The number one digit for an item in a supermarket is unevenly distributed – a smaller number has a clear numerical advantage. Where does this imbalance come from? What kind of supermarket, business or economic laws do these labels follow to present such strange results? Why are these top digits unevenly distributed? Shouldn't math treat all numbers equally?You're used to the atmosphere in the supermarket. You've been here hundreds, if not thousands. Sequential shelves, metal shelves, the regular sound of a checkout counter scanning a barcode, and customers walking around, unconsciously grabbing a bottle of milk or a few cans. But today, we are not here to shop, but to carry out observations. This place hides one of the most fascinating mathematical treasures. It's been in front of your eyes for so many years. It's not even the slightest cover, and you can see it at this moment. It's a small anomaly. It's one of those inconspicuous details right under your nose, seemingly useless, that might cause the prying observer to wonder about it. Get out your notebook or smartphone and get ready to take notes, and our investigation begins.
Take a look at the ** labels that line up the shelves in order. As we quickly swipe through the ** tabs one after the other, all of these numbers seem to be completely random. ** Ranges from a few cents to tens of euros. But it's not the details that we need to focus on. Forget about decimal points and decimals. Just look at the first significant digit of each **, which is the most important number, and it gives an approximation.
You see a bottle with a price tag of 154 of 530 grams of canned fruit, write down 1 on your notebook. A few more steps and a bottle with a price tag of 353 of 24-hour deodorant, denoted as 3. The price tag for one piece is 181 of 250 grams of cheese, denoted as 1. The price tag for one bite is 4590 of the non-stick pan, this ** has two digits, but it doesn't matter, we only focus on the first digit, which is recorded as 4. The list price of a pack is 0Roasted peanuts of 74, the first significant number of this ** is 7.
We walked around the supermarket for a few minutes, and the numbers were piling up. 1 3 1 4 7 9 2 2 1 7 9 8 1 1 3 1 1 1 8 1 1 2 1 2 1 1 9 1 4 7 1 6 1 5 9 2 2 1 3 2 2 2 1 2 2 6……But as the record continues, a small doubt arises. Don't you think there's something wrong with this string of numbers? It's as if there's some kind of imbalance in it. This string of numbers is mainly composed of the numbers 1 and 2, with occasional several and 9. It's as if we are naturally attracted to the lowest ** without realizing it. There's a problem here.
Then let's learn from statisticians and act with rigor: from now on, beware of our own biases and adopt a systematic approach. We randomly select several rows of shelves and record all the products on each row without exception. It's a lot of work, but you have to be aware of it.
An hour later, you'll have a whole page of numbers in your notebook. It's time to wrap up. After the calculations, the results are beyond doubt, and the trends presented are clear at a glance. You've documented more than a thousand products, and nearly a third of those numbers start with 1! More than a quarter of the numbers start with 2, and the higher the number, the fewer times it appears in the record.
Figure 11 is the percentage of the first digit that has been collated.
This time, we can no longer think of it as a simple random effect, or a biased choice of our own products. We have to admit that it's a fact: the top digit of goods in supermarkets is unevenly distributedSmaller numbers have a clear numerical advantage.
Where does this imbalance come from? That's the question I want to ask you. What kind of supermarket, business or economic laws do these labels follow to present such strange results? Why are these top digits unevenly distributed? Shouldn't math treat all numbers equally? Mathematics should be unbiased, unfavored, and unfavored. However, the facts are right in front of us, and they are clearly opposite to what we expected. In the supermarket, math has its own "darling", and the "darlings" are called 1 and 2.
We have observed it and confirmed it. Now, we need to think, analyze and peel back the dots. With the facts in our hands, it's time to investigate and draw conclusions.
In March 1938, the American engineer and physicist Frank Benford published "The Law of Anomalous Numbers," in which he analyzed digital data from more than 20,000 different observational sources. In his list, we can see the length of rivers around the world, the populations of different cities in the United States, the measured values of known atomic mass, randomly obtained numbers in newspapers, and even mathematical constants. For all of this data, Ben Ford has the same observation as ours every time: the top digits are unevenly distributed. About 30% of these numbers start with 1 and 18% with 2, and this percentage continues to decline until the number 9, with only 5% of numbers starting with 9 (Figure 1).2）。
Ben Ford didn't think of verifying his stats through the supermarket's ** label. But we have to admit that the results he got were surprisingly similar to ours – of course, there would be a slight change in percentages, but in terms of overall trends, the similarities were surprisingly high.
Benford's research shows that the data we collect is not unique. They are not specific to the way supermarkets operate, but are rooted in a broader trend. After 1938, many scientists observed the same distribution in increasingly extreme and diverse cases.
Take demographics, for example: out of 203 countries and regions on the planet, 62 (i.e. 305%). The first is China, with a population of about 1.4 billion. We will also find that out of these 62 countries, Mexico has about 1With a population of 2.2 billion, Senegal has a population of about 13 million and the islands of Tuvalu have a population of about 10,800. Conversely, only 14 countries and regions (i.e. 69%), the number of people starts with the number 9.
Do you prefer astronomy? Of the eight planets that orbit the Sun, four of them have equatorial diameters that start with 1. Jupiter has a diameter of about 142 984 km, Saturn about 120 536 km, Earth about 12 756 km, and Venus about 12 104 km. The diameter of the sun itself is about 1 392 000 km. If a sample of these nine objects isn't enough to arrive at a reliable trend, add dwarfs, moons, asteroids, and comets, and you'll always get the same observation: number 1 is overwhelmingly dominant.
Once we start paying attention to this, examples will follow. Take a list of numbers from any situation, analyze the first digit of those numbers, and you'll find that Ben Ford's distribution of numbers always appears over and over again. Far from being an exception, this statistical law seems to be completely natural and ubiquitous. Paradoxically, what we intuitively thought should be a more reasonable equilibrium does not seem to exist in the world.
At this level, there is nothing strange about the observations in the supermarket. What we have just revealed is a veritable law that governs not only many areas of human activity, but also nature itself in its most secret structures. To understand this law is to understand something deep about our world and how it works.
The impact of this law is so great that we can keep repeating it without even realizing it. People who price supermarket items don't necessarily negotiate with each other, and most of them have never heard of Frank Benford. However, they unconsciously obey Benford's law as if under the control of some power that surpasses them. The same is true for the population, the length of the river, and the diameter of the planet.
In 1938, Frank Benford named this distribution the "law of anomalies." However, this law is so ubiquitous that naming it "abnormal" doesn't sound appropriate. "Abnormality" is only a subjective judgment, and it exists only in the eyes of those who are surprised by it. On the contrary, nature seems to think that this law is all too common. A law is only "abnormal" if it is not understood by us. And we're going to find out about it.
So, which direction should you go? Which course of thought should we follow to lift the veil of anomalies and make the mystery obvious?
Benford's law is not complicated to understand, but it is not clear to explain in a few words. The math behind this law is simple and profound. We are not faced with a sudden epiphany and exclaiming, "Ah, I see, I see!" "The answer to the puzzle. What needs to change is how we understand numbers and how they are counted. If Benford's law doesn't seem obvious to us, it's because we don't think the right way. We have to learn to look at things that we think we know a lot from a different perspective, and we have to look at ourselves.
Take a walk into the world that Frank Benford has just opened up for us, and when you come from **, it won't be the same way. Benford's law changes you. Once you understand it, you'll never think the same way again.
The above **Turing News, excerpted from "Under the Umbrella of Mathematics", [Meet Mathematics] has been approved by **.Recommended Reading:
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