A True Number Can Tell a False Story
Some statements seem to have won an argument before anyone has even understood them. They contain a large number, a precise percentage, a result delivered with confidence. When we hear an opinion, we naturally feel free to question it; when we hear a figure, we are more inclined to think the matter has already been checked. As though writing down a number were enough to place reality beyond interpretation.
It is this trust that interests me. I do not think numbers are the problem: without them, we would be even more at the mercy of our impressions. But I do think that whoever chooses how to present them can steer our judgement without falsifying a single one. They can say something true and let us imagine something else, something far more convenient for them.
You do not always have to lie about numbers: sometimes it is enough to keep people from understanding what they measure. To think this through, I will use illustrative examples, without attributing them to real governments or companies, and a question that arose while I was reading my website's statistics. What interests me is learning to recognise a way of speaking we can encounter anywhere, and can slip into ourselves.
The Simple Trick of Multiplying by Five
The statement that started me thinking was this: “Seventy thousand more jobs in five years.” It was not a complicated calculation that struck me. It was that multiplication by five: describing a result through the total for a period long enough to make it look more impressive.
If, at the end of five years, there are 70,000 more jobs on a net basis than there were at the beginning, the average increase is 14,000 jobs a year. Both ways of putting it describe the same change, but they do not necessarily leave the same impression. “Seventy thousand” fills the stage; “in five years” risks becoming a tail end of the sentence to which we barely listen.
The average, of course, does not mean that exactly 14,000 jobs were added each year. There may have been years of growth and years of decline. Nor does it mean that only 70,000 people were hired: a net increase in jobs is not the same as the total number of hires or contracts. To understand the result, we need to know what is being counted, not simply perform a division.
There is nothing wrong with presenting a five-year result, as long as it is clear that it covers five years. It becomes misleading when it is compared, perhaps in passing, with the result for a single year, or when the audience is allowed to interpret it as an annual rate. The period is not a detail to be tucked away in small print: it is part of what the number means.
The same care is needed in the other direction: “It costs just one euro a day” can make an expense seem negligible when, over 365 days, it comes to 365 euros. That does not mean it is necessarily excessive. It means someone has chosen the unit in which they would prefer us to judge it. Just as 5,000 metres and 5 kilometres describe the same distance, a larger or smaller figure does not, by itself, change what we are talking about.
Percentages Need a Base
“We have doubled our result.” It sounds like extraordinary news. But if we have gone from one sale to two, the doubling is real and the absolute increase is a single item. I am not saying that this is insignificant: for someone just starting out, it may matter. I am saying that 100% does not, on its own, tell us the scale of the result. A percentage without its starting value leaves plenty of room for the imagination.
There is another source of confusion, too. Going from 3% to 6% means an increase of three percentage points and, at the same time, a doubling of the original percentage: a relative increase of 100%. These are not contradictory accounts, but two different ways of describing the change. Simply saying “it has increased by three per cent” does not accurately express that shift.
Even a small percentage can conceal a large number. “Only 2% experienced a problem,” out of ten million people, means two hundred thousand people. That “only” does not come from the calculation: it is a judgement added by the speaker. To understand how serious the problem is, we need at least to know its nature, its extent, and the number of people affected. Neither the percentage nor the total, taken alone, can replace all of that.
And successive percentage changes do not cancel each other out merely because they use the same number. A quantity that falls from 100 to 50 loses 50%; if it then grows by 50%, it reaches 75, not 100. To get back from 50 to 100, it must increase by 100%. So before I believe in a recovery, I need to check the base of the calculation. Plus and minus signs may look symmetrical; the quantities to which they are applied may not be.
The Average Almost No One Lives In
Imagine ten people: nine receive 1,000 euros each, and one receives 11,000. The total is 20,000 euros and the mean is 2,000. The calculation is correct. But nine people out of ten receive half that average, and no one receives exactly 2,000 euros. If I gave only the average figure, I could create a picture of circumstances that did not adequately describe life for most of the group.
This is not a flaw in mathematics. It is a feature of the measure we have chosen. The median, the middle value when we put the data in order, is 1,000 euros in this case: with ten values, we take the two central ones, both of which are 1,000. To understand that group properly, it would also help to know how far apart the values are, instead of appointing a single measure to stand as the whole portrait.
This makes me think of how often we say “we are better off” on the strength of an overall result, without asking who is better off. It can be true that the total has grown and, at the same time, that nothing has changed for many people. I do not want to use that possibility to deny every improvement. I would simply like a collective result not to become an automatic promise to each individual.
What Comes In Is Not the Net Result
Another statement that can impress is: “We have hired five thousand people.” Good. But if, during the same period and from the same workforce, seven thousand have left, and these are all the changes we need to account for, there are two thousand fewer people at the end. The five thousand hires remain a fact. What is false is the conclusion that the workforce must therefore have grown.
Counting who comes in is different from comparing how many people there are at the beginning and at the end. That is why “five thousand hires” and “five thousand more people in employment” are not interchangeable expressions. Someone may confuse them through carelessness; someone who knows the difference and exploits it is using the apparent precision of a figure to produce an imprecise impression.
This seems a useful rule to me beyond statistics, too: if someone tells me only what came in, I ask what went out. If they show me a gain, I ask which costs they have included. If they present a result, I check that the comparison covers the same things, over the same interval, and by the same criteria. Not to find fault at any cost, but to avoid applauding half the calculation as though it were the whole.
