indicated a bias in favor of boys who, as in the previous fictitious example, performed better than girls. However, the detailed study of admissions, faculty by faculty, showed no such thing: the faculties favoring boys were not more numerous than those favoring girls.

The explanation of the phenomenon was similar to that of our example: girls applied for more difficult faculties on average than those attempted by boys. Should we conclude that one thing and its opposite can be said to the statistics? Is there always a risk of encountering such situations? The answer is no: in the data aggregation that generates the Simpson paradox, we do not mix data corresponding to equal numbers for the sub-cases. If we aggregate the results of the exams of 100 girls taking biology, 100 girls taking physics, 100 boys taking biology, 100 boys taking physics, then the Simpson paradox disappears. If one wants to obtain sensible conclusions, the aggregation of the results must respect certain rules of homogeneity.

Note that the Simpson paradox is generalized by taking more than two categories of students. Data, depending on whether you look at it one way or another, can lead to exactly the opposite rankings. Let us continue with some examples proposed by Nicolas Gauvrit which show that paradoxes similar to that of Simpson are more frequent than one imagines.

Welcome to the Marchive company whose salary situation can be summarized as follows:

managerial workers 2016 salary 200 € 2,000 € workforce 1,000 100 2017 salary 180 € 1,800 € workforce 600,500

A conflict opposes the unions and the boss. The former say: “The wages of workers and managers have fallen this year by 10%. “The boss replies:” Our calculations indicate that the average salary in the company has increased. It went from 363.64 euros per week to 916.34 euros, which corresponds to an increase of 152%. Yet again no one is lying. How is it possible ?

The table responds: the weekly wages of workers, from 200 euros to 180 euros, fell by 10%. That of executives, from 2,000 to 1,800 euros, has also fallen by 10%. The unions are therefore right to claim that wages have fallen by 10%.

For his part, the boss does not cheat! The salary paid to the 1,100 employees of the company in 2006 was 400,000 euros each week (1,000 200 + 100 2,000), or 363.64 euros per employee. In 2007, the salary paid to 1,100 employees (the overall workforce is unchanged) amounted to 1,008,000 euros (180 600 + 1,800 500), or 916.34 euros per employee. The average salary has therefore increased by 152%.

The scam, because there is one, is that the workforce by category changed between 2016 and 2017. As a result, the decrease in salary in each category is offset by the increase in the number of executives who are better paid. A 10% reduction in the salary of each category of personnel is perfectly compatible with an increase in the average salary of employees.

This situation is not so fictitious as that, because each year the French State insists on the figures of the wage bill of civil servants and on the average salary of a civil servant which evolve more favorably than the index point used to pay. salaries of civil servants. This index point determines, for a fixed grade, the salary of a civil servant and of course it is to him that the unions prefer to trust. The increase in the average age of civil servants — due in particular to strong recruitment into education in the 1970s and 1980s — automatically leads to an increase in the average grade (as in the fictitious company in the example indicated above ) and therefore leads to a discrepancy between the figures relating to the average salary of a civil servant and those relating to the index point, which ministers and unions use in the way that suits them best.

Still on the subject of civil servants, let us see another apparent paradox. It can be said without contradicting ourselves that: (a) the average salary in the public service is higher than that in the private sector; (b) the majority of civil servants would earn more if they went into the private sector. The explanation lies again in the distribution of jobs in the public and private sectors: in the first, the number of skilled jobs is greater, which has the effect of increasing the average wage in the public sector, without for as much to contradict that with the same diploma one earns less in the public than in the private sector.

Statistics use indices with precise definitions which summarize in a single number a sometimes considerable mass of figures. These clues are inevitable: without them, we could not synthesize data tables. However, statisticians’ indices set traps and can generate completely absurd results. Either the one who examines them understands them imperfectly, or he does not take specific situations into account.

Life expectancy is a trick index. The poverty line as defined by INSEE is just as disturbing. By definition, the poverty line in a country is half of the median income, i.e. half of income X such that there are as many people earning more than X as people earning less in the country. . The poverty index is the percentage of people living below the poverty line.

