Are All Digits Equal?

This puzzle consists of some activities to try. They require collecting lots of data, so they are good to do with your whole class or even your whole school.

Pick up a book near you. Open the book to a random page and record the page number. Pick up a different book and flip to another random page. Record the first digit of that page. Try this with many different books until you have recorded 100 page numbers or more.
In a store, record the prices of 100 or more items. It doesn’t matter what kind of store it is. You can also do this by going to a Web site that sells things and recording the prices of the various items it offers.
Get the street addresses of everyone in your class. Have them write down the addresses of other people they know until you have 100 or more addresses.
Find out the populations of the 100 or more largest cities and towns in your country. Write these down.

For each set of data some numbers have 1 as the first digit, some have 2, some begin with 3, and so on. Do you think all the digits from 1 through 9 have about an equal chance of occurring, or does one of the digits appear more often?

Now total up how many times each digit occurs in each data set. Are you surprised? Do you have an explanation for what you found?

Course:

Integers [1]
Math [2]
Algebra [3]
Probability [4]

Result/Solution(s)

Solution: Are All Digits Equal? Math Puzzle

Here are some of the data we collected and the pattern we found:

Digit searched Number of hits
1 1,840,000,000
2 1,540,000,000
3 1,300,000,000
4 1,160,000,000
5 1,180,000,000
6 949,000,000
7 901,000,000
8 689,000,000
9 801,000,000

We went to addresses.com and looked for the addresses of all the people named M Smith in the United States where Smith is a common name. Of the first 100 addresses 32 began with the digit 1.
We found a Web site that listed the 156 longest rivers in the world, of which 95 had lengths in kilometers that began with 1.
We went to the Handbook of Chemistry and Physics and got the list of atomic masses of all the elements. It included multiple isotopes of many elements, so there were 346 items listed. Of these, 164 began with the number 1.
We did a Google search on the number 1 by entering “ 1” in the search window, that is, 1 preceded by a space in quotes. This picks up phrases that include a number that begins with the digit 1. We also did this for all the other digits. The results are shown in the table above.

The distribution of first digits is not random. The digit 1 seems to come up more often than the others in many situations.

This does not apply to truly random phenomena. For example, take nine ping-pong balls and write a single digit, 1 through 9, on each. Put them in a bag. Pick a ball and write down the number. Replace the ball, shake the bag, and pick again. If you do this many times, you should find that the digits come up in about equal numbers. But the data we are collecting is from the real world, and apparently not random.

This phenomenon is known as Benford’s law, also called the first digit law, first digit phenomenon, or leading digit phenomenon.

According to MathWorld, the probability that the first (decimal) digit is D is given by the logarithmic distribution
P_D = log₁₀ (1 + 1 / D)


Chart courtesy of MathWorld [5]


Digit searched	Number of hits
1	1,840,000,000
2	1,540,000,000
3	1,300,000,000
4	1,160,000,000
5	1,180,000,000
6	949,000,000
7	901,000,000
8	689,000,000
9	801,000,000