Ninety Per Cent of Whom?
“90% are satisfied.” It is a reassuring statement, until we ask an elementary question: ninety per cent of whom? Imagine inviting a thousand people to respond and receiving a hundred answers, ninety of them positive. We can say that 90% of those who responded said they were satisfied. We cannot automatically turn that result into the opinion of all one thousand people.
We do not know what the nine hundred who did not respond think. They might be more satisfied, less satisfied, or much like the others. Their silence warrants neither optimism nor pessimism. And changing the denominator does not solve the problem: ninety positive responses out of a thousand invitations is 9%, but that 9% does not measure overall satisfaction either. It tells us only how many of those invited gave a positive answer.
How people were selected matters as well, along with the question they were asked and the answers they were allowed to give. I am not saying that every survey is useless. I am saying that a result has limits to its validity, and presenting it without explaining how it was obtained can stretch those limits beyond what the data support. A very loud voice does not become everyone's voice merely because the others cannot be heard.
When a Graph Raises Its Voice
Sometimes the story begins before the words: all we have to do is look at a graph. Take two values, 50 and 51. The second is 2% higher than the first. But if I start the vertical axis of a bar chart at 49, the visible portions of the bars have heights of one and two. The second looks twice as tall, even though it represents a value nowhere near twice the first.
There may be good reason to magnify a small difference in order to study it. But we need to make that clear and choose a presentation that does not pass off the magnification as the actual size of the change. Not every graph has to tell us everything, and not every type of graph follows the same requirements. No one, though, should give the eye a message that the scale contradicts as soon as we read it.
A picture can shout what the data barely whisper. Colours, proportions, the interval chosen, and pieces of the story left out all influence what we notice. Looking at the scale and the dates does not spoil the clarity of a graph: it lets us check whether that clarity helps us understand or carries us too quickly towards a conclusion.
Even a Label Can Change the Story
A doubt of this kind arose for me while reading the statistics for my own website: a total for a period could be mistaken for a daily figure. I find this a useful example precisely because we do not have to imagine any intention to deceive. Sometimes an unclear label is enough for a correct number to be read the wrong way.
Take a counter that adds up estimated visitor counts for each day: if the total over seven days is 462, the daily average is 66. That does not mean 462 visitors every day. Nor can we conclude that 462 different people visited during the week: the same person may be counted on different days. These are estimates, too, not certain identifications of individuals. The counter's name should help us understand these distinctions, not leave us to guess them.
This experience reminds me of an important caution: misleading communication does not, by itself, prove an intention to deceive. There may be malice, but there may also be a mistake, a habit, or an inadequate explanation. The first thing to do is correct what is unclear. Judging intentions requires more information. Insisting on clarity should not make us unfair to people.
Small Questions That Put Big Announcements in Perspective
I do not believe we need to become statisticians to protect ourselves. Expertise matters, of course, but many checks begin with simple questions that we have every right to ask, even when the speaker sounds very sure. When faced with a figure, I would like to remember at least these:
- What are you counting: people, events, contracts, or amounts added up repeatedly over time?
- Over what period, and compared with what starting point?
- How many cases is the percentage based on, and what does it amount to in absolute numbers?
- Are you showing me a total, an average, or the net balance between what came in and what went out?
- Who is missing from the count, and what information might change the conclusion?
Not every answer makes the result less important. Sometimes it makes it more important. A small increase may be significant; an enormous figure may describe a phenomenon that truly is enormous. Checking should not begin with a desire to disprove. It should begin with a desire to make our judgement proportionate to what we know.
Above all, I would like to ask these questions of the numbers I like, too. It is easy to become meticulous about figures that contradict our ideas and careless about those that seem to confirm them. But if I change my standards to suit my own convenience, I am doing the very thing I reproach others for. Honesty does not consist only in exposing a misleading presentation: it means refusing to use one when it could work in my favour.
What I Would Like to Leave My Children
My children, I would like all three of you to keep your respect for numbers without being overawed by the people who speak them. Mathematics is a tool for seeing more clearly, not a language reserved for those who want to end an argument. When something does not add up, do not be ashamed to ask for a simple explanation. Someone who truly understands a figure should at least try to help you understand it.
Do not stop at the size of the number. Look for the scale of what it describes. Seventy thousand may be a lot or a little, depending on the period, the population, and the problem being discussed. Two per cent may concern a small imperfection or serious harm to a great many people. Before your enthusiasm and before your anger, give yourselves time to understand what you are reacting to.
I would also like you to be able to say “I do not know yet.” When the information you need is missing, that is not surrender. It is a way of keeping your judgement from falling into the hands of whoever spoke first or with the greatest confidence. Between believing everything and suspecting everything lies a more demanding path: checking, making distinctions, and, when necessary, changing your minds.
And when it is your turn to describe a result, give others what you expect for yourselves: the context, the period, the point of comparison. Do not hide the part that puts your success in perspective. Do not use a percentage to make people invisible. Being precise means more than getting a division right: it means not taking advantage of someone who has not yet had time to do it.
I would not want to teach you to win arguments with numbers. I would like to leave you the habit of using them to understand, and the honesty not to bend them to prove yourselves right. A figure becomes truly useful when it sheds light on something. If it serves only to dazzle our listeners, perhaps we have calculated well, but we are communicating badly.
Papy