If now someone announces to you that in such and such a country 60% of the people live below the poverty line, that is nonsense. By definition, this is not possible: half (to a few units) of people have income below the median income, and necessarily less than half have an income below half of the median income.

The dramatic consequence is that even in a country where everyone died of starvation, the number of poor would remain below 50%, while in contrast, in a country composed only of billionaires, there could very well be 45% of people living below the poverty line. Poverty and wealth are relative concepts, of course, but still!

# To be poor in plenty

Nicolas Gauvrit imagines the following short story which shows absurdly how dangerous the use of the poverty index is. In the land of plenty, an apple costs a thousandth of the floor. Accommodation costs 5. We eat well for 0.2 sol. We live in comfort for 100 sols per month, and like a mogul for 200 sols per month. There are two types of workers in this country: workers who have an income of 1,000 sols per month, and thinkers who have a monthly income of 3,000 sols. There are as many workers as there are thinkers, exactly. All of them live in harmony and in an opulence that neighboring countries envy. The only exception to the rule of two salaries: the president of the country, for his part, receives a salary of 2,800 sols. The median income is then 2,800 sol, as any statistician would confirm. Half of the median income is therefore 1,400 sols, and half of the population (not including the president) receives a lower salary. The poverty index is therefore 50% and it is the highest possible value of this index.

But one fine day in May 2068, the president agreed to lower his indemnities, because, he said, “there is no reason why I should claim more than a worker! I am a worker like the others ”. Taking into account his responsibilities, however, he allowed himself a small extension and declared himself satisfied with his new income of 1,400 sols per month.

The median income in Cocagne then drops immediately to 1,400 sols, and the median half-income to 700 sols, and by a breathtaking statistical miracle, the poverty index immediately drops, in May 2068, to 0, the minimum. possible. “

Created following a controversy over the poverty index, the BIP 40 or barometer of inequalities and poverty is a synthetic indicator of inequalities and poverty. This index was proposed by the Inequalities Alert Network in 2002 and it eliminates some of the problems mentioned above. However, it is quite complex, so it is difficult to understand its meaning and we cannot guarantee that it will avoid all the absurdities of simpler clues.

# A family matter

Among all the traps detailed in Nicolas Gauvrit’s book, one of them is quite subtle and deserves special attention, we will call it the paradox of the average number of children. An exhaustive survey carried out in a distant city indicates that families with children under 18 are distributed as follows: 10% families with 1 child, 50% with 2 children, 30% with 3 children, 10% with 4 children. The average number of children per family (among those with children) is therefore (10 + 100 + 90 + 40) / 100 = 2.4.

To check this statistic, the administrative authorities carry out a survey. We interview 1,000 carefully selected children under the age of 18 and ask them how many children there are in their families, including themselves. By taking the average of the answers, we get… 2.68! It sounds absurd. So we start the survey again, this time interviewing 10,000 children, we now find 2.67. A third poll of 100,000 children gives 2,668 again. Why this difference so important with the 2.4 of the statistic which took into account all the families with children?

The answer lies in the fact that by interviewing children at random, you will interview 4 times more children in families with 4 children than you will interview in families with 1 child, which skews the average. If there are 1000 families, there will be 100 only children, 1000 children belonging to a family of 2 children, 900 children belonging to a family of 3 children, 400 children belonging to a family of 4 children. In total, the answers given by these 2,400 children will lead to the result of 2,666… children per family.

The surveys conducted do not estimate the average number of children in a random family, but the average number of children found in the family of a random child. “Take a random family” and “Take a random child” are not the same thing.

We know since Condorcet for the votes, it is very difficult to synthesize a set of numbers in only one. We hope that the pitfalls of statistics, graphical representations, synthetic clues, polls presented here — and those you will find in Nicolas Gauvrit’s book — will help you better understand the truths behind the disturbing and fluctuating world of numbers.