Benford’s Law Will Make You Wonder For a While

By Anupum Pant

Benford’s law is a fairly simple law to grasp and it will blow your mind. It deals with the leading digits of numbers.

So, for example, you have the number 28 – The leading digit for it would be 2. Similarly, the leading digit for 934 would be 9. Just pick the first digit. Now…

In a data set you’d say – it is common sense to assume that the probability of leading digit one (1) appearing would be more or less equal to that of leading digit nine (9).
As there are 9 possible leading digits, you’d think that the probability of each leading digit would compute to something around 0.11
You’d imagine that it would be normal to assume a nearly straight graph of probability vs. leading digit. But this isn’t true.

Benford’s law says

Your common sense fails. What actually happens is that the likelihood of 1 appearing as the first digit in a data set is around 0.3
For the following digits, the probability keeps decreasing. And the following graph appears. You’ll see that the numbers rarely start with nine!


When does it work?

This counter-intuitive result applies to a wide variety of natural data sets. It works the best if your set spans quite a few orders of magnitude. Natural set of data like stock prices, electricity bills, populations, which could range from few single digit values to several digits work the best. Other data like the heights of people doesn’t work because it does not span “quite a few orders of magnitude”. Also, artificially tampered data fails to comply because the person who tampers does the same mistake everyone does. Therefore, Benford’s law is also used to detect frauds in data.


  1. Count the number of data points in a data set which have the leading digit 1 and write the number next to the number 1 in a table.
  2. Then, keep repeating it for all the numbers 2, 3, 4 and so on.
  3. Calculate the probabilities for each. In the end you’ll be left with a table that would look something like this. (Probability = Number of Data Points for that  digit / Total Data Points)
Leading Digit Digit Probability
1 0.301
2 0.17
3 0.125
4 0.097
5 0.079
6 0.067
7 0.058
8 0.051
9 0.046

How does it work?

Watch the  following video for the explanation:

Try it yourself: [Kirix]

Gompertz Law – The Dreadful Law of Death

By Anupum Pant

There is no astrologer in the world that can tell you for sure if you’ll die this year or not. But, thanks to Gompertz Law, if you ask me, there is one thing I can tell you for sure – Whatever may be the odds of you dying this year, in 8 years, the likelihood of you dying will double.

This dreadful law of death was named after the first person who noted it – Benjamin Gompertz, in the year 1825. The law rests on a general assumption that a person’s resistance to death decreases as he ages. The Gompertz Law of mortality, put simply in a sentence would compute to this:

Your probability of dying during a given year doubles every 8 years.

It is amazing, and no one knows how it works exactly. Why does nature pick the number 8, to double our likelihood of death? We’ll probably never know.

There is a whole table which relies on census data, and statistically notes the probabilities of people dying at different ages. And when it is plotted on a Probability of death vs. Age graph, you get an exponentially increasing mortality rate with age. That is death coming faster as you get older.

Gompertz Law can be verified for real-life data – the 2005 US census data. The following graph and the probability vs. age plotted using the law match almost perfectly. Amazingly, the law holds true for several other countries too.

gompertz law graph

That means, the probability of me, a 25-year-old dying during the next year is very small — about 1 in 3,000. When I become 33, this probability will grow to something around 1 in 1,500. In the next 8 years, the probability of me dying will be 1 in 750, and so on…At the age 100, the probability a person’s death will be about 1 out of 2 – fat chance of successfully moving on to 101!

Theoretically, using this data, it can be said with 99.999999% certainty that no human will ever live to the age of 130 (of course only if medicine doesn’t start tampering with human genes, or some other artificial factor). There is one thing for sure – there is almost no chance that you are going to beat Mr. Ming.

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